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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Exact Hidden Markovian Dynamics in Quantum Circuits</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2403.14807v2/"/></head> <body> <nav class="ltx_page_navbar"> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Exact Hidden Markovian Dynamics in Quantum Circuits</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">He-Ran Wang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Xiao-Yang Yang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China </span> <span class="ltx_contact ltx_role_affiliation">Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Zhong Wang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:wangzhongemail@tsinghua.edu.cn">wangzhongemail@tsinghua.edu.cn</a> </span> <span class="ltx_contact ltx_role_affiliation">Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Characterizing nonequilibrium dynamics in quantum many-body systems is a challenging frontier of physics. In this Letter, we systematically construct solvable nonintegrable quantum circuits that exhibit exact hidden Markovian subsystem dynamics. This feature thus enables accurately calculating local observables for arbitrary evolution time. Utilizing the influence matrix method, we show that the influence of the time-evolved global system on a finite subsystem can be analytically described by sequential, time-local quantum channels acting on the subsystem with an ancilla of finite Hilbert space dimension. The realization of exact hidden Markovian property is facilitated by a solvable condition on the underlying two-site gates in the quantum circuit. We further present several concrete examples with varying local Hilbert space dimensions to demonstrate our approach.</p> </div> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1">In isolated quantum many-body systems driven out of equilibrium, thermalization typically occurs, where local observables relax to their thermal-averaged expectation values after a finite time. Heuristically, the global system serves as a thermal bath for the local subsystem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib6" title="">6</a>]</cite>. On the other hand, various counterexamples of thermalization have been extensively studied, including integrable models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib7" title="">7</a>]</cite>, many-body localization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib9" title="">9</a>]</cite>, and quantum many-body scars <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib12" title="">12</a>]</cite>. However, for both scenarios (following or violating thermalization), it poses a formidable challenge to accurately quantify the influence of the time-evolved macroscopic many-body system on its own subsystem, due to the exponentially large Hilbert space dimension in the thermodynamic limit and the quantum memory effects brought by the non-Markovianity.</p> </div> <div class="ltx_para" id="p2"> <p class="ltx_p" id="p2.1">Recently, progress has been made in quantum circuits, where the unitary evolution is discretized to sequences of local unitary gates. In particular, Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib19" title="">19</a>]</cite> have developed an efficient tensor-network approach to trace out the system, and encode the influence on the subsystem into the fixed point of the spatial transfer matrix, which is also known as the <span class="ltx_text ltx_font_italic" id="p2.1.1">influence matrix</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib18" title="">18</a>]</cite>. However, the intrinsic complexity of many-body dynamics typically leads to complicated influence matrices as the evolution time grows, restricting rigorous numerical and analytical treatment within this approach <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib20" title="">20</a>]</cite>.</p> </div> <div class="ltx_para" id="p3"> <p class="ltx_p" id="p3.1">Here, we introduce a novel approach to systematically constructing 1+1 D nonintegrable quantum circuits exhibiting exact hidden Markovian subsystem dynamics. We introduce a solvable condition for the underlying unitary gates allowing for efficient contractions of quantum-circuit tensor networks for arbitrary evolution time. We show that the time-evolved system can be traced out to a closed-form influence matrix in the matrix product state (MPS) representation of finite bond dimension for arbitrary evolution time, thus enabling numerical calculations of subsystem dynamics in an exact fashion. Notably, we interpret the influence matrix as sequential quantum channels acting on the subsystem boundary. Hence, our work uncovers new principles leading to subsystem hidden Markovian property, and provides a promising testground to explore rich phenomena in quantum many-body dynamics through analytical tools. </p> </div> <figure class="ltx_figure" id="S0.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="268" id="S0.F1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Main results of this letter. Total time steps <math alttext="T=2" class="ltx_Math" display="inline" id="S0.F1.3.m1.1"><semantics id="S0.F1.3.m1.1b"><mrow id="S0.F1.3.m1.1.1" xref="S0.F1.3.m1.1.1.cmml"><mi id="S0.F1.3.m1.1.1.2" xref="S0.F1.3.m1.1.1.2.cmml">T</mi><mo id="S0.F1.3.m1.1.1.1" xref="S0.F1.3.m1.1.1.1.cmml">=</mo><mn id="S0.F1.3.m1.1.1.3" xref="S0.F1.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S0.F1.3.m1.1c"><apply id="S0.F1.3.m1.1.1.cmml" xref="S0.F1.3.m1.1.1"><eq id="S0.F1.3.m1.1.1.1.cmml" xref="S0.F1.3.m1.1.1.1"></eq><ci id="S0.F1.3.m1.1.1.2.cmml" xref="S0.F1.3.m1.1.1.2">𝑇</ci><cn id="S0.F1.3.m1.1.1.3.cmml" type="integer" xref="S0.F1.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F1.3.m1.1d">T=2</annotation><annotation encoding="application/x-llamapun" id="S0.F1.3.m1.1e">italic_T = 2</annotation></semantics></math>. (a) Tensor-network representation of a 1+1 D quantum circuit in the folded picture. The initial state <math alttext="\ket{\Psi_{\text{in}}}" class="ltx_Math" display="inline" id="S0.F1.4.m2.1"><semantics id="S0.F1.4.m2.1b"><mrow id="S0.F1.4.m2.1.1.3" xref="S0.F1.4.m2.1.1.2.cmml"><mo id="S0.F1.4.m2.1.1.3.1" stretchy="false" xref="S0.F1.4.m2.1.1.2.1.cmml">|</mo><msub id="S0.F1.4.m2.1.1.1.1" xref="S0.F1.4.m2.1.1.1.1.cmml"><mi id="S0.F1.4.m2.1.1.1.1.2" mathvariant="normal" xref="S0.F1.4.m2.1.1.1.1.2.cmml">Ψ</mi><mtext id="S0.F1.4.m2.1.1.1.1.3" xref="S0.F1.4.m2.1.1.1.1.3a.cmml">in</mtext></msub><mo id="S0.F1.4.m2.1.1.3.2" stretchy="false" xref="S0.F1.4.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S0.F1.4.m2.1c"><apply id="S0.F1.4.m2.1.1.2.cmml" xref="S0.F1.4.m2.1.1.3"><csymbol cd="latexml" id="S0.F1.4.m2.1.1.2.1.cmml" xref="S0.F1.4.m2.1.1.3.1">ket</csymbol><apply id="S0.F1.4.m2.1.1.1.1.cmml" xref="S0.F1.4.m2.1.1.1.1"><csymbol cd="ambiguous" id="S0.F1.4.m2.1.1.1.1.1.cmml" xref="S0.F1.4.m2.1.1.1.1">subscript</csymbol><ci id="S0.F1.4.m2.1.1.1.1.2.cmml" xref="S0.F1.4.m2.1.1.1.1.2">Ψ</ci><ci id="S0.F1.4.m2.1.1.1.1.3a.cmml" xref="S0.F1.4.m2.1.1.1.1.3"><mtext id="S0.F1.4.m2.1.1.1.1.3.cmml" mathsize="70%" xref="S0.F1.4.m2.1.1.1.1.3">in</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F1.4.m2.1d">\ket{\Psi_{\text{in}}}</annotation><annotation encoding="application/x-llamapun" id="S0.F1.4.m2.1e">| start_ARG roman_Ψ start_POSTSUBSCRIPT in end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> is evolved by applying four layers of two-site gates [purple squares, defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E1" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>)] in a brickwork architecture. (b) Illustration of the (left) influence matrix (IM). The system is initialized to a composite MPS over two regions: the left is a one-site shift invariant MPS [with local tensors in red, defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E2" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">2</span></a>)], and the right is a generic state. Thick lines correspond to the auxiliary Hilbert space. After attaching hollow dots on top outer legs in the left region, the tensor network in the light blue shaded region defines the influence matrix acting on the time slice (blue dotted line). (c) The exact influence matrix represented by MPS. (d) Open quantum system representation of the subsystem dynamics. Markovian property manifests when considering the joint dynamics of the ancilla in the auxiliary Hilbert space, and the subsystem. Two-site quantum channels are shown in red circles [defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E5" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">5</span></a>)]. </figcaption> </figure> <div class="ltx_para" id="p4"> <p class="ltx_p" id="p4.11"><span class="ltx_text ltx_font_italic" id="p4.11.1">The setup</span>—In this letter we consider quantum circuits on a 1D lattice, where each site is labeled by an integer <math alttext="x" class="ltx_Math" display="inline" id="p4.1.m1.1"><semantics id="p4.1.m1.1a"><mi id="p4.1.m1.1.1" xref="p4.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="p4.1.m1.1b"><ci id="p4.1.m1.1.1.cmml" xref="p4.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="p4.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="p4.1.m1.1d">italic_x</annotation></semantics></math>. We associate a <math alttext="q" class="ltx_Math" display="inline" id="p4.2.m2.1"><semantics id="p4.2.m2.1a"><mi id="p4.2.m2.1.1" xref="p4.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="p4.2.m2.1b"><ci id="p4.2.m2.1.1.cmml" xref="p4.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="p4.2.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="p4.2.m2.1d">italic_q</annotation></semantics></math>-dimensional Hilbert space <math alttext="\mathcal{H}_{q}" class="ltx_Math" display="inline" id="p4.3.m3.1"><semantics id="p4.3.m3.1a"><msub id="p4.3.m3.1.1" xref="p4.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p4.3.m3.1.1.2" xref="p4.3.m3.1.1.2.cmml">ℋ</mi><mi id="p4.3.m3.1.1.3" xref="p4.3.m3.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="p4.3.m3.1b"><apply id="p4.3.m3.1.1.cmml" xref="p4.3.m3.1.1"><csymbol cd="ambiguous" id="p4.3.m3.1.1.1.cmml" xref="p4.3.m3.1.1">subscript</csymbol><ci id="p4.3.m3.1.1.2.cmml" xref="p4.3.m3.1.1.2">ℋ</ci><ci id="p4.3.m3.1.1.3.cmml" xref="p4.3.m3.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p4.3.m3.1c">\mathcal{H}_{q}</annotation><annotation encoding="application/x-llamapun" id="p4.3.m3.1d">caligraphic_H start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> for each site, with a basis: <math alttext="\{\ket{a},a=0,1,\cdots,q-1\}" class="ltx_Math" display="inline" id="p4.4.m4.5"><semantics id="p4.4.m4.5a"><mrow id="p4.4.m4.5.5.1" xref="p4.4.m4.5.5.2.cmml"><mo id="p4.4.m4.5.5.1.2" stretchy="false" xref="p4.4.m4.5.5.2.cmml">{</mo><mrow id="p4.4.m4.5.5.1.1.2" xref="p4.4.m4.5.5.1.1.3.cmml"><mrow id="p4.4.m4.5.5.1.1.1.1" xref="p4.4.m4.5.5.1.1.1.1.cmml"><mrow id="p4.4.m4.5.5.1.1.1.1.2.2" xref="p4.4.m4.5.5.1.1.1.1.2.1.cmml"><mrow id="p4.4.m4.1.1.3" xref="p4.4.m4.1.1.2.cmml"><mo id="p4.4.m4.1.1.3.1" stretchy="false" xref="p4.4.m4.1.1.2.1.cmml">|</mo><mi id="p4.4.m4.1.1.1.1" xref="p4.4.m4.1.1.1.1.cmml">a</mi><mo id="p4.4.m4.1.1.3.2" stretchy="false" xref="p4.4.m4.1.1.2.1.cmml">⟩</mo></mrow><mo id="p4.4.m4.5.5.1.1.1.1.2.2.1" xref="p4.4.m4.5.5.1.1.1.1.2.1.cmml">,</mo><mi id="p4.4.m4.2.2" xref="p4.4.m4.2.2.cmml">a</mi></mrow><mo id="p4.4.m4.5.5.1.1.1.1.1" xref="p4.4.m4.5.5.1.1.1.1.1.cmml">=</mo><mn id="p4.4.m4.5.5.1.1.1.1.3" xref="p4.4.m4.5.5.1.1.1.1.3.cmml">0</mn></mrow><mo id="p4.4.m4.5.5.1.1.2.3" xref="p4.4.m4.5.5.1.1.3a.cmml">,</mo><mrow id="p4.4.m4.5.5.1.1.2.2.1" xref="p4.4.m4.5.5.1.1.2.2.2.cmml"><mn id="p4.4.m4.3.3" xref="p4.4.m4.3.3.cmml">1</mn><mo id="p4.4.m4.5.5.1.1.2.2.1.2" xref="p4.4.m4.5.5.1.1.2.2.2.cmml">,</mo><mi id="p4.4.m4.4.4" mathvariant="normal" xref="p4.4.m4.4.4.cmml">⋯</mi><mo id="p4.4.m4.5.5.1.1.2.2.1.3" xref="p4.4.m4.5.5.1.1.2.2.2.cmml">,</mo><mrow id="p4.4.m4.5.5.1.1.2.2.1.1" xref="p4.4.m4.5.5.1.1.2.2.1.1.cmml"><mi id="p4.4.m4.5.5.1.1.2.2.1.1.2" xref="p4.4.m4.5.5.1.1.2.2.1.1.2.cmml">q</mi><mo id="p4.4.m4.5.5.1.1.2.2.1.1.1" xref="p4.4.m4.5.5.1.1.2.2.1.1.1.cmml">−</mo><mn id="p4.4.m4.5.5.1.1.2.2.1.1.3" xref="p4.4.m4.5.5.1.1.2.2.1.1.3.cmml">1</mn></mrow></mrow></mrow><mo id="p4.4.m4.5.5.1.3" stretchy="false" xref="p4.4.m4.5.5.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="p4.4.m4.5b"><set id="p4.4.m4.5.5.2.cmml" xref="p4.4.m4.5.5.1"><apply id="p4.4.m4.5.5.1.1.3.cmml" xref="p4.4.m4.5.5.1.1.2"><csymbol cd="ambiguous" id="p4.4.m4.5.5.1.1.3a.cmml" xref="p4.4.m4.5.5.1.1.2.3">formulae-sequence</csymbol><apply id="p4.4.m4.5.5.1.1.1.1.cmml" xref="p4.4.m4.5.5.1.1.1.1"><eq id="p4.4.m4.5.5.1.1.1.1.1.cmml" xref="p4.4.m4.5.5.1.1.1.1.1"></eq><list id="p4.4.m4.5.5.1.1.1.1.2.1.cmml" xref="p4.4.m4.5.5.1.1.1.1.2.2"><apply id="p4.4.m4.1.1.2.cmml" xref="p4.4.m4.1.1.3"><csymbol cd="latexml" id="p4.4.m4.1.1.2.1.cmml" xref="p4.4.m4.1.1.3.1">ket</csymbol><ci id="p4.4.m4.1.1.1.1.cmml" xref="p4.4.m4.1.1.1.1">𝑎</ci></apply><ci id="p4.4.m4.2.2.cmml" xref="p4.4.m4.2.2">𝑎</ci></list><cn id="p4.4.m4.5.5.1.1.1.1.3.cmml" type="integer" xref="p4.4.m4.5.5.1.1.1.1.3">0</cn></apply><list id="p4.4.m4.5.5.1.1.2.2.2.cmml" xref="p4.4.m4.5.5.1.1.2.2.1"><cn id="p4.4.m4.3.3.cmml" type="integer" xref="p4.4.m4.3.3">1</cn><ci id="p4.4.m4.4.4.cmml" xref="p4.4.m4.4.4">⋯</ci><apply id="p4.4.m4.5.5.1.1.2.2.1.1.cmml" xref="p4.4.m4.5.5.1.1.2.2.1.1"><minus id="p4.4.m4.5.5.1.1.2.2.1.1.1.cmml" xref="p4.4.m4.5.5.1.1.2.2.1.1.1"></minus><ci id="p4.4.m4.5.5.1.1.2.2.1.1.2.cmml" xref="p4.4.m4.5.5.1.1.2.2.1.1.2">𝑞</ci><cn id="p4.4.m4.5.5.1.1.2.2.1.1.3.cmml" type="integer" xref="p4.4.m4.5.5.1.1.2.2.1.1.3">1</cn></apply></list></apply></set></annotation-xml><annotation encoding="application/x-tex" id="p4.4.m4.5c">\{\ket{a},a=0,1,\cdots,q-1\}</annotation><annotation encoding="application/x-llamapun" id="p4.4.m4.5d">{ | start_ARG italic_a end_ARG ⟩ , italic_a = 0 , 1 , ⋯ , italic_q - 1 }</annotation></semantics></math>. The system is prepared in the initial state <math alttext="\ket{\Psi_{\text{in}}}" class="ltx_Math" display="inline" id="p4.5.m5.1"><semantics id="p4.5.m5.1a"><mrow id="p4.5.m5.1.1.3" xref="p4.5.m5.1.1.2.cmml"><mo id="p4.5.m5.1.1.3.1" stretchy="false" xref="p4.5.m5.1.1.2.1.cmml">|</mo><msub id="p4.5.m5.1.1.1.1" xref="p4.5.m5.1.1.1.1.cmml"><mi id="p4.5.m5.1.1.1.1.2" mathvariant="normal" xref="p4.5.m5.1.1.1.1.2.cmml">Ψ</mi><mtext id="p4.5.m5.1.1.1.1.3" xref="p4.5.m5.1.1.1.1.3a.cmml">in</mtext></msub><mo id="p4.5.m5.1.1.3.2" stretchy="false" xref="p4.5.m5.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p4.5.m5.1b"><apply id="p4.5.m5.1.1.2.cmml" xref="p4.5.m5.1.1.3"><csymbol cd="latexml" id="p4.5.m5.1.1.2.1.cmml" xref="p4.5.m5.1.1.3.1">ket</csymbol><apply id="p4.5.m5.1.1.1.1.cmml" xref="p4.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="p4.5.m5.1.1.1.1.1.cmml" xref="p4.5.m5.1.1.1.1">subscript</csymbol><ci id="p4.5.m5.1.1.1.1.2.cmml" xref="p4.5.m5.1.1.1.1.2">Ψ</ci><ci id="p4.5.m5.1.1.1.1.3a.cmml" xref="p4.5.m5.1.1.1.1.3"><mtext id="p4.5.m5.1.1.1.1.3.cmml" mathsize="70%" xref="p4.5.m5.1.1.1.1.3">in</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p4.5.m5.1c">\ket{\Psi_{\text{in}}}</annotation><annotation encoding="application/x-llamapun" id="p4.5.m5.1d">| start_ARG roman_Ψ start_POSTSUBSCRIPT in end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and undergoes discrete time evolution. For each time step, the global unitary operator is <math alttext="\mathbb{U}=\mathbb{U}_{\text{odd}}\mathbb{U}_{\text{even}}" class="ltx_Math" display="inline" id="p4.6.m6.1"><semantics id="p4.6.m6.1a"><mrow id="p4.6.m6.1.1" xref="p4.6.m6.1.1.cmml"><mi id="p4.6.m6.1.1.2" xref="p4.6.m6.1.1.2.cmml">𝕌</mi><mo id="p4.6.m6.1.1.1" xref="p4.6.m6.1.1.1.cmml">=</mo><mrow id="p4.6.m6.1.1.3" xref="p4.6.m6.1.1.3.cmml"><msub id="p4.6.m6.1.1.3.2" xref="p4.6.m6.1.1.3.2.cmml"><mi id="p4.6.m6.1.1.3.2.2" xref="p4.6.m6.1.1.3.2.2.cmml">𝕌</mi><mtext id="p4.6.m6.1.1.3.2.3" xref="p4.6.m6.1.1.3.2.3a.cmml">odd</mtext></msub><mo id="p4.6.m6.1.1.3.1" xref="p4.6.m6.1.1.3.1.cmml">⁢</mo><msub id="p4.6.m6.1.1.3.3" xref="p4.6.m6.1.1.3.3.cmml"><mi id="p4.6.m6.1.1.3.3.2" xref="p4.6.m6.1.1.3.3.2.cmml">𝕌</mi><mtext id="p4.6.m6.1.1.3.3.3" xref="p4.6.m6.1.1.3.3.3a.cmml">even</mtext></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="p4.6.m6.1b"><apply id="p4.6.m6.1.1.cmml" xref="p4.6.m6.1.1"><eq id="p4.6.m6.1.1.1.cmml" xref="p4.6.m6.1.1.1"></eq><ci id="p4.6.m6.1.1.2.cmml" xref="p4.6.m6.1.1.2">𝕌</ci><apply id="p4.6.m6.1.1.3.cmml" xref="p4.6.m6.1.1.3"><times id="p4.6.m6.1.1.3.1.cmml" xref="p4.6.m6.1.1.3.1"></times><apply id="p4.6.m6.1.1.3.2.cmml" xref="p4.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="p4.6.m6.1.1.3.2.1.cmml" xref="p4.6.m6.1.1.3.2">subscript</csymbol><ci id="p4.6.m6.1.1.3.2.2.cmml" xref="p4.6.m6.1.1.3.2.2">𝕌</ci><ci id="p4.6.m6.1.1.3.2.3a.cmml" xref="p4.6.m6.1.1.3.2.3"><mtext id="p4.6.m6.1.1.3.2.3.cmml" mathsize="70%" xref="p4.6.m6.1.1.3.2.3">odd</mtext></ci></apply><apply id="p4.6.m6.1.1.3.3.cmml" xref="p4.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="p4.6.m6.1.1.3.3.1.cmml" xref="p4.6.m6.1.1.3.3">subscript</csymbol><ci id="p4.6.m6.1.1.3.3.2.cmml" xref="p4.6.m6.1.1.3.3.2">𝕌</ci><ci id="p4.6.m6.1.1.3.3.3a.cmml" xref="p4.6.m6.1.1.3.3.3"><mtext id="p4.6.m6.1.1.3.3.3.cmml" mathsize="70%" xref="p4.6.m6.1.1.3.3.3">even</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p4.6.m6.1c">\mathbb{U}=\mathbb{U}_{\text{odd}}\mathbb{U}_{\text{even}}</annotation><annotation encoding="application/x-llamapun" id="p4.6.m6.1d">blackboard_U = blackboard_U start_POSTSUBSCRIPT odd end_POSTSUBSCRIPT blackboard_U start_POSTSUBSCRIPT even end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\mathbb{U}_{\text{odd(even)}}=\otimes_{x\in{\text{odd(even)}}}U_{x,x+1}" class="ltx_math_unparsed" display="inline" id="p4.7.m7.2"><semantics id="p4.7.m7.2a"><mrow id="p4.7.m7.2b"><msub id="p4.7.m7.2.3"><mi id="p4.7.m7.2.3.2">𝕌</mi><mtext id="p4.7.m7.2.3.3">odd(even)</mtext></msub><mo id="p4.7.m7.2.4" rspace="0em">=</mo><msub id="p4.7.m7.2.5"><mo id="p4.7.m7.2.5.2" lspace="0em" rspace="0.222em">⊗</mo><mrow id="p4.7.m7.2.5.3"><mi id="p4.7.m7.2.5.3.2">x</mi><mo id="p4.7.m7.2.5.3.1">∈</mo><mtext id="p4.7.m7.2.5.3.3">odd(even)</mtext></mrow></msub><msub id="p4.7.m7.2.6"><mi id="p4.7.m7.2.6.2">U</mi><mrow id="p4.7.m7.2.2.2.2"><mi id="p4.7.m7.1.1.1.1">x</mi><mo id="p4.7.m7.2.2.2.2.2">,</mo><mrow id="p4.7.m7.2.2.2.2.1"><mi id="p4.7.m7.2.2.2.2.1.2">x</mi><mo id="p4.7.m7.2.2.2.2.1.1">+</mo><mn id="p4.7.m7.2.2.2.2.1.3">1</mn></mrow></mrow></msub></mrow><annotation encoding="application/x-tex" id="p4.7.m7.2c">\mathbb{U}_{\text{odd(even)}}=\otimes_{x\in{\text{odd(even)}}}U_{x,x+1}</annotation><annotation encoding="application/x-llamapun" id="p4.7.m7.2d">blackboard_U start_POSTSUBSCRIPT odd(even) end_POSTSUBSCRIPT = ⊗ start_POSTSUBSCRIPT italic_x ∈ odd(even) end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_x , italic_x + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="U_{x,x+1}" class="ltx_Math" display="inline" id="p4.8.m8.2"><semantics id="p4.8.m8.2a"><msub id="p4.8.m8.2.3" xref="p4.8.m8.2.3.cmml"><mi id="p4.8.m8.2.3.2" xref="p4.8.m8.2.3.2.cmml">U</mi><mrow id="p4.8.m8.2.2.2.2" xref="p4.8.m8.2.2.2.3.cmml"><mi id="p4.8.m8.1.1.1.1" xref="p4.8.m8.1.1.1.1.cmml">x</mi><mo id="p4.8.m8.2.2.2.2.2" xref="p4.8.m8.2.2.2.3.cmml">,</mo><mrow id="p4.8.m8.2.2.2.2.1" xref="p4.8.m8.2.2.2.2.1.cmml"><mi id="p4.8.m8.2.2.2.2.1.2" xref="p4.8.m8.2.2.2.2.1.2.cmml">x</mi><mo id="p4.8.m8.2.2.2.2.1.1" xref="p4.8.m8.2.2.2.2.1.1.cmml">+</mo><mn id="p4.8.m8.2.2.2.2.1.3" xref="p4.8.m8.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="p4.8.m8.2b"><apply id="p4.8.m8.2.3.cmml" xref="p4.8.m8.2.3"><csymbol cd="ambiguous" id="p4.8.m8.2.3.1.cmml" xref="p4.8.m8.2.3">subscript</csymbol><ci id="p4.8.m8.2.3.2.cmml" xref="p4.8.m8.2.3.2">𝑈</ci><list id="p4.8.m8.2.2.2.3.cmml" xref="p4.8.m8.2.2.2.2"><ci id="p4.8.m8.1.1.1.1.cmml" xref="p4.8.m8.1.1.1.1">𝑥</ci><apply id="p4.8.m8.2.2.2.2.1.cmml" xref="p4.8.m8.2.2.2.2.1"><plus id="p4.8.m8.2.2.2.2.1.1.cmml" xref="p4.8.m8.2.2.2.2.1.1"></plus><ci id="p4.8.m8.2.2.2.2.1.2.cmml" xref="p4.8.m8.2.2.2.2.1.2">𝑥</ci><cn id="p4.8.m8.2.2.2.2.1.3.cmml" type="integer" xref="p4.8.m8.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="p4.8.m8.2c">U_{x,x+1}</annotation><annotation encoding="application/x-llamapun" id="p4.8.m8.2d">italic_U start_POSTSUBSCRIPT italic_x , italic_x + 1 end_POSTSUBSCRIPT</annotation></semantics></math> are two-site gates acting locally on <math alttext="x" class="ltx_Math" display="inline" id="p4.9.m9.1"><semantics id="p4.9.m9.1a"><mi id="p4.9.m9.1.1" xref="p4.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="p4.9.m9.1b"><ci id="p4.9.m9.1.1.cmml" xref="p4.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="p4.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="p4.9.m9.1d">italic_x</annotation></semantics></math> and <math alttext="x+1" class="ltx_Math" display="inline" id="p4.10.m10.1"><semantics id="p4.10.m10.1a"><mrow 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As indicated by the form of <math alttext="\mathbb{U}" class="ltx_Math" display="inline" id="p4.11.m11.1"><semantics id="p4.11.m11.1a"><mi id="p4.11.m11.1.1" xref="p4.11.m11.1.1.cmml">𝕌</mi><annotation-xml encoding="MathML-Content" id="p4.11.m11.1b"><ci id="p4.11.m11.1.1.cmml" xref="p4.11.m11.1.1">𝕌</ci></annotation-xml><annotation encoding="application/x-tex" id="p4.11.m11.1c">\mathbb{U}</annotation><annotation encoding="application/x-llamapun" id="p4.11.m11.1d">blackboard_U</annotation></semantics></math>, the local gates are arranged in a brickwork architecture.</p> </div> <div class="ltx_para" id="p5"> <p class="ltx_p" id="p5.3">For the sake of convenience in depiction, we fold the forward and backward branches of time evolution [see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(a)]. By folding, each tensor is superimposed on its complex conjugate. 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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }% \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$=U_{ab}^{cd}\left(U_{a% ^{\prime}b^{\prime}}^{c^{\prime}d^{\prime}}\right)^{*}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}{}\pgfsys@moveto{126.0pt}{-149.54999pt}\pgfsys@stroke% \pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{127.125pt}{-156.62498pt}\pgfsys@invoke{ }\hbox{{\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }% \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$b,b^{\prime% }$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}{}\pgfsys@moveto{82.5pt}{-121.04999pt}\pgfsys@stroke\pgfsys@invoke% { 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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}{}\pgfsys@moveto{82.5pt}{-149.54999pt}\pgfsys@stroke\pgfsys@invoke% { }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1% .0}{83.625pt}{-154.77777pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor% }{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }% \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$a,a^{\prime% }$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}% \pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}% \lxSVG@closescope\endpgfpicture}},</annotation><annotation encoding="application/x-llamapun" id="S0.E1.m1.1d">= italic_U start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_d end_POSTSUPERSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_b , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_d , italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p5.2">where <math alttext="a,b,c,d" class="ltx_Math" display="inline" id="p5.1.m1.4"><semantics id="p5.1.m1.4a"><mrow id="p5.1.m1.4.5.2" xref="p5.1.m1.4.5.1.cmml"><mi id="p5.1.m1.1.1" xref="p5.1.m1.1.1.cmml">a</mi><mo id="p5.1.m1.4.5.2.1" xref="p5.1.m1.4.5.1.cmml">,</mo><mi id="p5.1.m1.2.2" xref="p5.1.m1.2.2.cmml">b</mi><mo id="p5.1.m1.4.5.2.2" xref="p5.1.m1.4.5.1.cmml">,</mo><mi id="p5.1.m1.3.3" xref="p5.1.m1.3.3.cmml">c</mi><mo id="p5.1.m1.4.5.2.3" xref="p5.1.m1.4.5.1.cmml">,</mo><mi id="p5.1.m1.4.4" xref="p5.1.m1.4.4.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="p5.1.m1.4b"><list id="p5.1.m1.4.5.1.cmml" xref="p5.1.m1.4.5.2"><ci id="p5.1.m1.1.1.cmml" xref="p5.1.m1.1.1">𝑎</ci><ci id="p5.1.m1.2.2.cmml" xref="p5.1.m1.2.2">𝑏</ci><ci id="p5.1.m1.3.3.cmml" xref="p5.1.m1.3.3">𝑐</ci><ci id="p5.1.m1.4.4.cmml" xref="p5.1.m1.4.4">𝑑</ci></list></annotation-xml><annotation encoding="application/x-tex" id="p5.1.m1.4c">a,b,c,d</annotation><annotation encoding="application/x-llamapun" 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id="p5.2.m2.3.3.3.3.1.cmml" xref="p5.2.m2.3.3.3.3">superscript</csymbol><ci id="p5.2.m2.3.3.3.3.2.cmml" xref="p5.2.m2.3.3.3.3.2">𝑐</ci><ci id="p5.2.m2.3.3.3.3.3.cmml" xref="p5.2.m2.3.3.3.3.3">′</ci></apply><apply id="p5.2.m2.4.4.4.4.cmml" xref="p5.2.m2.4.4.4.4"><csymbol cd="ambiguous" id="p5.2.m2.4.4.4.4.1.cmml" xref="p5.2.m2.4.4.4.4">superscript</csymbol><ci id="p5.2.m2.4.4.4.4.2.cmml" xref="p5.2.m2.4.4.4.4.2">𝑑</ci><ci id="p5.2.m2.4.4.4.4.3.cmml" xref="p5.2.m2.4.4.4.4.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="p5.2.m2.4c">a^{\prime},b^{\prime},c^{\prime},d^{\prime}</annotation><annotation encoding="application/x-llamapun" id="p5.2.m2.4d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>) denote the basis states in the forward (backward) branch.</p> </div> <div class="ltx_para" id="p6"> <p class="ltx_p" id="p6.20">We focus on a certain class of initial states on the composite system <math alttext="L" class="ltx_Math" display="inline" id="p6.1.m1.1"><semantics id="p6.1.m1.1a"><mi id="p6.1.m1.1.1" xref="p6.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p6.1.m1.1b"><ci id="p6.1.m1.1.1.cmml" xref="p6.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="p6.1.m1.1d">italic_L</annotation></semantics></math>-<math alttext="R" class="ltx_Math" display="inline" id="p6.2.m2.1"><semantics id="p6.2.m2.1a"><mi id="p6.2.m2.1.1" xref="p6.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p6.2.m2.1b"><ci id="p6.2.m2.1.1.cmml" xref="p6.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="p6.2.m2.1d">italic_R</annotation></semantics></math>, with <math alttext="L(R)" class="ltx_Math" display="inline" id="p6.3.m3.1"><semantics id="p6.3.m3.1a"><mrow id="p6.3.m3.1.2" xref="p6.3.m3.1.2.cmml"><mi id="p6.3.m3.1.2.2" xref="p6.3.m3.1.2.2.cmml">L</mi><mo id="p6.3.m3.1.2.1" xref="p6.3.m3.1.2.1.cmml">⁢</mo><mrow id="p6.3.m3.1.2.3.2" xref="p6.3.m3.1.2.cmml"><mo id="p6.3.m3.1.2.3.2.1" stretchy="false" xref="p6.3.m3.1.2.cmml">(</mo><mi id="p6.3.m3.1.1" xref="p6.3.m3.1.1.cmml">R</mi><mo id="p6.3.m3.1.2.3.2.2" stretchy="false" xref="p6.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p6.3.m3.1b"><apply id="p6.3.m3.1.2.cmml" xref="p6.3.m3.1.2"><times id="p6.3.m3.1.2.1.cmml" xref="p6.3.m3.1.2.1"></times><ci id="p6.3.m3.1.2.2.cmml" xref="p6.3.m3.1.2.2">𝐿</ci><ci id="p6.3.m3.1.1.cmml" xref="p6.3.m3.1.1">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.3.m3.1c">L(R)</annotation><annotation encoding="application/x-llamapun" id="p6.3.m3.1d">italic_L ( italic_R )</annotation></semantics></math> for the left (right), as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(b). For later convenience, we set the location of the leftmost site on R as the origin point <math alttext="x=0" class="ltx_Math" display="inline" id="p6.4.m4.1"><semantics id="p6.4.m4.1a"><mrow id="p6.4.m4.1.1" xref="p6.4.m4.1.1.cmml"><mi id="p6.4.m4.1.1.2" xref="p6.4.m4.1.1.2.cmml">x</mi><mo id="p6.4.m4.1.1.1" xref="p6.4.m4.1.1.1.cmml">=</mo><mn id="p6.4.m4.1.1.3" xref="p6.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="p6.4.m4.1b"><apply id="p6.4.m4.1.1.cmml" xref="p6.4.m4.1.1"><eq id="p6.4.m4.1.1.1.cmml" xref="p6.4.m4.1.1.1"></eq><ci id="p6.4.m4.1.1.2.cmml" xref="p6.4.m4.1.1.2">𝑥</ci><cn id="p6.4.m4.1.1.3.cmml" type="integer" xref="p6.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.4.m4.1c">x=0</annotation><annotation encoding="application/x-llamapun" id="p6.4.m4.1d">italic_x = 0</annotation></semantics></math>. The global unitary operator admits a decomposition as <math alttext="\mathbb{U}=\mathbb{U}_{\bar{R}}\mathbb{U}_{R}" class="ltx_Math" display="inline" id="p6.5.m5.1"><semantics id="p6.5.m5.1a"><mrow id="p6.5.m5.1.1" xref="p6.5.m5.1.1.cmml"><mi id="p6.5.m5.1.1.2" xref="p6.5.m5.1.1.2.cmml">𝕌</mi><mo id="p6.5.m5.1.1.1" xref="p6.5.m5.1.1.1.cmml">=</mo><mrow id="p6.5.m5.1.1.3" xref="p6.5.m5.1.1.3.cmml"><msub id="p6.5.m5.1.1.3.2" xref="p6.5.m5.1.1.3.2.cmml"><mi id="p6.5.m5.1.1.3.2.2" xref="p6.5.m5.1.1.3.2.2.cmml">𝕌</mi><mover accent="true" id="p6.5.m5.1.1.3.2.3" xref="p6.5.m5.1.1.3.2.3.cmml"><mi id="p6.5.m5.1.1.3.2.3.2" xref="p6.5.m5.1.1.3.2.3.2.cmml">R</mi><mo id="p6.5.m5.1.1.3.2.3.1" xref="p6.5.m5.1.1.3.2.3.1.cmml">¯</mo></mover></msub><mo id="p6.5.m5.1.1.3.1" xref="p6.5.m5.1.1.3.1.cmml">⁢</mo><msub id="p6.5.m5.1.1.3.3" xref="p6.5.m5.1.1.3.3.cmml"><mi id="p6.5.m5.1.1.3.3.2" xref="p6.5.m5.1.1.3.3.2.cmml">𝕌</mi><mi id="p6.5.m5.1.1.3.3.3" xref="p6.5.m5.1.1.3.3.3.cmml">R</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="p6.5.m5.1b"><apply id="p6.5.m5.1.1.cmml" xref="p6.5.m5.1.1"><eq id="p6.5.m5.1.1.1.cmml" xref="p6.5.m5.1.1.1"></eq><ci id="p6.5.m5.1.1.2.cmml" xref="p6.5.m5.1.1.2">𝕌</ci><apply id="p6.5.m5.1.1.3.cmml" xref="p6.5.m5.1.1.3"><times id="p6.5.m5.1.1.3.1.cmml" xref="p6.5.m5.1.1.3.1"></times><apply id="p6.5.m5.1.1.3.2.cmml" xref="p6.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="p6.5.m5.1.1.3.2.1.cmml" xref="p6.5.m5.1.1.3.2">subscript</csymbol><ci id="p6.5.m5.1.1.3.2.2.cmml" xref="p6.5.m5.1.1.3.2.2">𝕌</ci><apply id="p6.5.m5.1.1.3.2.3.cmml" xref="p6.5.m5.1.1.3.2.3"><ci id="p6.5.m5.1.1.3.2.3.1.cmml" xref="p6.5.m5.1.1.3.2.3.1">¯</ci><ci id="p6.5.m5.1.1.3.2.3.2.cmml" xref="p6.5.m5.1.1.3.2.3.2">𝑅</ci></apply></apply><apply id="p6.5.m5.1.1.3.3.cmml" xref="p6.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="p6.5.m5.1.1.3.3.1.cmml" xref="p6.5.m5.1.1.3.3">subscript</csymbol><ci id="p6.5.m5.1.1.3.3.2.cmml" xref="p6.5.m5.1.1.3.3.2">𝕌</ci><ci id="p6.5.m5.1.1.3.3.3.cmml" xref="p6.5.m5.1.1.3.3.3">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.5.m5.1c">\mathbb{U}=\mathbb{U}_{\bar{R}}\mathbb{U}_{R}</annotation><annotation encoding="application/x-llamapun" id="p6.5.m5.1d">blackboard_U = blackboard_U start_POSTSUBSCRIPT over¯ start_ARG italic_R end_ARG end_POSTSUBSCRIPT blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\mathbb{U}_{R}" class="ltx_Math" display="inline" id="p6.6.m6.1"><semantics id="p6.6.m6.1a"><msub id="p6.6.m6.1.1" xref="p6.6.m6.1.1.cmml"><mi id="p6.6.m6.1.1.2" xref="p6.6.m6.1.1.2.cmml">𝕌</mi><mi id="p6.6.m6.1.1.3" xref="p6.6.m6.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="p6.6.m6.1b"><apply id="p6.6.m6.1.1.cmml" xref="p6.6.m6.1.1"><csymbol cd="ambiguous" id="p6.6.m6.1.1.1.cmml" xref="p6.6.m6.1.1">subscript</csymbol><ci id="p6.6.m6.1.1.2.cmml" xref="p6.6.m6.1.1.2">𝕌</ci><ci id="p6.6.m6.1.1.3.cmml" xref="p6.6.m6.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.6.m6.1c">\mathbb{U}_{R}</annotation><annotation encoding="application/x-llamapun" id="p6.6.m6.1d">blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> acts nontrivially on <math alttext="R" class="ltx_Math" display="inline" id="p6.7.m7.1"><semantics id="p6.7.m7.1a"><mi id="p6.7.m7.1.1" xref="p6.7.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p6.7.m7.1b"><ci id="p6.7.m7.1.1.cmml" xref="p6.7.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.7.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="p6.7.m7.1d">italic_R</annotation></semantics></math>, and <math alttext="\mathbb{U}_{\bar{R}}" class="ltx_Math" display="inline" id="p6.8.m8.1"><semantics id="p6.8.m8.1a"><msub id="p6.8.m8.1.1" xref="p6.8.m8.1.1.cmml"><mi id="p6.8.m8.1.1.2" xref="p6.8.m8.1.1.2.cmml">𝕌</mi><mover accent="true" id="p6.8.m8.1.1.3" xref="p6.8.m8.1.1.3.cmml"><mi id="p6.8.m8.1.1.3.2" xref="p6.8.m8.1.1.3.2.cmml">R</mi><mo id="p6.8.m8.1.1.3.1" xref="p6.8.m8.1.1.3.1.cmml">¯</mo></mover></msub><annotation-xml encoding="MathML-Content" id="p6.8.m8.1b"><apply id="p6.8.m8.1.1.cmml" xref="p6.8.m8.1.1"><csymbol cd="ambiguous" id="p6.8.m8.1.1.1.cmml" xref="p6.8.m8.1.1">subscript</csymbol><ci id="p6.8.m8.1.1.2.cmml" xref="p6.8.m8.1.1.2">𝕌</ci><apply id="p6.8.m8.1.1.3.cmml" xref="p6.8.m8.1.1.3"><ci id="p6.8.m8.1.1.3.1.cmml" xref="p6.8.m8.1.1.3.1">¯</ci><ci id="p6.8.m8.1.1.3.2.cmml" xref="p6.8.m8.1.1.3.2">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.8.m8.1c">\mathbb{U}_{\bar{R}}</annotation><annotation encoding="application/x-llamapun" id="p6.8.m8.1d">blackboard_U start_POSTSUBSCRIPT over¯ start_ARG italic_R end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> acts on <math alttext="L" class="ltx_Math" display="inline" id="p6.9.m9.1"><semantics id="p6.9.m9.1a"><mi id="p6.9.m9.1.1" xref="p6.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p6.9.m9.1b"><ci id="p6.9.m9.1.1.cmml" xref="p6.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="p6.9.m9.1d">italic_L</annotation></semantics></math> and the boundary across the two regions. Meanwhile, we assume that the overall initial state can be decomposed as <math alttext="\ket{\Psi_{\text{in}}}=\sum_{j=1}^{\chi}\ket{\Psi_{L}^{j}}\otimes\ket{\Psi_{R}% ^{j}}" class="ltx_Math" display="inline" id="p6.10.m10.3"><semantics id="p6.10.m10.3a"><mrow id="p6.10.m10.3.4" xref="p6.10.m10.3.4.cmml"><mrow id="p6.10.m10.1.1.3" xref="p6.10.m10.1.1.2.cmml"><mo id="p6.10.m10.1.1.3.1" stretchy="false" xref="p6.10.m10.1.1.2.1.cmml">|</mo><msub id="p6.10.m10.1.1.1.1" xref="p6.10.m10.1.1.1.1.cmml"><mi id="p6.10.m10.1.1.1.1.2" mathvariant="normal" xref="p6.10.m10.1.1.1.1.2.cmml">Ψ</mi><mtext id="p6.10.m10.1.1.1.1.3" xref="p6.10.m10.1.1.1.1.3a.cmml">in</mtext></msub><mo id="p6.10.m10.1.1.3.2" stretchy="false" xref="p6.10.m10.1.1.2.1.cmml">⟩</mo></mrow><mo id="p6.10.m10.3.4.1" rspace="0.111em" xref="p6.10.m10.3.4.1.cmml">=</mo><mrow id="p6.10.m10.3.4.2" xref="p6.10.m10.3.4.2.cmml"><msubsup id="p6.10.m10.3.4.2.1" xref="p6.10.m10.3.4.2.1.cmml"><mo id="p6.10.m10.3.4.2.1.2.2" rspace="0em" xref="p6.10.m10.3.4.2.1.2.2.cmml">∑</mo><mrow id="p6.10.m10.3.4.2.1.2.3" xref="p6.10.m10.3.4.2.1.2.3.cmml"><mi id="p6.10.m10.3.4.2.1.2.3.2" xref="p6.10.m10.3.4.2.1.2.3.2.cmml">j</mi><mo id="p6.10.m10.3.4.2.1.2.3.1" xref="p6.10.m10.3.4.2.1.2.3.1.cmml">=</mo><mn id="p6.10.m10.3.4.2.1.2.3.3" xref="p6.10.m10.3.4.2.1.2.3.3.cmml">1</mn></mrow><mi id="p6.10.m10.3.4.2.1.3" xref="p6.10.m10.3.4.2.1.3.cmml">χ</mi></msubsup><mrow id="p6.10.m10.3.4.2.2" xref="p6.10.m10.3.4.2.2.cmml"><mrow id="p6.10.m10.2.2.3" xref="p6.10.m10.2.2.2.cmml"><mo id="p6.10.m10.2.2.3.1" stretchy="false" xref="p6.10.m10.2.2.2.1.cmml">|</mo><msubsup id="p6.10.m10.2.2.1.1" xref="p6.10.m10.2.2.1.1.cmml"><mi id="p6.10.m10.2.2.1.1.2.2" mathvariant="normal" xref="p6.10.m10.2.2.1.1.2.2.cmml">Ψ</mi><mi id="p6.10.m10.2.2.1.1.2.3" xref="p6.10.m10.2.2.1.1.2.3.cmml">L</mi><mi id="p6.10.m10.2.2.1.1.3" xref="p6.10.m10.2.2.1.1.3.cmml">j</mi></msubsup><mo id="p6.10.m10.2.2.3.2" rspace="0.055em" stretchy="false" xref="p6.10.m10.2.2.2.1.cmml">⟩</mo></mrow><mo id="p6.10.m10.3.4.2.2.1" rspace="0.222em" xref="p6.10.m10.3.4.2.2.1.cmml">⊗</mo><mrow id="p6.10.m10.3.3.3" xref="p6.10.m10.3.3.2.cmml"><mo id="p6.10.m10.3.3.3.1" stretchy="false" xref="p6.10.m10.3.3.2.1.cmml">|</mo><msubsup id="p6.10.m10.3.3.1.1" xref="p6.10.m10.3.3.1.1.cmml"><mi id="p6.10.m10.3.3.1.1.2.2" mathvariant="normal" xref="p6.10.m10.3.3.1.1.2.2.cmml">Ψ</mi><mi id="p6.10.m10.3.3.1.1.2.3" xref="p6.10.m10.3.3.1.1.2.3.cmml">R</mi><mi id="p6.10.m10.3.3.1.1.3" xref="p6.10.m10.3.3.1.1.3.cmml">j</mi></msubsup><mo id="p6.10.m10.3.3.3.2" stretchy="false" xref="p6.10.m10.3.3.2.1.cmml">⟩</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p6.10.m10.3b"><apply id="p6.10.m10.3.4.cmml" xref="p6.10.m10.3.4"><eq id="p6.10.m10.3.4.1.cmml" xref="p6.10.m10.3.4.1"></eq><apply id="p6.10.m10.1.1.2.cmml" xref="p6.10.m10.1.1.3"><csymbol cd="latexml" id="p6.10.m10.1.1.2.1.cmml" xref="p6.10.m10.1.1.3.1">ket</csymbol><apply id="p6.10.m10.1.1.1.1.cmml" xref="p6.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="p6.10.m10.1.1.1.1.1.cmml" xref="p6.10.m10.1.1.1.1">subscript</csymbol><ci id="p6.10.m10.1.1.1.1.2.cmml" xref="p6.10.m10.1.1.1.1.2">Ψ</ci><ci id="p6.10.m10.1.1.1.1.3a.cmml" xref="p6.10.m10.1.1.1.1.3"><mtext id="p6.10.m10.1.1.1.1.3.cmml" mathsize="70%" xref="p6.10.m10.1.1.1.1.3">in</mtext></ci></apply></apply><apply id="p6.10.m10.3.4.2.cmml" xref="p6.10.m10.3.4.2"><apply id="p6.10.m10.3.4.2.1.cmml" xref="p6.10.m10.3.4.2.1"><csymbol cd="ambiguous" id="p6.10.m10.3.4.2.1.1.cmml" xref="p6.10.m10.3.4.2.1">superscript</csymbol><apply id="p6.10.m10.3.4.2.1.2.cmml" xref="p6.10.m10.3.4.2.1"><csymbol cd="ambiguous" id="p6.10.m10.3.4.2.1.2.1.cmml" xref="p6.10.m10.3.4.2.1">subscript</csymbol><sum id="p6.10.m10.3.4.2.1.2.2.cmml" xref="p6.10.m10.3.4.2.1.2.2"></sum><apply id="p6.10.m10.3.4.2.1.2.3.cmml" xref="p6.10.m10.3.4.2.1.2.3"><eq id="p6.10.m10.3.4.2.1.2.3.1.cmml" xref="p6.10.m10.3.4.2.1.2.3.1"></eq><ci id="p6.10.m10.3.4.2.1.2.3.2.cmml" xref="p6.10.m10.3.4.2.1.2.3.2">𝑗</ci><cn id="p6.10.m10.3.4.2.1.2.3.3.cmml" type="integer" xref="p6.10.m10.3.4.2.1.2.3.3">1</cn></apply></apply><ci id="p6.10.m10.3.4.2.1.3.cmml" xref="p6.10.m10.3.4.2.1.3">𝜒</ci></apply><apply id="p6.10.m10.3.4.2.2.cmml" xref="p6.10.m10.3.4.2.2"><csymbol cd="latexml" id="p6.10.m10.3.4.2.2.1.cmml" xref="p6.10.m10.3.4.2.2.1">tensor-product</csymbol><apply id="p6.10.m10.2.2.2.cmml" xref="p6.10.m10.2.2.3"><csymbol cd="latexml" id="p6.10.m10.2.2.2.1.cmml" xref="p6.10.m10.2.2.3.1">ket</csymbol><apply id="p6.10.m10.2.2.1.1.cmml" xref="p6.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="p6.10.m10.2.2.1.1.1.cmml" xref="p6.10.m10.2.2.1.1">superscript</csymbol><apply id="p6.10.m10.2.2.1.1.2.cmml" xref="p6.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="p6.10.m10.2.2.1.1.2.1.cmml" xref="p6.10.m10.2.2.1.1">subscript</csymbol><ci id="p6.10.m10.2.2.1.1.2.2.cmml" xref="p6.10.m10.2.2.1.1.2.2">Ψ</ci><ci id="p6.10.m10.2.2.1.1.2.3.cmml" xref="p6.10.m10.2.2.1.1.2.3">𝐿</ci></apply><ci id="p6.10.m10.2.2.1.1.3.cmml" xref="p6.10.m10.2.2.1.1.3">𝑗</ci></apply></apply><apply id="p6.10.m10.3.3.2.cmml" xref="p6.10.m10.3.3.3"><csymbol cd="latexml" id="p6.10.m10.3.3.2.1.cmml" xref="p6.10.m10.3.3.3.1">ket</csymbol><apply id="p6.10.m10.3.3.1.1.cmml" xref="p6.10.m10.3.3.1.1"><csymbol cd="ambiguous" id="p6.10.m10.3.3.1.1.1.cmml" xref="p6.10.m10.3.3.1.1">superscript</csymbol><apply id="p6.10.m10.3.3.1.1.2.cmml" xref="p6.10.m10.3.3.1.1"><csymbol cd="ambiguous" id="p6.10.m10.3.3.1.1.2.1.cmml" xref="p6.10.m10.3.3.1.1">subscript</csymbol><ci id="p6.10.m10.3.3.1.1.2.2.cmml" xref="p6.10.m10.3.3.1.1.2.2">Ψ</ci><ci id="p6.10.m10.3.3.1.1.2.3.cmml" xref="p6.10.m10.3.3.1.1.2.3">𝑅</ci></apply><ci id="p6.10.m10.3.3.1.1.3.cmml" xref="p6.10.m10.3.3.1.1.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.10.m10.3c">\ket{\Psi_{\text{in}}}=\sum_{j=1}^{\chi}\ket{\Psi_{L}^{j}}\otimes\ket{\Psi_{R}% ^{j}}</annotation><annotation encoding="application/x-llamapun" id="p6.10.m10.3d">| start_ARG roman_Ψ start_POSTSUBSCRIPT in end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_χ end_POSTSUPERSCRIPT | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩ ⊗ | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> (where each <math alttext="\ket{\Psi_{R/L}^{j}}" class="ltx_Math" display="inline" id="p6.11.m11.1"><semantics id="p6.11.m11.1a"><mrow id="p6.11.m11.1.1.3" xref="p6.11.m11.1.1.2.cmml"><mo id="p6.11.m11.1.1.3.1" stretchy="false" xref="p6.11.m11.1.1.2.1.cmml">|</mo><msubsup id="p6.11.m11.1.1.1.1" xref="p6.11.m11.1.1.1.1.cmml"><mi id="p6.11.m11.1.1.1.1.2.2" mathvariant="normal" xref="p6.11.m11.1.1.1.1.2.2.cmml">Ψ</mi><mrow id="p6.11.m11.1.1.1.1.2.3" xref="p6.11.m11.1.1.1.1.2.3.cmml"><mi id="p6.11.m11.1.1.1.1.2.3.2" xref="p6.11.m11.1.1.1.1.2.3.2.cmml">R</mi><mo id="p6.11.m11.1.1.1.1.2.3.1" xref="p6.11.m11.1.1.1.1.2.3.1.cmml">/</mo><mi id="p6.11.m11.1.1.1.1.2.3.3" xref="p6.11.m11.1.1.1.1.2.3.3.cmml">L</mi></mrow><mi id="p6.11.m11.1.1.1.1.3" xref="p6.11.m11.1.1.1.1.3.cmml">j</mi></msubsup><mo id="p6.11.m11.1.1.3.2" stretchy="false" xref="p6.11.m11.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p6.11.m11.1b"><apply id="p6.11.m11.1.1.2.cmml" xref="p6.11.m11.1.1.3"><csymbol cd="latexml" id="p6.11.m11.1.1.2.1.cmml" xref="p6.11.m11.1.1.3.1">ket</csymbol><apply id="p6.11.m11.1.1.1.1.cmml" xref="p6.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="p6.11.m11.1.1.1.1.1.cmml" xref="p6.11.m11.1.1.1.1">superscript</csymbol><apply id="p6.11.m11.1.1.1.1.2.cmml" xref="p6.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="p6.11.m11.1.1.1.1.2.1.cmml" xref="p6.11.m11.1.1.1.1">subscript</csymbol><ci id="p6.11.m11.1.1.1.1.2.2.cmml" xref="p6.11.m11.1.1.1.1.2.2">Ψ</ci><apply id="p6.11.m11.1.1.1.1.2.3.cmml" xref="p6.11.m11.1.1.1.1.2.3"><divide id="p6.11.m11.1.1.1.1.2.3.1.cmml" xref="p6.11.m11.1.1.1.1.2.3.1"></divide><ci id="p6.11.m11.1.1.1.1.2.3.2.cmml" xref="p6.11.m11.1.1.1.1.2.3.2">𝑅</ci><ci id="p6.11.m11.1.1.1.1.2.3.3.cmml" xref="p6.11.m11.1.1.1.1.2.3.3">𝐿</ci></apply></apply><ci id="p6.11.m11.1.1.1.1.3.cmml" xref="p6.11.m11.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.11.m11.1c">\ket{\Psi_{R/L}^{j}}</annotation><annotation encoding="application/x-llamapun" id="p6.11.m11.1d">| start_ARG roman_Ψ start_POSTSUBSCRIPT italic_R / italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> is not required to be normalized). Here, we consider generic <math alttext="\ket{\Psi_{R}^{j}}" class="ltx_Math" display="inline" id="p6.12.m12.1"><semantics id="p6.12.m12.1a"><mrow id="p6.12.m12.1.1.3" xref="p6.12.m12.1.1.2.cmml"><mo id="p6.12.m12.1.1.3.1" stretchy="false" xref="p6.12.m12.1.1.2.1.cmml">|</mo><msubsup id="p6.12.m12.1.1.1.1" xref="p6.12.m12.1.1.1.1.cmml"><mi id="p6.12.m12.1.1.1.1.2.2" mathvariant="normal" xref="p6.12.m12.1.1.1.1.2.2.cmml">Ψ</mi><mi id="p6.12.m12.1.1.1.1.2.3" xref="p6.12.m12.1.1.1.1.2.3.cmml">R</mi><mi id="p6.12.m12.1.1.1.1.3" xref="p6.12.m12.1.1.1.1.3.cmml">j</mi></msubsup><mo id="p6.12.m12.1.1.3.2" stretchy="false" xref="p6.12.m12.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p6.12.m12.1b"><apply id="p6.12.m12.1.1.2.cmml" xref="p6.12.m12.1.1.3"><csymbol cd="latexml" id="p6.12.m12.1.1.2.1.cmml" xref="p6.12.m12.1.1.3.1">ket</csymbol><apply id="p6.12.m12.1.1.1.1.cmml" xref="p6.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="p6.12.m12.1.1.1.1.1.cmml" xref="p6.12.m12.1.1.1.1">superscript</csymbol><apply id="p6.12.m12.1.1.1.1.2.cmml" xref="p6.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="p6.12.m12.1.1.1.1.2.1.cmml" xref="p6.12.m12.1.1.1.1">subscript</csymbol><ci id="p6.12.m12.1.1.1.1.2.2.cmml" xref="p6.12.m12.1.1.1.1.2.2">Ψ</ci><ci id="p6.12.m12.1.1.1.1.2.3.cmml" xref="p6.12.m12.1.1.1.1.2.3">𝑅</ci></apply><ci id="p6.12.m12.1.1.1.1.3.cmml" xref="p6.12.m12.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.12.m12.1c">\ket{\Psi_{R}^{j}}</annotation><annotation encoding="application/x-llamapun" id="p6.12.m12.1d">| start_ARG roman_Ψ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> on the right, while <math alttext="\ket{\Psi_{L}^{j}}" class="ltx_Math" display="inline" id="p6.13.m13.1"><semantics id="p6.13.m13.1a"><mrow id="p6.13.m13.1.1.3" xref="p6.13.m13.1.1.2.cmml"><mo id="p6.13.m13.1.1.3.1" stretchy="false" xref="p6.13.m13.1.1.2.1.cmml">|</mo><msubsup id="p6.13.m13.1.1.1.1" xref="p6.13.m13.1.1.1.1.cmml"><mi id="p6.13.m13.1.1.1.1.2.2" mathvariant="normal" xref="p6.13.m13.1.1.1.1.2.2.cmml">Ψ</mi><mi id="p6.13.m13.1.1.1.1.2.3" xref="p6.13.m13.1.1.1.1.2.3.cmml">L</mi><mi id="p6.13.m13.1.1.1.1.3" xref="p6.13.m13.1.1.1.1.3.cmml">j</mi></msubsup><mo id="p6.13.m13.1.1.3.2" stretchy="false" xref="p6.13.m13.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p6.13.m13.1b"><apply id="p6.13.m13.1.1.2.cmml" xref="p6.13.m13.1.1.3"><csymbol cd="latexml" id="p6.13.m13.1.1.2.1.cmml" xref="p6.13.m13.1.1.3.1">ket</csymbol><apply id="p6.13.m13.1.1.1.1.cmml" xref="p6.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="p6.13.m13.1.1.1.1.1.cmml" xref="p6.13.m13.1.1.1.1">superscript</csymbol><apply id="p6.13.m13.1.1.1.1.2.cmml" xref="p6.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="p6.13.m13.1.1.1.1.2.1.cmml" xref="p6.13.m13.1.1.1.1">subscript</csymbol><ci id="p6.13.m13.1.1.1.1.2.2.cmml" xref="p6.13.m13.1.1.1.1.2.2">Ψ</ci><ci id="p6.13.m13.1.1.1.1.2.3.cmml" xref="p6.13.m13.1.1.1.1.2.3">𝐿</ci></apply><ci id="p6.13.m13.1.1.1.1.3.cmml" xref="p6.13.m13.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.13.m13.1c">\ket{\Psi_{L}^{j}}</annotation><annotation encoding="application/x-llamapun" id="p6.13.m13.1d">| start_ARG roman_Ψ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> can be written as a one-site shift-invariant MPS of bond dimension <math alttext="\chi" class="ltx_Math" display="inline" id="p6.14.m14.1"><semantics id="p6.14.m14.1a"><mi id="p6.14.m14.1.1" xref="p6.14.m14.1.1.cmml">χ</mi><annotation-xml encoding="MathML-Content" id="p6.14.m14.1b"><ci id="p6.14.m14.1.1.cmml" xref="p6.14.m14.1.1">𝜒</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.14.m14.1c">\chi</annotation><annotation encoding="application/x-llamapun" id="p6.14.m14.1d">italic_χ</annotation></semantics></math> in terms of the three-leg tensor <math alttext="A^{(a)}_{jk}" class="ltx_Math" display="inline" id="p6.15.m15.1"><semantics id="p6.15.m15.1a"><msubsup id="p6.15.m15.1.2" xref="p6.15.m15.1.2.cmml"><mi id="p6.15.m15.1.2.2.2" xref="p6.15.m15.1.2.2.2.cmml">A</mi><mrow id="p6.15.m15.1.2.3" xref="p6.15.m15.1.2.3.cmml"><mi id="p6.15.m15.1.2.3.2" xref="p6.15.m15.1.2.3.2.cmml">j</mi><mo id="p6.15.m15.1.2.3.1" xref="p6.15.m15.1.2.3.1.cmml">⁢</mo><mi id="p6.15.m15.1.2.3.3" xref="p6.15.m15.1.2.3.3.cmml">k</mi></mrow><mrow id="p6.15.m15.1.1.1.3" xref="p6.15.m15.1.2.cmml"><mo id="p6.15.m15.1.1.1.3.1" stretchy="false" xref="p6.15.m15.1.2.cmml">(</mo><mi id="p6.15.m15.1.1.1.1" xref="p6.15.m15.1.1.1.1.cmml">a</mi><mo id="p6.15.m15.1.1.1.3.2" stretchy="false" xref="p6.15.m15.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="p6.15.m15.1b"><apply id="p6.15.m15.1.2.cmml" xref="p6.15.m15.1.2"><csymbol cd="ambiguous" id="p6.15.m15.1.2.1.cmml" xref="p6.15.m15.1.2">subscript</csymbol><apply id="p6.15.m15.1.2.2.cmml" xref="p6.15.m15.1.2"><csymbol cd="ambiguous" id="p6.15.m15.1.2.2.1.cmml" xref="p6.15.m15.1.2">superscript</csymbol><ci id="p6.15.m15.1.2.2.2.cmml" xref="p6.15.m15.1.2.2.2">𝐴</ci><ci id="p6.15.m15.1.1.1.1.cmml" xref="p6.15.m15.1.1.1.1">𝑎</ci></apply><apply id="p6.15.m15.1.2.3.cmml" xref="p6.15.m15.1.2.3"><times id="p6.15.m15.1.2.3.1.cmml" xref="p6.15.m15.1.2.3.1"></times><ci id="p6.15.m15.1.2.3.2.cmml" xref="p6.15.m15.1.2.3.2">𝑗</ci><ci id="p6.15.m15.1.2.3.3.cmml" xref="p6.15.m15.1.2.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.15.m15.1c">A^{(a)}_{jk}</annotation><annotation encoding="application/x-llamapun" id="p6.15.m15.1d">italic_A start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="j=0,1,\cdots,\chi-1" class="ltx_Math" display="inline" id="p6.16.m16.4"><semantics id="p6.16.m16.4a"><mrow id="p6.16.m16.4.4" xref="p6.16.m16.4.4.cmml"><mi id="p6.16.m16.4.4.3" xref="p6.16.m16.4.4.3.cmml">j</mi><mo id="p6.16.m16.4.4.2" xref="p6.16.m16.4.4.2.cmml">=</mo><mrow id="p6.16.m16.4.4.1.1" xref="p6.16.m16.4.4.1.2.cmml"><mn id="p6.16.m16.1.1" xref="p6.16.m16.1.1.cmml">0</mn><mo id="p6.16.m16.4.4.1.1.2" xref="p6.16.m16.4.4.1.2.cmml">,</mo><mn id="p6.16.m16.2.2" xref="p6.16.m16.2.2.cmml">1</mn><mo id="p6.16.m16.4.4.1.1.3" xref="p6.16.m16.4.4.1.2.cmml">,</mo><mi id="p6.16.m16.3.3" mathvariant="normal" xref="p6.16.m16.3.3.cmml">⋯</mi><mo id="p6.16.m16.4.4.1.1.4" xref="p6.16.m16.4.4.1.2.cmml">,</mo><mrow id="p6.16.m16.4.4.1.1.1" xref="p6.16.m16.4.4.1.1.1.cmml"><mi id="p6.16.m16.4.4.1.1.1.2" xref="p6.16.m16.4.4.1.1.1.2.cmml">χ</mi><mo id="p6.16.m16.4.4.1.1.1.1" xref="p6.16.m16.4.4.1.1.1.1.cmml">−</mo><mn id="p6.16.m16.4.4.1.1.1.3" xref="p6.16.m16.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p6.16.m16.4b"><apply id="p6.16.m16.4.4.cmml" xref="p6.16.m16.4.4"><eq id="p6.16.m16.4.4.2.cmml" xref="p6.16.m16.4.4.2"></eq><ci id="p6.16.m16.4.4.3.cmml" xref="p6.16.m16.4.4.3">𝑗</ci><list id="p6.16.m16.4.4.1.2.cmml" xref="p6.16.m16.4.4.1.1"><cn id="p6.16.m16.1.1.cmml" type="integer" xref="p6.16.m16.1.1">0</cn><cn id="p6.16.m16.2.2.cmml" type="integer" xref="p6.16.m16.2.2">1</cn><ci id="p6.16.m16.3.3.cmml" xref="p6.16.m16.3.3">⋯</ci><apply id="p6.16.m16.4.4.1.1.1.cmml" xref="p6.16.m16.4.4.1.1.1"><minus id="p6.16.m16.4.4.1.1.1.1.cmml" xref="p6.16.m16.4.4.1.1.1.1"></minus><ci id="p6.16.m16.4.4.1.1.1.2.cmml" xref="p6.16.m16.4.4.1.1.1.2">𝜒</ci><cn id="p6.16.m16.4.4.1.1.1.3.cmml" type="integer" xref="p6.16.m16.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.16.m16.4c">j=0,1,\cdots,\chi-1</annotation><annotation encoding="application/x-llamapun" id="p6.16.m16.4d">italic_j = 0 , 1 , ⋯ , italic_χ - 1</annotation></semantics></math>. The matrices <math alttext="\{A^{(a)}\}" class="ltx_Math" display="inline" id="p6.17.m17.2"><semantics id="p6.17.m17.2a"><mrow id="p6.17.m17.2.2.1" xref="p6.17.m17.2.2.2.cmml"><mo id="p6.17.m17.2.2.1.2" stretchy="false" xref="p6.17.m17.2.2.2.cmml">{</mo><msup id="p6.17.m17.2.2.1.1" xref="p6.17.m17.2.2.1.1.cmml"><mi id="p6.17.m17.2.2.1.1.2" xref="p6.17.m17.2.2.1.1.2.cmml">A</mi><mrow id="p6.17.m17.1.1.1.3" xref="p6.17.m17.2.2.1.1.cmml"><mo id="p6.17.m17.1.1.1.3.1" stretchy="false" xref="p6.17.m17.2.2.1.1.cmml">(</mo><mi id="p6.17.m17.1.1.1.1" xref="p6.17.m17.1.1.1.1.cmml">a</mi><mo id="p6.17.m17.1.1.1.3.2" stretchy="false" xref="p6.17.m17.2.2.1.1.cmml">)</mo></mrow></msup><mo id="p6.17.m17.2.2.1.3" stretchy="false" xref="p6.17.m17.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="p6.17.m17.2b"><set id="p6.17.m17.2.2.2.cmml" xref="p6.17.m17.2.2.1"><apply id="p6.17.m17.2.2.1.1.cmml" xref="p6.17.m17.2.2.1.1"><csymbol cd="ambiguous" id="p6.17.m17.2.2.1.1.1.cmml" xref="p6.17.m17.2.2.1.1">superscript</csymbol><ci id="p6.17.m17.2.2.1.1.2.cmml" xref="p6.17.m17.2.2.1.1.2">𝐴</ci><ci id="p6.17.m17.1.1.1.1.cmml" xref="p6.17.m17.1.1.1.1">𝑎</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="p6.17.m17.2c">\{A^{(a)}\}</annotation><annotation encoding="application/x-llamapun" id="p6.17.m17.2d">{ italic_A start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT }</annotation></semantics></math> act on the auxiliary Hilbert space <math alttext="\mathcal{H}_{\chi}" class="ltx_Math" display="inline" id="p6.18.m18.1"><semantics id="p6.18.m18.1a"><msub id="p6.18.m18.1.1" xref="p6.18.m18.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p6.18.m18.1.1.2" xref="p6.18.m18.1.1.2.cmml">ℋ</mi><mi id="p6.18.m18.1.1.3" xref="p6.18.m18.1.1.3.cmml">χ</mi></msub><annotation-xml encoding="MathML-Content" id="p6.18.m18.1b"><apply id="p6.18.m18.1.1.cmml" xref="p6.18.m18.1.1"><csymbol cd="ambiguous" id="p6.18.m18.1.1.1.cmml" xref="p6.18.m18.1.1">subscript</csymbol><ci id="p6.18.m18.1.1.2.cmml" xref="p6.18.m18.1.1.2">ℋ</ci><ci id="p6.18.m18.1.1.3.cmml" xref="p6.18.m18.1.1.3">𝜒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p6.18.m18.1c">\mathcal{H}_{\chi}</annotation><annotation encoding="application/x-llamapun" id="p6.18.m18.1d">caligraphic_H start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT</annotation></semantics></math> spanned by the basis <math alttext="\{|j)\}_{j=0}^{\chi-1}" class="ltx_math_unparsed" display="inline" id="p6.19.m19.1"><semantics id="p6.19.m19.1a"><mrow id="p6.19.m19.1b"><mrow id="p6.19.m19.1.1"><mo id="p6.19.m19.1.1.1" stretchy="false">{</mo><mo fence="false" id="p6.19.m19.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="p6.19.m19.1.1.3">j</mi><mo id="p6.19.m19.1.1.4" stretchy="false">)</mo></mrow><mo id="p6.19.m19.1.2" stretchy="false">}</mo><msub id="p6.19.m19.1.3"><mi id="p6.19.m19.1.3a"></mi><mrow id="p6.19.m19.1.3.1"><mi id="p6.19.m19.1.3.1.2">j</mi><mo id="p6.19.m19.1.3.1.1">=</mo><mn id="p6.19.m19.1.3.1.3">0</mn></mrow></msub><msup id="p6.19.m19.1.4"><mi id="p6.19.m19.1.4a"></mi><mrow id="p6.19.m19.1.4.1"><mi id="p6.19.m19.1.4.1.2">χ</mi><mo id="p6.19.m19.1.4.1.1">−</mo><mn id="p6.19.m19.1.4.1.3">1</mn></mrow></msup></mrow><annotation encoding="application/x-tex" id="p6.19.m19.1c">\{|j)\}_{j=0}^{\chi-1}</annotation><annotation encoding="application/x-llamapun" id="p6.19.m19.1d">{ | italic_j ) } start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_χ - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Graphically, we can represent the folded tensor <math alttext="A" class="ltx_Math" display="inline" id="p6.20.m20.1"><semantics id="p6.20.m20.1a"><mi id="p6.20.m20.1.1" xref="p6.20.m20.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="p6.20.m20.1b"><ci id="p6.20.m20.1.1.cmml" xref="p6.20.m20.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="p6.20.m20.1c">A</annotation><annotation encoding="application/x-llamapun" id="p6.20.m20.1d">italic_A</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S0.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\leavevmode\hbox to99pt{\vbox to31.09pt{\pgfpicture\makeatletter\hbox{\hskip-7% 5.67499pt\lower-114.87749pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}% \pgfsys@invoke{ 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href="https://arxiv.org/html/2403.14807v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib23" title="">23</a>]</cite> <math alttext="\sum_{a=0}^{q-1}{A^{(a)}}^{\dagger}A^{(a)}=I_{\chi}" class="ltx_Math" display="inline" id="p6.22.m1.2"><semantics id="p6.22.m1.2a"><mrow id="p6.22.m1.2.3" xref="p6.22.m1.2.3.cmml"><mrow id="p6.22.m1.2.3.2" xref="p6.22.m1.2.3.2.cmml"><msubsup id="p6.22.m1.2.3.2.1" xref="p6.22.m1.2.3.2.1.cmml"><mo id="p6.22.m1.2.3.2.1.2.2" xref="p6.22.m1.2.3.2.1.2.2.cmml">∑</mo><mrow id="p6.22.m1.2.3.2.1.2.3" xref="p6.22.m1.2.3.2.1.2.3.cmml"><mi id="p6.22.m1.2.3.2.1.2.3.2" xref="p6.22.m1.2.3.2.1.2.3.2.cmml">a</mi><mo id="p6.22.m1.2.3.2.1.2.3.1" xref="p6.22.m1.2.3.2.1.2.3.1.cmml">=</mo><mn id="p6.22.m1.2.3.2.1.2.3.3" xref="p6.22.m1.2.3.2.1.2.3.3.cmml">0</mn></mrow><mrow id="p6.22.m1.2.3.2.1.3" xref="p6.22.m1.2.3.2.1.3.cmml"><mi 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=\delta_{j,j^{\prime}}.</annotation><annotation encoding="application/x-llamapun" id="S0.E3.m1.1.1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.m1.3d">= italic_δ start_POSTSUBSCRIPT italic_j , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .</annotation></semantics></math></foreignobject></g></g></g></svg></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="0"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="p7"> <p class="ltx_p" id="p7.8"><span class="ltx_text ltx_font_italic" id="p7.8.1">Exact influence matrix</span>—We would like to explore the influence of an infinitely large system on its own subsystem. To this end, in our setup we take the region <math alttext="L" class="ltx_Math" display="inline" id="p7.1.m1.1"><semantics id="p7.1.m1.1a"><mi id="p7.1.m1.1.1" xref="p7.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p7.1.m1.1b"><ci id="p7.1.m1.1.1.cmml" xref="p7.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="p7.1.m1.1d">italic_L</annotation></semantics></math> to be semi-infinite. Next, we trace out the region <math alttext="L" class="ltx_Math" display="inline" id="p7.2.m2.1"><semantics id="p7.2.m2.1a"><mi id="p7.2.m2.1.1" xref="p7.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p7.2.m2.1b"><ci id="p7.2.m2.1.1.cmml" xref="p7.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="p7.2.m2.1d">italic_L</annotation></semantics></math> after <math alttext="T" class="ltx_Math" display="inline" id="p7.3.m3.1"><semantics id="p7.3.m3.1a"><mi id="p7.3.m3.1.1" xref="p7.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="p7.3.m3.1b"><ci id="p7.3.m3.1.1.cmml" xref="p7.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="p7.3.m3.1d">italic_T</annotation></semantics></math> time steps of evolution, and thus obtain the reduced density matrix on <math alttext="R" class="ltx_Math" display="inline" id="p7.4.m4.1"><semantics id="p7.4.m4.1a"><mi id="p7.4.m4.1.1" xref="p7.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p7.4.m4.1b"><ci id="p7.4.m4.1.1.cmml" xref="p7.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="p7.4.m4.1d">italic_R</annotation></semantics></math>: <math alttext="\rho_{R}(T)=\textnormal{Tr}_{\text{L}}[\mathbb{U}^{T}\ket{\Psi_{\text{in}}}% \bra{\Psi_{\text{in}}}{\mathbb{U}^{\dagger}}^{T}]" class="ltx_Math" display="inline" id="p7.5.m5.4"><semantics id="p7.5.m5.4a"><mrow id="p7.5.m5.4.4" xref="p7.5.m5.4.4.cmml"><mrow id="p7.5.m5.4.4.3" xref="p7.5.m5.4.4.3.cmml"><msub id="p7.5.m5.4.4.3.2" xref="p7.5.m5.4.4.3.2.cmml"><mi id="p7.5.m5.4.4.3.2.2" xref="p7.5.m5.4.4.3.2.2.cmml">ρ</mi><mi id="p7.5.m5.4.4.3.2.3" xref="p7.5.m5.4.4.3.2.3.cmml">R</mi></msub><mo id="p7.5.m5.4.4.3.1" xref="p7.5.m5.4.4.3.1.cmml">⁢</mo><mrow id="p7.5.m5.4.4.3.3.2" xref="p7.5.m5.4.4.3.cmml"><mo id="p7.5.m5.4.4.3.3.2.1" stretchy="false" xref="p7.5.m5.4.4.3.cmml">(</mo><mi id="p7.5.m5.3.3" xref="p7.5.m5.3.3.cmml">T</mi><mo id="p7.5.m5.4.4.3.3.2.2" stretchy="false" xref="p7.5.m5.4.4.3.cmml">)</mo></mrow></mrow><mo id="p7.5.m5.4.4.2" xref="p7.5.m5.4.4.2.cmml">=</mo><mrow id="p7.5.m5.4.4.1" xref="p7.5.m5.4.4.1.cmml"><msub id="p7.5.m5.4.4.1.3" xref="p7.5.m5.4.4.1.3.cmml"><mtext id="p7.5.m5.4.4.1.3.2" xref="p7.5.m5.4.4.1.3.2a.cmml">Tr</mtext><mtext id="p7.5.m5.4.4.1.3.3" xref="p7.5.m5.4.4.1.3.3a.cmml">L</mtext></msub><mo id="p7.5.m5.4.4.1.2" xref="p7.5.m5.4.4.1.2.cmml">⁢</mo><mrow id="p7.5.m5.4.4.1.1.1" xref="p7.5.m5.4.4.1.1.2.cmml"><mo id="p7.5.m5.4.4.1.1.1.2" stretchy="false" xref="p7.5.m5.4.4.1.1.2.1.cmml">[</mo><mrow id="p7.5.m5.4.4.1.1.1.1" xref="p7.5.m5.4.4.1.1.1.1.cmml"><msup id="p7.5.m5.4.4.1.1.1.1.2" xref="p7.5.m5.4.4.1.1.1.1.2.cmml"><mi id="p7.5.m5.4.4.1.1.1.1.2.2" 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xref="p7.5.m5.4.4.1.1.1.1.3">superscript</csymbol><ci id="p7.5.m5.4.4.1.1.1.1.3.2.2.cmml" xref="p7.5.m5.4.4.1.1.1.1.3.2.2">𝕌</ci><ci id="p7.5.m5.4.4.1.1.1.1.3.2.3.cmml" xref="p7.5.m5.4.4.1.1.1.1.3.2.3">†</ci></apply><ci id="p7.5.m5.4.4.1.1.1.1.3.3.cmml" xref="p7.5.m5.4.4.1.1.1.1.3.3">𝑇</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p7.5.m5.4c">\rho_{R}(T)=\textnormal{Tr}_{\text{L}}[\mathbb{U}^{T}\ket{\Psi_{\text{in}}}% \bra{\Psi_{\text{in}}}{\mathbb{U}^{\dagger}}^{T}]</annotation><annotation encoding="application/x-llamapun" id="p7.5.m5.4d">italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_T ) = Tr start_POSTSUBSCRIPT L end_POSTSUBSCRIPT [ blackboard_U start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT | start_ARG roman_Ψ start_POSTSUBSCRIPT in end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG roman_Ψ start_POSTSUBSCRIPT in end_POSTSUBSCRIPT end_ARG | blackboard_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ]</annotation></semantics></math>. In Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(b), the partial trace operation after time evolution is carried out by attaching hollow dots to the outer legs in <math alttext="L" class="ltx_Math" display="inline" id="p7.6.m6.1"><semantics id="p7.6.m6.1a"><mi id="p7.6.m6.1.1" xref="p7.6.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p7.6.m6.1b"><ci id="p7.6.m6.1.1.cmml" xref="p7.6.m6.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.6.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="p7.6.m6.1d">italic_L</annotation></semantics></math>. Consequently, the evolution of the subsystem on <math alttext="R" class="ltx_Math" display="inline" id="p7.7.m7.1"><semantics id="p7.7.m7.1a"><mi id="p7.7.m7.1.1" xref="p7.7.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p7.7.m7.1b"><ci id="p7.7.m7.1.1.cmml" xref="p7.7.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p7.7.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="p7.7.m7.1d">italic_R</annotation></semantics></math> can be expressed in terms of the internal dynamics <math alttext="\mathbb{U}_{R}" class="ltx_Math" display="inline" id="p7.8.m8.1"><semantics id="p7.8.m8.1a"><msub id="p7.8.m8.1.1" xref="p7.8.m8.1.1.cmml"><mi id="p7.8.m8.1.1.2" xref="p7.8.m8.1.1.2.cmml">𝕌</mi><mi id="p7.8.m8.1.1.3" xref="p7.8.m8.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="p7.8.m8.1b"><apply id="p7.8.m8.1.1.cmml" xref="p7.8.m8.1.1"><csymbol cd="ambiguous" id="p7.8.m8.1.1.1.cmml" xref="p7.8.m8.1.1">subscript</csymbol><ci id="p7.8.m8.1.1.2.cmml" xref="p7.8.m8.1.1.2">𝕌</ci><ci id="p7.8.m8.1.1.3.cmml" xref="p7.8.m8.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p7.8.m8.1c">\mathbb{U}_{R}</annotation><annotation encoding="application/x-llamapun" id="p7.8.m8.1d">blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math>, together with the action of a multitime operator accounting for the temporal correlations in the left bath. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(b), this operator lies on the multitime Hilbert space, which is obtained by tensoring those local Hilbert spaces carried by the legs cut by the time slice. This operator is also referred to as the influence matrix (IM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib18" title="">18</a>]</cite> or the process tensor <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib25" title="">25</a>]</cite>.</p> </div> <div class="ltx_para" id="p8"> <p class="ltx_p" id="p8.1">We vectorize the IM to a quantum state in the doubled multitime Hilbert space <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib17" title="">17</a>]</cite>. Despite the class of dual unitary circuits and their generalizations that generate product-state IMs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib32" title="">32</a>]</cite>, examples of analytically tractable IMs are limited to some exactly solvable models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib35" title="">35</a>]</cite>. On the other hand, in generic quantum circuits, the long-time IM usually becomes complicated, characterized by the bipartite entanglement entropy growing linearly with respect to the evolution time <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib20" title="">20</a>]</cite>.</p> </div> <div class="ltx_para" id="p9"> <p class="ltx_p" id="p9.3">Here, we find a solvable condition of the IM for arbitrarily long time. Notice that here the solvable condition serves as a sufficient but not necessary condition for obtaining closed-form influence matrices. 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unitary gate acts on the tensor <math alttext="A" class="ltx_Math" display="inline" id="p9.1.m1.1"><semantics id="p9.1.m1.1a"><mi id="p9.1.m1.1.1" xref="p9.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="p9.1.m1.1b"><ci id="p9.1.m1.1.1.cmml" xref="p9.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="p9.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="p9.1.m1.1d">italic_A</annotation></semantics></math> as the <math alttext="{\mathrm{SWAP}}" class="ltx_Math" display="inline" id="p9.2.m2.1"><semantics id="p9.2.m2.1a"><mi id="p9.2.m2.1.1" mathsize="70%" xref="p9.2.m2.1.1.cmml">SWAP</mi><annotation-xml encoding="MathML-Content" id="p9.2.m2.1b"><ci id="p9.2.m2.1.1.cmml" xref="p9.2.m2.1.1">SWAP</ci></annotation-xml><annotation encoding="application/x-tex" id="p9.2.m2.1c">{\mathrm{SWAP}}</annotation><annotation encoding="application/x-llamapun" id="p9.2.m2.1d">roman_SWAP</annotation></semantics></math> gate in the spatial direction. Indeed, the solvable condition can be viewed as a refined version of zipper conditions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib36" title="">36</a>]</cite>. Previously, the zipper condition served as an ansatz to solve MPS influence matrices in integrable models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib35" title="">35</a>]</cite>. In contrast to those isolated examples, here we use the solvable condition as a criterion to construct generic nonintegrable quantum circuits with exact influence matrices <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib37" title="">37</a>]</cite>.</p> </div> <div class="ltx_para" id="p10"> <p class="ltx_p" id="p10.2">In Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(c), we directly present the exact form of the IM under the solvable condition (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite> for detailed derivations). An intuitive picture is that, by tracing out the left region, the initial MPS is rotated by <math alttext="\pi/2" class="ltx_Math" display="inline" id="p10.1.m1.1"><semantics id="p10.1.m1.1a"><mrow id="p10.1.m1.1.1" xref="p10.1.m1.1.1.cmml"><mi id="p10.1.m1.1.1.2" xref="p10.1.m1.1.1.2.cmml">π</mi><mo id="p10.1.m1.1.1.1" xref="p10.1.m1.1.1.1.cmml">/</mo><mn id="p10.1.m1.1.1.3" xref="p10.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="p10.1.m1.1b"><apply id="p10.1.m1.1.1.cmml" xref="p10.1.m1.1.1"><divide id="p10.1.m1.1.1.1.cmml" xref="p10.1.m1.1.1.1"></divide><ci id="p10.1.m1.1.1.2.cmml" xref="p10.1.m1.1.1.2">𝜋</ci><cn id="p10.1.m1.1.1.3.cmml" type="integer" xref="p10.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p10.1.m1.1c">\pi/2</annotation><annotation encoding="application/x-llamapun" id="p10.1.m1.1d">italic_π / 2</annotation></semantics></math> and lies along the time slice, showing a novel manifestation of spacetime duality. The IM is represented by a one-time-step shift-invariant MPS of bond dimension <math alttext="\chi^{2}" class="ltx_Math" display="inline" id="p10.2.m2.1"><semantics id="p10.2.m2.1a"><msup id="p10.2.m2.1.1" xref="p10.2.m2.1.1.cmml"><mi id="p10.2.m2.1.1.2" xref="p10.2.m2.1.1.2.cmml">χ</mi><mn id="p10.2.m2.1.1.3" xref="p10.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="p10.2.m2.1b"><apply id="p10.2.m2.1.1.cmml" xref="p10.2.m2.1.1"><csymbol cd="ambiguous" id="p10.2.m2.1.1.1.cmml" xref="p10.2.m2.1.1">superscript</csymbol><ci id="p10.2.m2.1.1.2.cmml" xref="p10.2.m2.1.1.2">𝜒</ci><cn id="p10.2.m2.1.1.3.cmml" type="integer" xref="p10.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p10.2.m2.1c">\chi^{2}</annotation><annotation encoding="application/x-llamapun" id="p10.2.m2.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with appropriate boundary conditions. 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represented explicitly in terms of <math alttext="A" class="ltx_Math" display="inline" id="p10.3.m1.1"><semantics id="p10.3.m1.1a"><mi id="p10.3.m1.1.1" xref="p10.3.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="p10.3.m1.1b"><ci id="p10.3.m1.1.1.cmml" xref="p10.3.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="p10.3.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="p10.3.m1.1d">italic_A</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite>, and replicates <math alttext="T" class="ltx_Math" display="inline" id="p10.4.m2.1"><semantics id="p10.4.m2.1a"><mi id="p10.4.m2.1.1" xref="p10.4.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="p10.4.m2.1b"><ci id="p10.4.m2.1.1.cmml" xref="p10.4.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="p10.4.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="p10.4.m2.1d">italic_T</annotation></semantics></math> times in the IM. At the lower boundary <math alttext="t=0" class="ltx_Math" display="inline" id="p10.5.m3.1"><semantics id="p10.5.m3.1a"><mrow id="p10.5.m3.1.1" xref="p10.5.m3.1.1.cmml"><mi id="p10.5.m3.1.1.2" xref="p10.5.m3.1.1.2.cmml">t</mi><mo id="p10.5.m3.1.1.1" xref="p10.5.m3.1.1.1.cmml">=</mo><mn id="p10.5.m3.1.1.3" xref="p10.5.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="p10.5.m3.1b"><apply id="p10.5.m3.1.1.cmml" xref="p10.5.m3.1.1"><eq id="p10.5.m3.1.1.1.cmml" xref="p10.5.m3.1.1.1"></eq><ci id="p10.5.m3.1.1.2.cmml" xref="p10.5.m3.1.1.2">𝑡</ci><cn id="p10.5.m3.1.1.3.cmml" type="integer" xref="p10.5.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p10.5.m3.1c">t=0</annotation><annotation encoding="application/x-llamapun" id="p10.5.m3.1d">italic_t = 0</annotation></semantics></math>, the auxiliary leg is connected to the initial state in the region <math alttext="R" class="ltx_Math" display="inline" id="p10.6.m4.1"><semantics id="p10.6.m4.1a"><mi id="p10.6.m4.1.1" xref="p10.6.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p10.6.m4.1b"><ci id="p10.6.m4.1.1.cmml" xref="p10.6.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p10.6.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="p10.6.m4.1d">italic_R</annotation></semantics></math>; at the upper boundary <math alttext="t=T" class="ltx_Math" display="inline" id="p10.7.m5.1"><semantics id="p10.7.m5.1a"><mrow id="p10.7.m5.1.1" xref="p10.7.m5.1.1.cmml"><mi id="p10.7.m5.1.1.2" xref="p10.7.m5.1.1.2.cmml">t</mi><mo id="p10.7.m5.1.1.1" xref="p10.7.m5.1.1.1.cmml">=</mo><mi id="p10.7.m5.1.1.3" xref="p10.7.m5.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="p10.7.m5.1b"><apply id="p10.7.m5.1.1.cmml" xref="p10.7.m5.1.1"><eq id="p10.7.m5.1.1.1.cmml" xref="p10.7.m5.1.1.1"></eq><ci id="p10.7.m5.1.1.2.cmml" xref="p10.7.m5.1.1.2">𝑡</ci><ci id="p10.7.m5.1.1.3.cmml" xref="p10.7.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p10.7.m5.1c">t=T</annotation><annotation encoding="application/x-llamapun" id="p10.7.m5.1d">italic_t = italic_T</annotation></semantics></math>, the auxiliary Hilbert space is traced out. Interestingly, the form of the IM indicates the nonsignaling property of the subsystem dynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib42" title="">42</a>]</cite>: the information flowing from the subsystem to the environment is completely discarded, as implied by the trace operator in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E5" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">5</span></a>), and can never flow back into the subsystem. However, the environment retains the memory of initial correlations between <math alttext="L" class="ltx_Math" display="inline" id="p10.8.m6.1"><semantics id="p10.8.m6.1a"><mi id="p10.8.m6.1.1" xref="p10.8.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="p10.8.m6.1b"><ci id="p10.8.m6.1.1.cmml" xref="p10.8.m6.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="p10.8.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="p10.8.m6.1d">italic_L</annotation></semantics></math> and <math alttext="R" class="ltx_Math" display="inline" id="p10.9.m7.1"><semantics id="p10.9.m7.1a"><mi id="p10.9.m7.1.1" xref="p10.9.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p10.9.m7.1b"><ci id="p10.9.m7.1.1.cmml" xref="p10.9.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p10.9.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="p10.9.m7.1d">italic_R</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="p11"> <p class="ltx_p" id="p11.6"><span class="ltx_text ltx_font_italic" id="p11.6.1">Exact hidden Markovian subsystem dynamics</span>—Next, we show that the exact IM gives rise to hidden Markovian dynamics of the subsystem. As implied by the MPS representation, the IM is correlated in the time direction (when <math alttext="\chi>1" class="ltx_Math" display="inline" id="p11.1.m1.1"><semantics id="p11.1.m1.1a"><mrow id="p11.1.m1.1.1" xref="p11.1.m1.1.1.cmml"><mi id="p11.1.m1.1.1.2" xref="p11.1.m1.1.1.2.cmml">χ</mi><mo id="p11.1.m1.1.1.1" xref="p11.1.m1.1.1.1.cmml">></mo><mn id="p11.1.m1.1.1.3" xref="p11.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="p11.1.m1.1b"><apply id="p11.1.m1.1.1.cmml" xref="p11.1.m1.1.1"><gt id="p11.1.m1.1.1.1.cmml" xref="p11.1.m1.1.1.1"></gt><ci id="p11.1.m1.1.1.2.cmml" xref="p11.1.m1.1.1.2">𝜒</ci><cn id="p11.1.m1.1.1.3.cmml" type="integer" xref="p11.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.1.m1.1c">\chi>1</annotation><annotation encoding="application/x-llamapun" id="p11.1.m1.1d">italic_χ > 1</annotation></semantics></math>), and thus the Markovian property is lacking <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib44" title="">44</a>]</cite>. Nevertheless, we can incorporate an ancilla of <span class="ltx_text ltx_font_italic" id="p11.6.2">finite dimension</span> into the subsystem, which renders the joint dynamics to be governed by sequential quantum channels. The ancilla lives in the auxiliary Hilbert space <math alttext="\mathcal{H}_{\chi}" class="ltx_Math" display="inline" id="p11.2.m2.1"><semantics id="p11.2.m2.1a"><msub id="p11.2.m2.1.1" xref="p11.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p11.2.m2.1.1.2" xref="p11.2.m2.1.1.2.cmml">ℋ</mi><mi id="p11.2.m2.1.1.3" xref="p11.2.m2.1.1.3.cmml">χ</mi></msub><annotation-xml encoding="MathML-Content" id="p11.2.m2.1b"><apply id="p11.2.m2.1.1.cmml" xref="p11.2.m2.1.1"><csymbol cd="ambiguous" id="p11.2.m2.1.1.1.cmml" xref="p11.2.m2.1.1">subscript</csymbol><ci id="p11.2.m2.1.1.2.cmml" xref="p11.2.m2.1.1.2">ℋ</ci><ci id="p11.2.m2.1.1.3.cmml" xref="p11.2.m2.1.1.3">𝜒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.2.m2.1c">\mathcal{H}_{\chi}</annotation><annotation encoding="application/x-llamapun" id="p11.2.m2.1d">caligraphic_H start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT</annotation></semantics></math>. We term such dynamics as <span class="ltx_text ltx_font_italic" id="p11.6.3">hidden Markovian</span>, aligning with its classical counterpart <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib47" title="">47</a>]</cite>. At <math alttext="t=0" class="ltx_Math" display="inline" id="p11.3.m3.1"><semantics id="p11.3.m3.1a"><mrow id="p11.3.m3.1.1" xref="p11.3.m3.1.1.cmml"><mi id="p11.3.m3.1.1.2" xref="p11.3.m3.1.1.2.cmml">t</mi><mo id="p11.3.m3.1.1.1" xref="p11.3.m3.1.1.1.cmml">=</mo><mn id="p11.3.m3.1.1.3" xref="p11.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="p11.3.m3.1b"><apply id="p11.3.m3.1.1.cmml" xref="p11.3.m3.1.1"><eq id="p11.3.m3.1.1.1.cmml" xref="p11.3.m3.1.1.1"></eq><ci id="p11.3.m3.1.1.2.cmml" xref="p11.3.m3.1.1.2">𝑡</ci><cn id="p11.3.m3.1.1.3.cmml" type="integer" xref="p11.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.3.m3.1c">t=0</annotation><annotation encoding="application/x-llamapun" id="p11.3.m3.1d">italic_t = 0</annotation></semantics></math>, we prepare the joint system in the pure state as <math alttext="\ket{\tilde{\Psi}_{R}}=\sum_{j=0}^{\chi-1}|j)\bigotimes\ket{\Psi_{R}^{j}}" class="ltx_math_unparsed" display="inline" id="p11.4.m4.2"><semantics id="p11.4.m4.2a"><mrow id="p11.4.m4.2b"><mrow id="p11.4.m4.1.1.3"><mo id="p11.4.m4.1.1.3.1" stretchy="false">|</mo><msub id="p11.4.m4.1.1.1.1"><mover accent="true" id="p11.4.m4.1.1.1.1.2"><mi id="p11.4.m4.1.1.1.1.2.2" mathvariant="normal">Ψ</mi><mo id="p11.4.m4.1.1.1.1.2.1">~</mo></mover><mi id="p11.4.m4.1.1.1.1.3">R</mi></msub><mo id="p11.4.m4.1.1.3.2" stretchy="false">⟩</mo></mrow><mo id="p11.4.m4.2.3" rspace="0.111em">=</mo><msubsup id="p11.4.m4.2.4"><mo id="p11.4.m4.2.4.2.2" rspace="0em">∑</mo><mrow id="p11.4.m4.2.4.2.3"><mi id="p11.4.m4.2.4.2.3.2">j</mi><mo id="p11.4.m4.2.4.2.3.1">=</mo><mn id="p11.4.m4.2.4.2.3.3">0</mn></mrow><mrow id="p11.4.m4.2.4.3"><mi id="p11.4.m4.2.4.3.2">χ</mi><mo id="p11.4.m4.2.4.3.1">−</mo><mn id="p11.4.m4.2.4.3.3">1</mn></mrow></msubsup><mo fence="false" id="p11.4.m4.2.5" rspace="0.167em" stretchy="false">|</mo><mi id="p11.4.m4.2.6">j</mi><mo id="p11.4.m4.2.7" stretchy="false">)</mo><mo id="p11.4.m4.2.8" rspace="0em">⨂</mo><mrow id="p11.4.m4.2.2.3"><mo id="p11.4.m4.2.2.3.1" stretchy="false">|</mo><msubsup id="p11.4.m4.2.2.1.1"><mi id="p11.4.m4.2.2.1.1.2.2" mathvariant="normal">Ψ</mi><mi id="p11.4.m4.2.2.1.1.2.3">R</mi><mi id="p11.4.m4.2.2.1.1.3">j</mi></msubsup><mo id="p11.4.m4.2.2.3.2" stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex" id="p11.4.m4.2c">\ket{\tilde{\Psi}_{R}}=\sum_{j=0}^{\chi-1}|j)\bigotimes\ket{\Psi_{R}^{j}}</annotation><annotation encoding="application/x-llamapun" id="p11.4.m4.2d">| start_ARG over~ start_ARG roman_Ψ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_χ - 1 end_POSTSUPERSCRIPT | italic_j ) ⨂ | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math>, which gives the same subsystem reduced density matrix when tracing out the ancilla. Here, <math alttext="\bigotimes" class="ltx_Math" display="inline" id="p11.5.m5.1"><semantics id="p11.5.m5.1a"><mo id="p11.5.m5.1.1" xref="p11.5.m5.1.1.cmml">⨂</mo><annotation-xml encoding="MathML-Content" id="p11.5.m5.1b"><csymbol cd="latexml" id="p11.5.m5.1.1.cmml" xref="p11.5.m5.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="p11.5.m5.1c">\bigotimes</annotation><annotation encoding="application/x-llamapun" id="p11.5.m5.1d">⨂</annotation></semantics></math> denotes tensor products between the auxiliary and physical Hilbert space. Hence, the initial correlations between two regions are captured by this ancilla. As illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F1" title="Figure 1 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">1</span></a>(d), for each time step, the joint system is evolved by a layer of local gates, followed by a two-site quantum channel <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="p11.6.m6.1"><semantics id="p11.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="p11.6.m6.1.1" xref="p11.6.m6.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="p11.6.m6.1b"><ci id="p11.6.m6.1.1.cmml" xref="p11.6.m6.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="p11.6.m6.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="p11.6.m6.1d">caligraphic_M</annotation></semantics></math> acting on the ancilla and the leftmost site,</p> <table class="ltx_equation ltx_eqn_table" id="S0.E6"> <tbody><tr class="ltx_equation ltx_eqn_row 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xref="S0.E6.m1.2.2.1.1.2.1.1.1.2.3">𝑅</ci></apply><apply id="S0.E6.m1.2.2.1.1.2.1.1.1.3.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="S0.E6.m1.2.2.1.1.2.1.1.1.3.1.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3">subscript</csymbol><apply id="S0.E6.m1.2.2.1.1.2.1.1.1.3.2.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3.2"><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.3.2.1.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3.2.1">~</ci><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.3.2.2.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3.2.2">𝜌</ci></apply><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.3.3.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.3.3">𝑅</ci></apply><ci id="S0.E6.m1.1.1.cmml" xref="S0.E6.m1.1.1">𝑡</ci><apply id="S0.E6.m1.2.2.1.1.2.1.1.1.5.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5"><csymbol cd="ambiguous" id="S0.E6.m1.2.2.1.1.2.1.1.1.5.1.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5">superscript</csymbol><apply id="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5"><csymbol cd="ambiguous" id="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.1.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5">subscript</csymbol><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.2.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.2">𝕌</ci><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.3.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5.2.3">𝑅</ci></apply><ci id="S0.E6.m1.2.2.1.1.2.1.1.1.5.3.cmml" xref="S0.E6.m1.2.2.1.1.2.1.1.1.5.3">†</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.E6.m1.2c">\tilde{\rho}_{R}(t+1)=\mathcal{M}[\mathbb{U}_{R}\tilde{\rho}_{R}(t)\mathbb{U}_% {R}^{\dagger}],</annotation><annotation encoding="application/x-llamapun" id="S0.E6.m1.2d">over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t + 1 ) = caligraphic_M [ blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t ) blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p11.13">where <math alttext="\tilde{\rho}_{R}(t)" class="ltx_Math" display="inline" id="p11.7.m1.1"><semantics id="p11.7.m1.1a"><mrow id="p11.7.m1.1.2" xref="p11.7.m1.1.2.cmml"><msub id="p11.7.m1.1.2.2" xref="p11.7.m1.1.2.2.cmml"><mover accent="true" id="p11.7.m1.1.2.2.2" xref="p11.7.m1.1.2.2.2.cmml"><mi id="p11.7.m1.1.2.2.2.2" xref="p11.7.m1.1.2.2.2.2.cmml">ρ</mi><mo id="p11.7.m1.1.2.2.2.1" xref="p11.7.m1.1.2.2.2.1.cmml">~</mo></mover><mi id="p11.7.m1.1.2.2.3" xref="p11.7.m1.1.2.2.3.cmml">R</mi></msub><mo id="p11.7.m1.1.2.1" xref="p11.7.m1.1.2.1.cmml">⁢</mo><mrow id="p11.7.m1.1.2.3.2" xref="p11.7.m1.1.2.cmml"><mo id="p11.7.m1.1.2.3.2.1" stretchy="false" xref="p11.7.m1.1.2.cmml">(</mo><mi id="p11.7.m1.1.1" xref="p11.7.m1.1.1.cmml">t</mi><mo id="p11.7.m1.1.2.3.2.2" stretchy="false" xref="p11.7.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p11.7.m1.1b"><apply id="p11.7.m1.1.2.cmml" xref="p11.7.m1.1.2"><times id="p11.7.m1.1.2.1.cmml" xref="p11.7.m1.1.2.1"></times><apply id="p11.7.m1.1.2.2.cmml" xref="p11.7.m1.1.2.2"><csymbol cd="ambiguous" id="p11.7.m1.1.2.2.1.cmml" xref="p11.7.m1.1.2.2">subscript</csymbol><apply id="p11.7.m1.1.2.2.2.cmml" xref="p11.7.m1.1.2.2.2"><ci id="p11.7.m1.1.2.2.2.1.cmml" xref="p11.7.m1.1.2.2.2.1">~</ci><ci id="p11.7.m1.1.2.2.2.2.cmml" xref="p11.7.m1.1.2.2.2.2">𝜌</ci></apply><ci id="p11.7.m1.1.2.2.3.cmml" xref="p11.7.m1.1.2.2.3">𝑅</ci></apply><ci id="p11.7.m1.1.1.cmml" xref="p11.7.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.7.m1.1c">\tilde{\rho}_{R}(t)</annotation><annotation encoding="application/x-llamapun" id="p11.7.m1.1d">over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is the joint system density matrix at time <math alttext="t" class="ltx_Math" display="inline" id="p11.8.m2.1"><semantics id="p11.8.m2.1a"><mi id="p11.8.m2.1.1" xref="p11.8.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="p11.8.m2.1b"><ci id="p11.8.m2.1.1.cmml" xref="p11.8.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="p11.8.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="p11.8.m2.1d">italic_t</annotation></semantics></math>. The dynamics of <math alttext="\tilde{\rho}_{R}(t)" class="ltx_Math" display="inline" id="p11.9.m3.1"><semantics id="p11.9.m3.1a"><mrow id="p11.9.m3.1.2" xref="p11.9.m3.1.2.cmml"><msub id="p11.9.m3.1.2.2" xref="p11.9.m3.1.2.2.cmml"><mover accent="true" id="p11.9.m3.1.2.2.2" xref="p11.9.m3.1.2.2.2.cmml"><mi id="p11.9.m3.1.2.2.2.2" xref="p11.9.m3.1.2.2.2.2.cmml">ρ</mi><mo id="p11.9.m3.1.2.2.2.1" xref="p11.9.m3.1.2.2.2.1.cmml">~</mo></mover><mi id="p11.9.m3.1.2.2.3" xref="p11.9.m3.1.2.2.3.cmml">R</mi></msub><mo id="p11.9.m3.1.2.1" xref="p11.9.m3.1.2.1.cmml">⁢</mo><mrow id="p11.9.m3.1.2.3.2" xref="p11.9.m3.1.2.cmml"><mo id="p11.9.m3.1.2.3.2.1" stretchy="false" xref="p11.9.m3.1.2.cmml">(</mo><mi id="p11.9.m3.1.1" xref="p11.9.m3.1.1.cmml">t</mi><mo id="p11.9.m3.1.2.3.2.2" stretchy="false" xref="p11.9.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p11.9.m3.1b"><apply id="p11.9.m3.1.2.cmml" xref="p11.9.m3.1.2"><times id="p11.9.m3.1.2.1.cmml" xref="p11.9.m3.1.2.1"></times><apply id="p11.9.m3.1.2.2.cmml" xref="p11.9.m3.1.2.2"><csymbol cd="ambiguous" id="p11.9.m3.1.2.2.1.cmml" xref="p11.9.m3.1.2.2">subscript</csymbol><apply id="p11.9.m3.1.2.2.2.cmml" xref="p11.9.m3.1.2.2.2"><ci id="p11.9.m3.1.2.2.2.1.cmml" xref="p11.9.m3.1.2.2.2.1">~</ci><ci id="p11.9.m3.1.2.2.2.2.cmml" xref="p11.9.m3.1.2.2.2.2">𝜌</ci></apply><ci id="p11.9.m3.1.2.2.3.cmml" xref="p11.9.m3.1.2.2.3">𝑅</ci></apply><ci id="p11.9.m3.1.1.cmml" xref="p11.9.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.9.m3.1c">\tilde{\rho}_{R}(t)</annotation><annotation encoding="application/x-llamapun" id="p11.9.m3.1d">over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is explicitly Markovian <span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>This step can be traced back to the method of Markovian embedding, by incorporating additional degrees of freedom into the non-Markovian system to create the Markovian joint dynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib76" title="">76</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib77" title="">77</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib78" title="">78</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib79" title="">79</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib80" title="">80</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib81" title="">81</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib82" title="">82</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib83" title="">83</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib84" title="">84</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib85" title="">85</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib86" title="">86</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib87" title="">87</a>]</cite>.</span></span></span>, where the state at the (<math alttext="t+1" class="ltx_Math" display="inline" id="p11.10.m4.1"><semantics id="p11.10.m4.1a"><mrow id="p11.10.m4.1.1" xref="p11.10.m4.1.1.cmml"><mi id="p11.10.m4.1.1.2" xref="p11.10.m4.1.1.2.cmml">t</mi><mo id="p11.10.m4.1.1.1" xref="p11.10.m4.1.1.1.cmml">+</mo><mn id="p11.10.m4.1.1.3" xref="p11.10.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="p11.10.m4.1b"><apply id="p11.10.m4.1.1.cmml" xref="p11.10.m4.1.1"><plus id="p11.10.m4.1.1.1.cmml" xref="p11.10.m4.1.1.1"></plus><ci id="p11.10.m4.1.1.2.cmml" xref="p11.10.m4.1.1.2">𝑡</ci><cn id="p11.10.m4.1.1.3.cmml" type="integer" xref="p11.10.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.10.m4.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="p11.10.m4.1d">italic_t + 1</annotation></semantics></math>)th time step only depends on the state at <math alttext="t" class="ltx_Math" display="inline" id="p11.11.m5.1"><semantics id="p11.11.m5.1a"><mi id="p11.11.m5.1.1" xref="p11.11.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="p11.11.m5.1b"><ci id="p11.11.m5.1.1.cmml" xref="p11.11.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="p11.11.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="p11.11.m5.1d">italic_t</annotation></semantics></math>. The quantum channel <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="p11.12.m6.1"><semantics id="p11.12.m6.1a"><mi class="ltx_font_mathcaligraphic" id="p11.12.m6.1.1" xref="p11.12.m6.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="p11.12.m6.1b"><ci id="p11.12.m6.1.1.cmml" xref="p11.12.m6.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="p11.12.m6.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="p11.12.m6.1d">caligraphic_M</annotation></semantics></math> is given by the four-leg tensor defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E5" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">5</span></a>), which can be written in the standard Kraus form <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite>, <math alttext="\mathcal{M}(\tilde{\rho}_{R})=\sum_{\mu}K_{\mu}\tilde{\rho}_{R}K_{\mu}^{\dagger}" class="ltx_Math" display="inline" id="p11.13.m7.1"><semantics id="p11.13.m7.1a"><mrow id="p11.13.m7.1.1" xref="p11.13.m7.1.1.cmml"><mrow id="p11.13.m7.1.1.1" xref="p11.13.m7.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p11.13.m7.1.1.1.3" xref="p11.13.m7.1.1.1.3.cmml">ℳ</mi><mo id="p11.13.m7.1.1.1.2" xref="p11.13.m7.1.1.1.2.cmml">⁢</mo><mrow id="p11.13.m7.1.1.1.1.1" xref="p11.13.m7.1.1.1.1.1.1.cmml"><mo id="p11.13.m7.1.1.1.1.1.2" 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id="p11.13.m7.1c">\mathcal{M}(\tilde{\rho}_{R})=\sum_{\mu}K_{\mu}\tilde{\rho}_{R}K_{\mu}^{\dagger}</annotation><annotation encoding="application/x-llamapun" id="p11.13.m7.1d">caligraphic_M ( over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT</annotation></semantics></math>, where</p> <table class="ltx_equation ltx_eqn_table" id="S0.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="K_{\mu}=K_{a,a^{\prime}}=\sum_{b=0}^{q-1}A^{(b)}A^{(a)}\bigotimes(\ket{b}\bra{% a^{\prime}})_{x=0}." class="ltx_Math" display="block" id="S0.E7.m1.7"><semantics 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p11.19">Here <math alttext="a,a^{\prime}" class="ltx_Math" display="inline" id="p11.14.m1.2"><semantics id="p11.14.m1.2a"><mrow id="p11.14.m1.2.2.1" xref="p11.14.m1.2.2.2.cmml"><mi id="p11.14.m1.1.1" xref="p11.14.m1.1.1.cmml">a</mi><mo id="p11.14.m1.2.2.1.2" xref="p11.14.m1.2.2.2.cmml">,</mo><msup id="p11.14.m1.2.2.1.1" xref="p11.14.m1.2.2.1.1.cmml"><mi id="p11.14.m1.2.2.1.1.2" xref="p11.14.m1.2.2.1.1.2.cmml">a</mi><mo id="p11.14.m1.2.2.1.1.3" xref="p11.14.m1.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="p11.14.m1.2b"><list id="p11.14.m1.2.2.2.cmml" xref="p11.14.m1.2.2.1"><ci id="p11.14.m1.1.1.cmml" xref="p11.14.m1.1.1">𝑎</ci><apply id="p11.14.m1.2.2.1.1.cmml" xref="p11.14.m1.2.2.1.1"><csymbol cd="ambiguous" id="p11.14.m1.2.2.1.1.1.cmml" xref="p11.14.m1.2.2.1.1">superscript</csymbol><ci id="p11.14.m1.2.2.1.1.2.cmml" xref="p11.14.m1.2.2.1.1.2">𝑎</ci><ci id="p11.14.m1.2.2.1.1.3.cmml" xref="p11.14.m1.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="p11.14.m1.2c">a,a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="p11.14.m1.2d">italic_a , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> run over <math alttext="0" class="ltx_Math" display="inline" id="p11.15.m2.1"><semantics id="p11.15.m2.1a"><mn id="p11.15.m2.1.1" xref="p11.15.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="p11.15.m2.1b"><cn id="p11.15.m2.1.1.cmml" type="integer" xref="p11.15.m2.1.1">0</cn></annotation-xml></semantics></math> to <math alttext="q-1" class="ltx_Math" display="inline" id="p11.16.m3.1"><semantics id="p11.16.m3.1a"><mrow id="p11.16.m3.1.1" xref="p11.16.m3.1.1.cmml"><mi id="p11.16.m3.1.1.2" xref="p11.16.m3.1.1.2.cmml">q</mi><mo id="p11.16.m3.1.1.1" xref="p11.16.m3.1.1.1.cmml">−</mo><mn id="p11.16.m3.1.1.3" xref="p11.16.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="p11.16.m3.1b"><apply id="p11.16.m3.1.1.cmml" xref="p11.16.m3.1.1"><minus id="p11.16.m3.1.1.1.cmml" xref="p11.16.m3.1.1.1"></minus><ci id="p11.16.m3.1.1.2.cmml" xref="p11.16.m3.1.1.2">𝑞</ci><cn id="p11.16.m3.1.1.3.cmml" type="integer" xref="p11.16.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p11.16.m3.1c">q-1</annotation><annotation encoding="application/x-llamapun" id="p11.16.m3.1d">italic_q - 1</annotation></semantics></math>. Provided the Kraus-form representation, the quantum channel <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="p11.17.m4.1"><semantics id="p11.17.m4.1a"><mi class="ltx_font_mathcaligraphic" id="p11.17.m4.1.1" xref="p11.17.m4.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="p11.17.m4.1b"><ci id="p11.17.m4.1.1.cmml" xref="p11.17.m4.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="p11.17.m4.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="p11.17.m4.1d">caligraphic_M</annotation></semantics></math> is completely positive and trace-preserving <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib50" title="">50</a>]</cite>. 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]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="p12"> <p class="ltx_p" id="p12.10">A few remarks are in order. First, we can prepare the left initial state in a matrix product density operator, instead of a MPS pure state. All the analysis works as well, except that the expression of Kraus form should be slightly modified <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite>. Second, the left initial states can be extended to a certain class of two-site shift-invariant MPS, while keeping the solvability of the influence matrix <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite>. Third, due to the chiral structure in the solvable condition, it only allows efficient contractions of tensor networks from left to right, but not vice versa. 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Quantum circuits equipped with two solvable conditions of opposite chirality facilitate analytical descriptions for more physical quantities, e.g., R<math alttext="\acute{\text{e}}" class="ltx_Math" display="inline" id="p12.1.m1.1"><semantics id="p12.1.m1.1a"><mover accent="true" id="p12.1.m1.1.1" xref="p12.1.m1.1.1.cmml"><mtext id="p12.1.m1.1.1.2" xref="p12.1.m1.1.1.2a.cmml">e</mtext><mo id="p12.1.m1.1.1.1" xref="p12.1.m1.1.1.1.cmml">´</mo></mover><annotation-xml encoding="MathML-Content" id="p12.1.m1.1b"><apply id="p12.1.m1.1.1.cmml" xref="p12.1.m1.1.1"><ci id="p12.1.m1.1.1.1.cmml" xref="p12.1.m1.1.1.1">´</ci><ci id="p12.1.m1.1.1.2a.cmml" xref="p12.1.m1.1.1.2"><mtext id="p12.1.m1.1.1.2.cmml" xref="p12.1.m1.1.1.2">e</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p12.1.m1.1c">\acute{\text{e}}</annotation><annotation encoding="application/x-llamapun" id="p12.1.m1.1d">over´ start_ARG e end_ARG</annotation></semantics></math>nyi entropy dynamics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib51" title="">51</a>]</cite>. 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start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib52" title="">52</a>]</cite> – the asymptotic growing rate of the <math alttext="n" class="ltx_Math" display="inline" id="p12.4.m4.1"><semantics id="p12.4.m4.1a"><mi id="p12.4.m4.1.1" xref="p12.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="p12.4.m4.1b"><ci id="p12.4.m4.1.1.cmml" xref="p12.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="p12.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="p12.4.m4.1d">italic_n</annotation></semantics></math>th R<math alttext="\acute{\text{e}}" class="ltx_Math" display="inline" id="p12.5.m5.1"><semantics id="p12.5.m5.1a"><mover accent="true" id="p12.5.m5.1.1" xref="p12.5.m5.1.1.cmml"><mtext id="p12.5.m5.1.1.2" xref="p12.5.m5.1.1.2a.cmml">e</mtext><mo id="p12.5.m5.1.1.1" 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xref="S0.E9.m1.11.11.4.2">𝑞</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.E9.m1.13c">v_{E}^{(n)}\equiv\lim_{t\to\infty}\frac{\ln(\textnormal{Tr}[\rho_{R}^{n}(t)])/% (1-n)}{t\ln(q)}=\frac{2\ln(\lambda_{n})}{(1-n)\ln(q)}.</annotation><annotation encoding="application/x-llamapun" id="S0.E9.m1.13d">italic_v start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ≡ roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT divide start_ARG roman_ln ( Tr [ italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_t ) ] ) / ( 1 - italic_n ) end_ARG start_ARG italic_t roman_ln ( italic_q ) end_ARG = divide start_ARG 2 roman_ln ( italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_ARG start_ARG ( 1 - italic_n ) roman_ln ( italic_q ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p12.9">The entanglement velocity explicitly depends on the R<math alttext="\acute{\text{e}}" class="ltx_Math" display="inline" id="p12.8.m1.1"><semantics id="p12.8.m1.1a"><mover accent="true" id="p12.8.m1.1.1" xref="p12.8.m1.1.1.cmml"><mtext id="p12.8.m1.1.1.2" xref="p12.8.m1.1.1.2a.cmml">e</mtext><mo id="p12.8.m1.1.1.1" xref="p12.8.m1.1.1.1.cmml">´</mo></mover><annotation-xml encoding="MathML-Content" id="p12.8.m1.1b"><apply id="p12.8.m1.1.1.cmml" xref="p12.8.m1.1.1"><ci id="p12.8.m1.1.1.1.cmml" xref="p12.8.m1.1.1.1">´</ci><ci id="p12.8.m1.1.1.2a.cmml" xref="p12.8.m1.1.1.2"><mtext id="p12.8.m1.1.1.2.cmml" xref="p12.8.m1.1.1.2">e</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p12.8.m1.1c">\acute{\text{e}}</annotation><annotation encoding="application/x-llamapun" id="p12.8.m1.1d">over´ start_ARG e end_ARG</annotation></semantics></math>nyi index <math alttext="n" class="ltx_Math" display="inline" id="p12.9.m2.1"><semantics id="p12.9.m2.1a"><mi id="p12.9.m2.1.1" xref="p12.9.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="p12.9.m2.1b"><ci id="p12.9.m2.1.1.cmml" xref="p12.9.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="p12.9.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="p12.9.m2.1d">italic_n</annotation></semantics></math>, which is clearly different from dual-unitary gates with solvable initial state <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib27" title="">27</a>]</cite>.</p> </div> <div class="ltx_para" id="p13"> <p class="ltx_p" id="p13.7">Next, we demonstrate that solutions (up to one-site unitaries) of Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E4" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">4</span></a>) actually depend only on two characteristic values, <math alttext="q" class="ltx_Math" display="inline" id="p13.1.m1.1"><semantics id="p13.1.m1.1a"><mi id="p13.1.m1.1.1" xref="p13.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="p13.1.m1.1b"><ci id="p13.1.m1.1.1.cmml" xref="p13.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="p13.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="p13.1.m1.1d">italic_q</annotation></semantics></math> and <math alttext="\tilde{q}" class="ltx_Math" display="inline" id="p13.2.m2.1"><semantics id="p13.2.m2.1a"><mover accent="true" id="p13.2.m2.1.1" xref="p13.2.m2.1.1.cmml"><mi id="p13.2.m2.1.1.2" xref="p13.2.m2.1.1.2.cmml">q</mi><mo id="p13.2.m2.1.1.1" xref="p13.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="p13.2.m2.1b"><apply id="p13.2.m2.1.1.cmml" xref="p13.2.m2.1.1"><ci id="p13.2.m2.1.1.1.cmml" xref="p13.2.m2.1.1.1">~</ci><ci id="p13.2.m2.1.1.2.cmml" xref="p13.2.m2.1.1.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.2.m2.1c">\tilde{q}</annotation><annotation encoding="application/x-llamapun" id="p13.2.m2.1d">over~ start_ARG italic_q end_ARG</annotation></semantics></math>, of the local tensor <math alttext="A^{(a)}_{jk}" class="ltx_Math" display="inline" id="p13.3.m3.1"><semantics id="p13.3.m3.1a"><msubsup id="p13.3.m3.1.2" xref="p13.3.m3.1.2.cmml"><mi id="p13.3.m3.1.2.2.2" xref="p13.3.m3.1.2.2.2.cmml">A</mi><mrow id="p13.3.m3.1.2.3" xref="p13.3.m3.1.2.3.cmml"><mi id="p13.3.m3.1.2.3.2" xref="p13.3.m3.1.2.3.2.cmml">j</mi><mo id="p13.3.m3.1.2.3.1" xref="p13.3.m3.1.2.3.1.cmml">⁢</mo><mi id="p13.3.m3.1.2.3.3" xref="p13.3.m3.1.2.3.3.cmml">k</mi></mrow><mrow id="p13.3.m3.1.1.1.3" xref="p13.3.m3.1.2.cmml"><mo id="p13.3.m3.1.1.1.3.1" stretchy="false" xref="p13.3.m3.1.2.cmml">(</mo><mi id="p13.3.m3.1.1.1.1" xref="p13.3.m3.1.1.1.1.cmml">a</mi><mo id="p13.3.m3.1.1.1.3.2" stretchy="false" xref="p13.3.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="p13.3.m3.1b"><apply id="p13.3.m3.1.2.cmml" xref="p13.3.m3.1.2"><csymbol cd="ambiguous" id="p13.3.m3.1.2.1.cmml" xref="p13.3.m3.1.2">subscript</csymbol><apply id="p13.3.m3.1.2.2.cmml" xref="p13.3.m3.1.2"><csymbol cd="ambiguous" id="p13.3.m3.1.2.2.1.cmml" xref="p13.3.m3.1.2">superscript</csymbol><ci id="p13.3.m3.1.2.2.2.cmml" xref="p13.3.m3.1.2.2.2">𝐴</ci><ci id="p13.3.m3.1.1.1.1.cmml" xref="p13.3.m3.1.1.1.1">𝑎</ci></apply><apply id="p13.3.m3.1.2.3.cmml" xref="p13.3.m3.1.2.3"><times id="p13.3.m3.1.2.3.1.cmml" xref="p13.3.m3.1.2.3.1"></times><ci id="p13.3.m3.1.2.3.2.cmml" xref="p13.3.m3.1.2.3.2">𝑗</ci><ci id="p13.3.m3.1.2.3.3.cmml" xref="p13.3.m3.1.2.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.3.m3.1c">A^{(a)}_{jk}</annotation><annotation encoding="application/x-llamapun" id="p13.3.m3.1d">italic_A start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, rather than on exact values of all the tensor elements. Here, <math alttext="q" class="ltx_Math" display="inline" id="p13.4.m4.1"><semantics id="p13.4.m4.1a"><mi id="p13.4.m4.1.1" xref="p13.4.m4.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="p13.4.m4.1b"><ci id="p13.4.m4.1.1.cmml" xref="p13.4.m4.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="p13.4.m4.1c">q</annotation><annotation encoding="application/x-llamapun" id="p13.4.m4.1d">italic_q</annotation></semantics></math> is the local Hilbert space dimension, and <math alttext="\tilde{q}" class="ltx_Math" display="inline" id="p13.5.m5.1"><semantics id="p13.5.m5.1a"><mover accent="true" id="p13.5.m5.1.1" xref="p13.5.m5.1.1.cmml"><mi id="p13.5.m5.1.1.2" xref="p13.5.m5.1.1.2.cmml">q</mi><mo id="p13.5.m5.1.1.1" xref="p13.5.m5.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="p13.5.m5.1b"><apply id="p13.5.m5.1.1.cmml" xref="p13.5.m5.1.1"><ci id="p13.5.m5.1.1.1.cmml" xref="p13.5.m5.1.1.1">~</ci><ci id="p13.5.m5.1.1.2.cmml" xref="p13.5.m5.1.1.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.5.m5.1c">\tilde{q}</annotation><annotation encoding="application/x-llamapun" id="p13.5.m5.1d">over~ start_ARG italic_q end_ARG</annotation></semantics></math> is the one-site MPS dimension which will be clarified later. For convenience, we define the reshuffled operator <math alttext="U^{R}" class="ltx_Math" display="inline" id="p13.6.m6.1"><semantics id="p13.6.m6.1a"><msup id="p13.6.m6.1.1" xref="p13.6.m6.1.1.cmml"><mi id="p13.6.m6.1.1.2" xref="p13.6.m6.1.1.2.cmml">U</mi><mi id="p13.6.m6.1.1.3" xref="p13.6.m6.1.1.3.cmml">R</mi></msup><annotation-xml encoding="MathML-Content" id="p13.6.m6.1b"><apply id="p13.6.m6.1.1.cmml" xref="p13.6.m6.1.1"><csymbol cd="ambiguous" id="p13.6.m6.1.1.1.cmml" xref="p13.6.m6.1.1">superscript</csymbol><ci id="p13.6.m6.1.1.2.cmml" xref="p13.6.m6.1.1.2">𝑈</ci><ci id="p13.6.m6.1.1.3.cmml" xref="p13.6.m6.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.6.m6.1c">U^{R}</annotation><annotation encoding="application/x-llamapun" id="p13.6.m6.1d">italic_U start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT</annotation></semantics></math> as <math alttext="(U^{R})_{ab}^{cd}=U_{ac}^{bd}" class="ltx_Math" display="inline" id="p13.7.m7.1"><semantics 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xref="p13.7.m7.1.1.1.3.2.cmml">c</mi><mo id="p13.7.m7.1.1.1.3.1" xref="p13.7.m7.1.1.1.3.1.cmml">⁢</mo><mi id="p13.7.m7.1.1.1.3.3" xref="p13.7.m7.1.1.1.3.3.cmml">d</mi></mrow></msubsup><mo id="p13.7.m7.1.1.2" xref="p13.7.m7.1.1.2.cmml">=</mo><msubsup id="p13.7.m7.1.1.3" xref="p13.7.m7.1.1.3.cmml"><mi id="p13.7.m7.1.1.3.2.2" xref="p13.7.m7.1.1.3.2.2.cmml">U</mi><mrow id="p13.7.m7.1.1.3.2.3" xref="p13.7.m7.1.1.3.2.3.cmml"><mi id="p13.7.m7.1.1.3.2.3.2" xref="p13.7.m7.1.1.3.2.3.2.cmml">a</mi><mo id="p13.7.m7.1.1.3.2.3.1" xref="p13.7.m7.1.1.3.2.3.1.cmml">⁢</mo><mi id="p13.7.m7.1.1.3.2.3.3" xref="p13.7.m7.1.1.3.2.3.3.cmml">c</mi></mrow><mrow id="p13.7.m7.1.1.3.3" xref="p13.7.m7.1.1.3.3.cmml"><mi id="p13.7.m7.1.1.3.3.2" xref="p13.7.m7.1.1.3.3.2.cmml">b</mi><mo id="p13.7.m7.1.1.3.3.1" xref="p13.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="p13.7.m7.1.1.3.3.3" xref="p13.7.m7.1.1.3.3.3.cmml">d</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="p13.7.m7.1b"><apply 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xref="p13.7.m7.1.1.1.1.3.3">𝑏</ci></apply></apply><apply id="p13.7.m7.1.1.1.3.cmml" xref="p13.7.m7.1.1.1.3"><times id="p13.7.m7.1.1.1.3.1.cmml" xref="p13.7.m7.1.1.1.3.1"></times><ci id="p13.7.m7.1.1.1.3.2.cmml" xref="p13.7.m7.1.1.1.3.2">𝑐</ci><ci id="p13.7.m7.1.1.1.3.3.cmml" xref="p13.7.m7.1.1.1.3.3">𝑑</ci></apply></apply><apply id="p13.7.m7.1.1.3.cmml" xref="p13.7.m7.1.1.3"><csymbol cd="ambiguous" id="p13.7.m7.1.1.3.1.cmml" xref="p13.7.m7.1.1.3">superscript</csymbol><apply id="p13.7.m7.1.1.3.2.cmml" xref="p13.7.m7.1.1.3"><csymbol cd="ambiguous" id="p13.7.m7.1.1.3.2.1.cmml" xref="p13.7.m7.1.1.3">subscript</csymbol><ci id="p13.7.m7.1.1.3.2.2.cmml" xref="p13.7.m7.1.1.3.2.2">𝑈</ci><apply id="p13.7.m7.1.1.3.2.3.cmml" xref="p13.7.m7.1.1.3.2.3"><times id="p13.7.m7.1.1.3.2.3.1.cmml" xref="p13.7.m7.1.1.3.2.3.1"></times><ci id="p13.7.m7.1.1.3.2.3.2.cmml" xref="p13.7.m7.1.1.3.2.3.2">𝑎</ci><ci id="p13.7.m7.1.1.3.2.3.3.cmml" xref="p13.7.m7.1.1.3.2.3.3">𝑐</ci></apply></apply><apply id="p13.7.m7.1.1.3.3.cmml" xref="p13.7.m7.1.1.3.3"><times id="p13.7.m7.1.1.3.3.1.cmml" xref="p13.7.m7.1.1.3.3.1"></times><ci id="p13.7.m7.1.1.3.3.2.cmml" xref="p13.7.m7.1.1.3.3.2">𝑏</ci><ci id="p13.7.m7.1.1.3.3.3.cmml" xref="p13.7.m7.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.7.m7.1c">(U^{R})_{ab}^{cd}=U_{ac}^{bd}</annotation><annotation encoding="application/x-llamapun" id="p13.7.m7.1d">( italic_U start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_d end_POSTSUPERSCRIPT = italic_U start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. When read along the space direction, the solvable condition can be expressed as</p> <table class="ltx_equation ltx_eqn_table" id="S0.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="U^{R}(\ket{A_{jk}}\bra{A_{j^{\prime}k^{\prime}}}\otimes I_{q})(U^{R})^{\dagger% }=I_{q}\otimes\ket{A_{jk}}\bra{A_{j^{\prime}k^{\prime}}}," class="ltx_Math" display="block" id="S0.E10.m1.5"><semantics id="S0.E10.m1.5a"><mrow id="S0.E10.m1.5.5.1" xref="S0.E10.m1.5.5.1.1.cmml"><mrow id="S0.E10.m1.5.5.1.1" xref="S0.E10.m1.5.5.1.1.cmml"><mrow id="S0.E10.m1.5.5.1.1.2" xref="S0.E10.m1.5.5.1.1.2.cmml"><msup id="S0.E10.m1.5.5.1.1.2.4" xref="S0.E10.m1.5.5.1.1.2.4.cmml"><mi id="S0.E10.m1.5.5.1.1.2.4.2" xref="S0.E10.m1.5.5.1.1.2.4.2.cmml">U</mi><mi id="S0.E10.m1.5.5.1.1.2.4.3" xref="S0.E10.m1.5.5.1.1.2.4.3.cmml">R</mi></msup><mo id="S0.E10.m1.5.5.1.1.2.3" 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xref="S0.E10.m1.4.4.1.1.3.3"><csymbol cd="ambiguous" id="S0.E10.m1.4.4.1.1.3.3.1.cmml" xref="S0.E10.m1.4.4.1.1.3.3">superscript</csymbol><ci id="S0.E10.m1.4.4.1.1.3.3.2.cmml" xref="S0.E10.m1.4.4.1.1.3.3.2">𝑘</ci><ci id="S0.E10.m1.4.4.1.1.3.3.3.cmml" xref="S0.E10.m1.4.4.1.1.3.3.3">′</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.E10.m1.5c">U^{R}(\ket{A_{jk}}\bra{A_{j^{\prime}k^{\prime}}}\otimes I_{q})(U^{R})^{\dagger% }=I_{q}\otimes\ket{A_{jk}}\bra{A_{j^{\prime}k^{\prime}}},</annotation><annotation encoding="application/x-llamapun" id="S0.E10.m1.5d">italic_U start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT ( | start_ARG italic_A start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_A start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_ARG | ⊗ italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ( italic_U start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ⊗ | start_ARG italic_A start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_A start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_ARG | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p13.14">for all combinations of <math alttext="j,j^{\prime},k,k^{\prime}" class="ltx_Math" display="inline" id="p13.8.m1.4"><semantics id="p13.8.m1.4a"><mrow id="p13.8.m1.4.4.2" xref="p13.8.m1.4.4.3.cmml"><mi id="p13.8.m1.1.1" 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Here, <math alttext="I_{q}" class="ltx_Math" display="inline" id="p13.9.m2.1"><semantics id="p13.9.m2.1a"><msub id="p13.9.m2.1.1" xref="p13.9.m2.1.1.cmml"><mi id="p13.9.m2.1.1.2" xref="p13.9.m2.1.1.2.cmml">I</mi><mi id="p13.9.m2.1.1.3" xref="p13.9.m2.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="p13.9.m2.1b"><apply id="p13.9.m2.1.1.cmml" xref="p13.9.m2.1.1"><csymbol cd="ambiguous" id="p13.9.m2.1.1.1.cmml" xref="p13.9.m2.1.1">subscript</csymbol><ci id="p13.9.m2.1.1.2.cmml" xref="p13.9.m2.1.1.2">𝐼</ci><ci id="p13.9.m2.1.1.3.cmml" xref="p13.9.m2.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.9.m2.1c">I_{q}</annotation><annotation encoding="application/x-llamapun" id="p13.9.m2.1d">italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> is the identity operator in the local Hilbert space, and the one-site MPS is defined as <math alttext="\ket{A_{jk}}=\sum_{a=0}^{q-1}A^{(a)}_{jk}\ket{a}" 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xref="p13.10.m3.3.4.2.2.2"><csymbol cd="ambiguous" id="p13.10.m3.3.4.2.2.2.1.cmml" xref="p13.10.m3.3.4.2.2.2">subscript</csymbol><apply id="p13.10.m3.3.4.2.2.2.2.cmml" xref="p13.10.m3.3.4.2.2.2"><csymbol cd="ambiguous" id="p13.10.m3.3.4.2.2.2.2.1.cmml" xref="p13.10.m3.3.4.2.2.2">superscript</csymbol><ci id="p13.10.m3.3.4.2.2.2.2.2.cmml" xref="p13.10.m3.3.4.2.2.2.2.2">𝐴</ci><ci id="p13.10.m3.3.3.1.1.cmml" xref="p13.10.m3.3.3.1.1">𝑎</ci></apply><apply id="p13.10.m3.3.4.2.2.2.3.cmml" xref="p13.10.m3.3.4.2.2.2.3"><times id="p13.10.m3.3.4.2.2.2.3.1.cmml" xref="p13.10.m3.3.4.2.2.2.3.1"></times><ci id="p13.10.m3.3.4.2.2.2.3.2.cmml" xref="p13.10.m3.3.4.2.2.2.3.2">𝑗</ci><ci id="p13.10.m3.3.4.2.2.2.3.3.cmml" xref="p13.10.m3.3.4.2.2.2.3.3">𝑘</ci></apply></apply><apply id="p13.10.m3.2.2.2.cmml" xref="p13.10.m3.2.2.3"><csymbol cd="latexml" id="p13.10.m3.2.2.2.1.cmml" xref="p13.10.m3.2.2.3.1">ket</csymbol><ci id="p13.10.m3.2.2.1.1.cmml" xref="p13.10.m3.2.2.1.1">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.10.m3.3c">\ket{A_{jk}}=\sum_{a=0}^{q-1}A^{(a)}_{jk}\ket{a}</annotation><annotation encoding="application/x-llamapun" id="p13.10.m3.3d">| start_ARG italic_A start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_a = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q - 1 end_POSTSUPERSCRIPT italic_A start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT | start_ARG italic_a end_ARG ⟩</annotation></semantics></math>. This expression suggests that the set of solutions <math alttext="U" class="ltx_Math" display="inline" id="p13.11.m4.1"><semantics id="p13.11.m4.1a"><mi id="p13.11.m4.1.1" xref="p13.11.m4.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="p13.11.m4.1b"><ci id="p13.11.m4.1.1.cmml" xref="p13.11.m4.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="p13.11.m4.1c">U</annotation><annotation encoding="application/x-llamapun" id="p13.11.m4.1d">italic_U</annotation></semantics></math> essentially depends on the Hilbert subspace <math alttext="\mathcal{H}_{A}" class="ltx_Math" display="inline" id="p13.12.m5.1"><semantics id="p13.12.m5.1a"><msub id="p13.12.m5.1.1" xref="p13.12.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p13.12.m5.1.1.2" xref="p13.12.m5.1.1.2.cmml">ℋ</mi><mi id="p13.12.m5.1.1.3" xref="p13.12.m5.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="p13.12.m5.1b"><apply id="p13.12.m5.1.1.cmml" xref="p13.12.m5.1.1"><csymbol cd="ambiguous" id="p13.12.m5.1.1.1.cmml" xref="p13.12.m5.1.1">subscript</csymbol><ci id="p13.12.m5.1.1.2.cmml" xref="p13.12.m5.1.1.2">ℋ</ci><ci id="p13.12.m5.1.1.3.cmml" xref="p13.12.m5.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.12.m5.1c">\mathcal{H}_{A}</annotation><annotation encoding="application/x-llamapun" id="p13.12.m5.1d">caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> spanned by <math alttext="\{\ket{A_{jk}}\}_{j,k=1}^{\chi}" class="ltx_Math" display="inline" id="p13.13.m6.3"><semantics id="p13.13.m6.3a"><msubsup id="p13.13.m6.3.4" xref="p13.13.m6.3.4.cmml"><mrow id="p13.13.m6.3.4.2.2.2" xref="p13.13.m6.3.4.2.2.1.cmml"><mo id="p13.13.m6.3.4.2.2.2.1" stretchy="false" xref="p13.13.m6.3.4.2.2.1.cmml">{</mo><mrow id="p13.13.m6.1.1.3" xref="p13.13.m6.1.1.2.cmml"><mo id="p13.13.m6.1.1.3.1" stretchy="false" xref="p13.13.m6.1.1.2.1.cmml">|</mo><msub id="p13.13.m6.1.1.1.1" xref="p13.13.m6.1.1.1.1.cmml"><mi id="p13.13.m6.1.1.1.1.2" xref="p13.13.m6.1.1.1.1.2.cmml">A</mi><mrow id="p13.13.m6.1.1.1.1.3" xref="p13.13.m6.1.1.1.1.3.cmml"><mi id="p13.13.m6.1.1.1.1.3.2" xref="p13.13.m6.1.1.1.1.3.2.cmml">j</mi><mo id="p13.13.m6.1.1.1.1.3.1" xref="p13.13.m6.1.1.1.1.3.1.cmml">⁢</mo><mi id="p13.13.m6.1.1.1.1.3.3" xref="p13.13.m6.1.1.1.1.3.3.cmml">k</mi></mrow></msub><mo id="p13.13.m6.1.1.3.2" stretchy="false" xref="p13.13.m6.1.1.2.1.cmml">⟩</mo></mrow><mo id="p13.13.m6.3.4.2.2.2.2" stretchy="false" xref="p13.13.m6.3.4.2.2.1.cmml">}</mo></mrow><mrow id="p13.13.m6.3.3.2" xref="p13.13.m6.3.3.2.cmml"><mrow id="p13.13.m6.3.3.2.4.2" xref="p13.13.m6.3.3.2.4.1.cmml"><mi id="p13.13.m6.2.2.1.1" xref="p13.13.m6.2.2.1.1.cmml">j</mi><mo id="p13.13.m6.3.3.2.4.2.1" xref="p13.13.m6.3.3.2.4.1.cmml">,</mo><mi id="p13.13.m6.3.3.2.2" xref="p13.13.m6.3.3.2.2.cmml">k</mi></mrow><mo id="p13.13.m6.3.3.2.3" xref="p13.13.m6.3.3.2.3.cmml">=</mo><mn id="p13.13.m6.3.3.2.5" xref="p13.13.m6.3.3.2.5.cmml">1</mn></mrow><mi id="p13.13.m6.3.4.3" xref="p13.13.m6.3.4.3.cmml">χ</mi></msubsup><annotation-xml encoding="MathML-Content" id="p13.13.m6.3b"><apply id="p13.13.m6.3.4.cmml" xref="p13.13.m6.3.4"><csymbol cd="ambiguous" id="p13.13.m6.3.4.1.cmml" xref="p13.13.m6.3.4">superscript</csymbol><apply id="p13.13.m6.3.4.2.cmml" xref="p13.13.m6.3.4"><csymbol cd="ambiguous" id="p13.13.m6.3.4.2.1.cmml" xref="p13.13.m6.3.4">subscript</csymbol><set id="p13.13.m6.3.4.2.2.1.cmml" xref="p13.13.m6.3.4.2.2.2"><apply id="p13.13.m6.1.1.2.cmml" xref="p13.13.m6.1.1.3"><csymbol cd="latexml" id="p13.13.m6.1.1.2.1.cmml" xref="p13.13.m6.1.1.3.1">ket</csymbol><apply id="p13.13.m6.1.1.1.1.cmml" xref="p13.13.m6.1.1.1.1"><csymbol cd="ambiguous" id="p13.13.m6.1.1.1.1.1.cmml" xref="p13.13.m6.1.1.1.1">subscript</csymbol><ci id="p13.13.m6.1.1.1.1.2.cmml" xref="p13.13.m6.1.1.1.1.2">𝐴</ci><apply id="p13.13.m6.1.1.1.1.3.cmml" xref="p13.13.m6.1.1.1.1.3"><times id="p13.13.m6.1.1.1.1.3.1.cmml" 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1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_χ end_POSTSUPERSCRIPT</annotation></semantics></math>. Furthermore, due to the isomorphism between different Hilbert subspaces with the same dimension, the solutions should solely rely on the subspace dimension <math alttext="\tilde{q}=\text{dim}(\mathcal{H}_{A})" class="ltx_Math" display="inline" id="p13.14.m7.1"><semantics id="p13.14.m7.1a"><mrow id="p13.14.m7.1.1" xref="p13.14.m7.1.1.cmml"><mover accent="true" id="p13.14.m7.1.1.3" xref="p13.14.m7.1.1.3.cmml"><mi id="p13.14.m7.1.1.3.2" xref="p13.14.m7.1.1.3.2.cmml">q</mi><mo id="p13.14.m7.1.1.3.1" xref="p13.14.m7.1.1.3.1.cmml">~</mo></mover><mo id="p13.14.m7.1.1.2" xref="p13.14.m7.1.1.2.cmml">=</mo><mrow id="p13.14.m7.1.1.1" xref="p13.14.m7.1.1.1.cmml"><mtext id="p13.14.m7.1.1.1.3" xref="p13.14.m7.1.1.1.3a.cmml">dim</mtext><mo id="p13.14.m7.1.1.1.2" xref="p13.14.m7.1.1.1.2.cmml">⁢</mo><mrow id="p13.14.m7.1.1.1.1.1" xref="p13.14.m7.1.1.1.1.1.1.cmml"><mo id="p13.14.m7.1.1.1.1.1.2" stretchy="false" xref="p13.14.m7.1.1.1.1.1.1.cmml">(</mo><msub id="p13.14.m7.1.1.1.1.1.1" xref="p13.14.m7.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p13.14.m7.1.1.1.1.1.1.2" xref="p13.14.m7.1.1.1.1.1.1.2.cmml">ℋ</mi><mi id="p13.14.m7.1.1.1.1.1.1.3" xref="p13.14.m7.1.1.1.1.1.1.3.cmml">A</mi></msub><mo id="p13.14.m7.1.1.1.1.1.3" stretchy="false" xref="p13.14.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p13.14.m7.1b"><apply id="p13.14.m7.1.1.cmml" xref="p13.14.m7.1.1"><eq id="p13.14.m7.1.1.2.cmml" xref="p13.14.m7.1.1.2"></eq><apply id="p13.14.m7.1.1.3.cmml" xref="p13.14.m7.1.1.3"><ci id="p13.14.m7.1.1.3.1.cmml" xref="p13.14.m7.1.1.3.1">~</ci><ci id="p13.14.m7.1.1.3.2.cmml" xref="p13.14.m7.1.1.3.2">𝑞</ci></apply><apply id="p13.14.m7.1.1.1.cmml" xref="p13.14.m7.1.1.1"><times id="p13.14.m7.1.1.1.2.cmml" xref="p13.14.m7.1.1.1.2"></times><ci id="p13.14.m7.1.1.1.3a.cmml" xref="p13.14.m7.1.1.1.3"><mtext id="p13.14.m7.1.1.1.3.cmml" xref="p13.14.m7.1.1.1.3">dim</mtext></ci><apply id="p13.14.m7.1.1.1.1.1.1.cmml" xref="p13.14.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="p13.14.m7.1.1.1.1.1.1.1.cmml" xref="p13.14.m7.1.1.1.1.1">subscript</csymbol><ci id="p13.14.m7.1.1.1.1.1.1.2.cmml" xref="p13.14.m7.1.1.1.1.1.1.2">ℋ</ci><ci id="p13.14.m7.1.1.1.1.1.1.3.cmml" xref="p13.14.m7.1.1.1.1.1.1.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p13.14.m7.1c">\tilde{q}=\text{dim}(\mathcal{H}_{A})</annotation><annotation encoding="application/x-llamapun" id="p13.14.m7.1d">over~ start_ARG italic_q end_ARG = dim ( caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT )</annotation></semantics></math>, up to one-site unitaries.</p> </div> <div class="ltx_para" id="p14"> <p class="ltx_p" id="p14.2">Since the set of solutions applies to all initial states with the same <math alttext="q" class="ltx_Math" display="inline" id="p14.1.m1.1"><semantics id="p14.1.m1.1a"><mi id="p14.1.m1.1.1" xref="p14.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="p14.1.m1.1b"><ci id="p14.1.m1.1.1.cmml" xref="p14.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="p14.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="p14.1.m1.1d">italic_q</annotation></semantics></math> and <math alttext="\tilde{q}" class="ltx_Math" display="inline" id="p14.2.m2.1"><semantics id="p14.2.m2.1a"><mover accent="true" id="p14.2.m2.1.1" xref="p14.2.m2.1.1.cmml"><mi id="p14.2.m2.1.1.2" xref="p14.2.m2.1.1.2.cmml">q</mi><mo id="p14.2.m2.1.1.1" xref="p14.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="p14.2.m2.1b"><apply id="p14.2.m2.1.1.cmml" xref="p14.2.m2.1.1"><ci id="p14.2.m2.1.1.1.cmml" xref="p14.2.m2.1.1.1">~</ci><ci id="p14.2.m2.1.1.2.cmml" xref="p14.2.m2.1.1.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p14.2.m2.1c">\tilde{q}</annotation><annotation encoding="application/x-llamapun" id="p14.2.m2.1d">over~ start_ARG italic_q end_ARG</annotation></semantics></math>, we will label the solutions by these two characteristic values in the following.</p> </div> <div class="ltx_para" id="p15"> <p class="ltx_p" id="p15.4"><span class="ltx_text ltx_font_italic" id="p15.4.1">Example I</span>—In this and the next part, we will show that the solvable condition indeed leads to a wide range of solutions for quantum circuits. First, we present the solutions of <math alttext="q=2,\tilde{q}=1" class="ltx_Math" display="inline" id="p15.1.m1.2"><semantics id="p15.1.m1.2a"><mrow id="p15.1.m1.2.2.2" xref="p15.1.m1.2.2.3.cmml"><mrow id="p15.1.m1.1.1.1.1" xref="p15.1.m1.1.1.1.1.cmml"><mi id="p15.1.m1.1.1.1.1.2" xref="p15.1.m1.1.1.1.1.2.cmml">q</mi><mo id="p15.1.m1.1.1.1.1.1" xref="p15.1.m1.1.1.1.1.1.cmml">=</mo><mn id="p15.1.m1.1.1.1.1.3" xref="p15.1.m1.1.1.1.1.3.cmml">2</mn></mrow><mo id="p15.1.m1.2.2.2.3" xref="p15.1.m1.2.2.3a.cmml">,</mo><mrow id="p15.1.m1.2.2.2.2" xref="p15.1.m1.2.2.2.2.cmml"><mover accent="true" id="p15.1.m1.2.2.2.2.2" xref="p15.1.m1.2.2.2.2.2.cmml"><mi id="p15.1.m1.2.2.2.2.2.2" xref="p15.1.m1.2.2.2.2.2.2.cmml">q</mi><mo id="p15.1.m1.2.2.2.2.2.1" xref="p15.1.m1.2.2.2.2.2.1.cmml">~</mo></mover><mo id="p15.1.m1.2.2.2.2.1" xref="p15.1.m1.2.2.2.2.1.cmml">=</mo><mn id="p15.1.m1.2.2.2.2.3" xref="p15.1.m1.2.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="p15.1.m1.2b"><apply id="p15.1.m1.2.2.3.cmml" xref="p15.1.m1.2.2.2"><csymbol cd="ambiguous" id="p15.1.m1.2.2.3a.cmml" xref="p15.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="p15.1.m1.1.1.1.1.cmml" xref="p15.1.m1.1.1.1.1"><eq id="p15.1.m1.1.1.1.1.1.cmml" xref="p15.1.m1.1.1.1.1.1"></eq><ci id="p15.1.m1.1.1.1.1.2.cmml" xref="p15.1.m1.1.1.1.1.2">𝑞</ci><cn id="p15.1.m1.1.1.1.1.3.cmml" type="integer" xref="p15.1.m1.1.1.1.1.3">2</cn></apply><apply id="p15.1.m1.2.2.2.2.cmml" xref="p15.1.m1.2.2.2.2"><eq id="p15.1.m1.2.2.2.2.1.cmml" xref="p15.1.m1.2.2.2.2.1"></eq><apply id="p15.1.m1.2.2.2.2.2.cmml" xref="p15.1.m1.2.2.2.2.2"><ci id="p15.1.m1.2.2.2.2.2.1.cmml" xref="p15.1.m1.2.2.2.2.2.1">~</ci><ci id="p15.1.m1.2.2.2.2.2.2.cmml" xref="p15.1.m1.2.2.2.2.2.2">𝑞</ci></apply><cn id="p15.1.m1.2.2.2.2.3.cmml" type="integer" xref="p15.1.m1.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.1.m1.2c">q=2,\tilde{q}=1</annotation><annotation encoding="application/x-llamapun" id="p15.1.m1.2d">italic_q = 2 , over~ start_ARG italic_q end_ARG = 1</annotation></semantics></math>. For the reason demonstrated above, we can take a specific left initial state <math alttext="\otimes_{x<0}\ket{0}_{x}" class="ltx_Math" display="inline" id="p15.2.m2.1"><semantics id="p15.2.m2.1a"><mrow id="p15.2.m2.1.2" xref="p15.2.m2.1.2.cmml"><mi id="p15.2.m2.1.2.2" xref="p15.2.m2.1.2.2.cmml"></mi><msub id="p15.2.m2.1.2.1" xref="p15.2.m2.1.2.1.cmml"><mo id="p15.2.m2.1.2.1.2" lspace="0.222em" rspace="0.222em" xref="p15.2.m2.1.2.1.2.cmml">⊗</mo><mrow id="p15.2.m2.1.2.1.3" xref="p15.2.m2.1.2.1.3.cmml"><mi id="p15.2.m2.1.2.1.3.2" xref="p15.2.m2.1.2.1.3.2.cmml">x</mi><mo id="p15.2.m2.1.2.1.3.1" xref="p15.2.m2.1.2.1.3.1.cmml"><</mo><mn id="p15.2.m2.1.2.1.3.3" xref="p15.2.m2.1.2.1.3.3.cmml">0</mn></mrow></msub><msub id="p15.2.m2.1.2.3" xref="p15.2.m2.1.2.3.cmml"><mrow id="p15.2.m2.1.1.3" xref="p15.2.m2.1.1.2.cmml"><mo id="p15.2.m2.1.1.3.1" stretchy="false" xref="p15.2.m2.1.1.2.1.cmml">|</mo><mn id="p15.2.m2.1.1.1.1" xref="p15.2.m2.1.1.1.1.cmml">0</mn><mo id="p15.2.m2.1.1.3.2" stretchy="false" xref="p15.2.m2.1.1.2.1.cmml">⟩</mo></mrow><mi id="p15.2.m2.1.2.3.2" xref="p15.2.m2.1.2.3.2.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="p15.2.m2.1b"><apply id="p15.2.m2.1.2.cmml" xref="p15.2.m2.1.2"><apply id="p15.2.m2.1.2.1.cmml" xref="p15.2.m2.1.2.1"><csymbol cd="ambiguous" id="p15.2.m2.1.2.1.1.cmml" xref="p15.2.m2.1.2.1">subscript</csymbol><csymbol cd="latexml" id="p15.2.m2.1.2.1.2.cmml" xref="p15.2.m2.1.2.1.2">tensor-product</csymbol><apply id="p15.2.m2.1.2.1.3.cmml" xref="p15.2.m2.1.2.1.3"><lt id="p15.2.m2.1.2.1.3.1.cmml" xref="p15.2.m2.1.2.1.3.1"></lt><ci id="p15.2.m2.1.2.1.3.2.cmml" xref="p15.2.m2.1.2.1.3.2">𝑥</ci><cn id="p15.2.m2.1.2.1.3.3.cmml" type="integer" xref="p15.2.m2.1.2.1.3.3">0</cn></apply></apply><csymbol cd="latexml" id="p15.2.m2.1.2.2.cmml" xref="p15.2.m2.1.2.2">absent</csymbol><apply id="p15.2.m2.1.2.3.cmml" xref="p15.2.m2.1.2.3"><csymbol cd="ambiguous" id="p15.2.m2.1.2.3.1.cmml" xref="p15.2.m2.1.2.3">subscript</csymbol><apply id="p15.2.m2.1.1.2.cmml" xref="p15.2.m2.1.1.3"><csymbol cd="latexml" id="p15.2.m2.1.1.2.1.cmml" xref="p15.2.m2.1.1.3.1">ket</csymbol><cn id="p15.2.m2.1.1.1.1.cmml" type="integer" xref="p15.2.m2.1.1.1.1">0</cn></apply><ci id="p15.2.m2.1.2.3.2.cmml" xref="p15.2.m2.1.2.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.2.m2.1c">\otimes_{x<0}\ket{0}_{x}</annotation><annotation encoding="application/x-llamapun" id="p15.2.m2.1d">⊗ start_POSTSUBSCRIPT italic_x < 0 end_POSTSUBSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. In the context of MPS, the tensors are given by <math alttext="A^{(0)}=1,A^{(1)}=0" class="ltx_Math" display="inline" id="p15.3.m3.4"><semantics id="p15.3.m3.4a"><mrow id="p15.3.m3.4.4.2" xref="p15.3.m3.4.4.3.cmml"><mrow id="p15.3.m3.3.3.1.1" xref="p15.3.m3.3.3.1.1.cmml"><msup id="p15.3.m3.3.3.1.1.2" xref="p15.3.m3.3.3.1.1.2.cmml"><mi id="p15.3.m3.3.3.1.1.2.2" xref="p15.3.m3.3.3.1.1.2.2.cmml">A</mi><mrow id="p15.3.m3.1.1.1.3" xref="p15.3.m3.3.3.1.1.2.cmml"><mo id="p15.3.m3.1.1.1.3.1" stretchy="false" xref="p15.3.m3.3.3.1.1.2.cmml">(</mo><mn id="p15.3.m3.1.1.1.1" xref="p15.3.m3.1.1.1.1.cmml">0</mn><mo id="p15.3.m3.1.1.1.3.2" stretchy="false" xref="p15.3.m3.3.3.1.1.2.cmml">)</mo></mrow></msup><mo id="p15.3.m3.3.3.1.1.1" xref="p15.3.m3.3.3.1.1.1.cmml">=</mo><mn id="p15.3.m3.3.3.1.1.3" xref="p15.3.m3.3.3.1.1.3.cmml">1</mn></mrow><mo id="p15.3.m3.4.4.2.3" xref="p15.3.m3.4.4.3a.cmml">,</mo><mrow id="p15.3.m3.4.4.2.2" xref="p15.3.m3.4.4.2.2.cmml"><msup id="p15.3.m3.4.4.2.2.2" xref="p15.3.m3.4.4.2.2.2.cmml"><mi id="p15.3.m3.4.4.2.2.2.2" xref="p15.3.m3.4.4.2.2.2.2.cmml">A</mi><mrow id="p15.3.m3.2.2.1.3" xref="p15.3.m3.4.4.2.2.2.cmml"><mo id="p15.3.m3.2.2.1.3.1" stretchy="false" xref="p15.3.m3.4.4.2.2.2.cmml">(</mo><mn id="p15.3.m3.2.2.1.1" xref="p15.3.m3.2.2.1.1.cmml">1</mn><mo id="p15.3.m3.2.2.1.3.2" stretchy="false" xref="p15.3.m3.4.4.2.2.2.cmml">)</mo></mrow></msup><mo id="p15.3.m3.4.4.2.2.1" xref="p15.3.m3.4.4.2.2.1.cmml">=</mo><mn id="p15.3.m3.4.4.2.2.3" xref="p15.3.m3.4.4.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="p15.3.m3.4b"><apply id="p15.3.m3.4.4.3.cmml" xref="p15.3.m3.4.4.2"><csymbol cd="ambiguous" id="p15.3.m3.4.4.3a.cmml" xref="p15.3.m3.4.4.2.3">formulae-sequence</csymbol><apply id="p15.3.m3.3.3.1.1.cmml" xref="p15.3.m3.3.3.1.1"><eq id="p15.3.m3.3.3.1.1.1.cmml" xref="p15.3.m3.3.3.1.1.1"></eq><apply id="p15.3.m3.3.3.1.1.2.cmml" xref="p15.3.m3.3.3.1.1.2"><csymbol cd="ambiguous" id="p15.3.m3.3.3.1.1.2.1.cmml" xref="p15.3.m3.3.3.1.1.2">superscript</csymbol><ci id="p15.3.m3.3.3.1.1.2.2.cmml" xref="p15.3.m3.3.3.1.1.2.2">𝐴</ci><cn id="p15.3.m3.1.1.1.1.cmml" type="integer" xref="p15.3.m3.1.1.1.1">0</cn></apply><cn id="p15.3.m3.3.3.1.1.3.cmml" type="integer" xref="p15.3.m3.3.3.1.1.3">1</cn></apply><apply id="p15.3.m3.4.4.2.2.cmml" xref="p15.3.m3.4.4.2.2"><eq id="p15.3.m3.4.4.2.2.1.cmml" xref="p15.3.m3.4.4.2.2.1"></eq><apply id="p15.3.m3.4.4.2.2.2.cmml" xref="p15.3.m3.4.4.2.2.2"><csymbol cd="ambiguous" id="p15.3.m3.4.4.2.2.2.1.cmml" xref="p15.3.m3.4.4.2.2.2">superscript</csymbol><ci id="p15.3.m3.4.4.2.2.2.2.cmml" xref="p15.3.m3.4.4.2.2.2.2">𝐴</ci><cn id="p15.3.m3.2.2.1.1.cmml" type="integer" xref="p15.3.m3.2.2.1.1">1</cn></apply><cn id="p15.3.m3.4.4.2.2.3.cmml" type="integer" xref="p15.3.m3.4.4.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.3.m3.4c">A^{(0)}=1,A^{(1)}=0</annotation><annotation encoding="application/x-llamapun" id="p15.3.m3.4d">italic_A start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT = 1 , italic_A start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = 0</annotation></semantics></math>, and thus <math alttext="\mathcal{H}_{A}=\{\ket{0}\}" class="ltx_Math" display="inline" id="p15.4.m4.1"><semantics id="p15.4.m4.1a"><mrow id="p15.4.m4.1.2" xref="p15.4.m4.1.2.cmml"><msub id="p15.4.m4.1.2.2" xref="p15.4.m4.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="p15.4.m4.1.2.2.2" xref="p15.4.m4.1.2.2.2.cmml">ℋ</mi><mi id="p15.4.m4.1.2.2.3" xref="p15.4.m4.1.2.2.3.cmml">A</mi></msub><mo id="p15.4.m4.1.2.1" xref="p15.4.m4.1.2.1.cmml">=</mo><mrow id="p15.4.m4.1.2.3.2" xref="p15.4.m4.1.2.3.1.cmml"><mo id="p15.4.m4.1.2.3.2.1" stretchy="false" xref="p15.4.m4.1.2.3.1.cmml">{</mo><mrow id="p15.4.m4.1.1.3" xref="p15.4.m4.1.1.2.cmml"><mo id="p15.4.m4.1.1.3.1" stretchy="false" xref="p15.4.m4.1.1.2.1.cmml">|</mo><mn id="p15.4.m4.1.1.1.1" xref="p15.4.m4.1.1.1.1.cmml">0</mn><mo id="p15.4.m4.1.1.3.2" stretchy="false" xref="p15.4.m4.1.1.2.1.cmml">⟩</mo></mrow><mo id="p15.4.m4.1.2.3.2.2" stretchy="false" xref="p15.4.m4.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p15.4.m4.1b"><apply id="p15.4.m4.1.2.cmml" xref="p15.4.m4.1.2"><eq id="p15.4.m4.1.2.1.cmml" xref="p15.4.m4.1.2.1"></eq><apply id="p15.4.m4.1.2.2.cmml" xref="p15.4.m4.1.2.2"><csymbol cd="ambiguous" id="p15.4.m4.1.2.2.1.cmml" xref="p15.4.m4.1.2.2">subscript</csymbol><ci id="p15.4.m4.1.2.2.2.cmml" xref="p15.4.m4.1.2.2.2">ℋ</ci><ci id="p15.4.m4.1.2.2.3.cmml" xref="p15.4.m4.1.2.2.3">𝐴</ci></apply><set id="p15.4.m4.1.2.3.1.cmml" xref="p15.4.m4.1.2.3.2"><apply id="p15.4.m4.1.1.2.cmml" xref="p15.4.m4.1.1.3"><csymbol cd="latexml" id="p15.4.m4.1.1.2.1.cmml" xref="p15.4.m4.1.1.3.1">ket</csymbol><cn id="p15.4.m4.1.1.1.1.cmml" type="integer" xref="p15.4.m4.1.1.1.1">0</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.4.m4.1c">\mathcal{H}_{A}=\{\ket{0}\}</annotation><annotation encoding="application/x-llamapun" id="p15.4.m4.1d">caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = { | start_ARG 0 end_ARG ⟩ }</annotation></semantics></math>. We provide an exhaustive parametrization for the solutions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib38" title="">38</a>]</cite></p> <table class="ltx_equation ltx_eqn_table" id="S0.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="U=e^{i\phi}(u\otimes e^{-i\epsilon\sigma^{3}})V[J](e^{-i\eta\sigma^{3}}\otimes v)," class="ltx_Math" display="block" id="S0.E11.m1.2"><semantics id="S0.E11.m1.2a"><mrow id="S0.E11.m1.2.2.1" xref="S0.E11.m1.2.2.1.1.cmml"><mrow id="S0.E11.m1.2.2.1.1" xref="S0.E11.m1.2.2.1.1.cmml"><mi id="S0.E11.m1.2.2.1.1.4" xref="S0.E11.m1.2.2.1.1.4.cmml">U</mi><mo id="S0.E11.m1.2.2.1.1.3" xref="S0.E11.m1.2.2.1.1.3.cmml">=</mo><mrow id="S0.E11.m1.2.2.1.1.2" xref="S0.E11.m1.2.2.1.1.2.cmml"><msup id="S0.E11.m1.2.2.1.1.2.4" xref="S0.E11.m1.2.2.1.1.2.4.cmml"><mi id="S0.E11.m1.2.2.1.1.2.4.2" xref="S0.E11.m1.2.2.1.1.2.4.2.cmml">e</mi><mrow id="S0.E11.m1.2.2.1.1.2.4.3" xref="S0.E11.m1.2.2.1.1.2.4.3.cmml"><mi id="S0.E11.m1.2.2.1.1.2.4.3.2" xref="S0.E11.m1.2.2.1.1.2.4.3.2.cmml">i</mi><mo id="S0.E11.m1.2.2.1.1.2.4.3.1" xref="S0.E11.m1.2.2.1.1.2.4.3.1.cmml">⁢</mo><mi id="S0.E11.m1.2.2.1.1.2.4.3.3" xref="S0.E11.m1.2.2.1.1.2.4.3.3.cmml">ϕ</mi></mrow></msup><mo id="S0.E11.m1.2.2.1.1.2.3" xref="S0.E11.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S0.E11.m1.2.2.1.1.1.1.1" xref="S0.E11.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S0.E11.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S0.E11.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S0.E11.m1.2.2.1.1.1.1.1.1" xref="S0.E11.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S0.E11.m1.2.2.1.1.1.1.1.1.2" xref="S0.E11.m1.2.2.1.1.1.1.1.1.2.cmml">u</mi><mo id="S0.E11.m1.2.2.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S0.E11.m1.2.2.1.1.1.1.1.1.1.cmml">⊗</mo><msup id="S0.E11.m1.2.2.1.1.1.1.1.1.3" xref="S0.E11.m1.2.2.1.1.1.1.1.1.3.cmml"><mi 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xref="S0.E11.m1.2.2.1.1.1.1.1.1.3.3.2.4.3.cmml">3</mn></msup></mrow></mrow></msup></mrow><mo id="S0.E11.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S0.E11.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S0.E11.m1.2.2.1.1.2.3a" xref="S0.E11.m1.2.2.1.1.2.3.cmml">⁢</mo><mi id="S0.E11.m1.2.2.1.1.2.5" xref="S0.E11.m1.2.2.1.1.2.5.cmml">V</mi><mo id="S0.E11.m1.2.2.1.1.2.3b" xref="S0.E11.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S0.E11.m1.2.2.1.1.2.6.2" xref="S0.E11.m1.2.2.1.1.2.6.1.cmml"><mo id="S0.E11.m1.2.2.1.1.2.6.2.1" stretchy="false" xref="S0.E11.m1.2.2.1.1.2.6.1.1.cmml">[</mo><mi id="S0.E11.m1.1.1" xref="S0.E11.m1.1.1.cmml">J</mi><mo id="S0.E11.m1.2.2.1.1.2.6.2.2" stretchy="false" xref="S0.E11.m1.2.2.1.1.2.6.1.1.cmml">]</mo></mrow><mo id="S0.E11.m1.2.2.1.1.2.3c" xref="S0.E11.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S0.E11.m1.2.2.1.1.2.2.1" xref="S0.E11.m1.2.2.1.1.2.2.1.1.cmml"><mo id="S0.E11.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="S0.E11.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="S0.E11.m1.2.2.1.1.2.2.1.1" xref="S0.E11.m1.2.2.1.1.2.2.1.1.cmml"><msup id="S0.E11.m1.2.2.1.1.2.2.1.1.2" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.cmml"><mi id="S0.E11.m1.2.2.1.1.2.2.1.1.2.2" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.2.cmml">e</mi><mrow id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.cmml"><mo id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3a" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.cmml">−</mo><mrow id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.cmml"><mi id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.2" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.2.cmml">i</mi><mo id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.1" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.1.cmml">⁢</mo><mi id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.3" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.3.cmml">η</mi><mo id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.1a" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.1.cmml">⁢</mo><msup id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.4" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.4.cmml"><mi 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id="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.4.3.cmml" type="integer" xref="S0.E11.m1.2.2.1.1.2.2.1.1.2.3.2.4.3">3</cn></apply></apply></apply></apply><ci id="S0.E11.m1.2.2.1.1.2.2.1.1.3.cmml" xref="S0.E11.m1.2.2.1.1.2.2.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.E11.m1.2c">U=e^{i\phi}(u\otimes e^{-i\epsilon\sigma^{3}})V[J](e^{-i\eta\sigma^{3}}\otimes v),</annotation><annotation encoding="application/x-llamapun" id="S0.E11.m1.2d">italic_U = italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ end_POSTSUPERSCRIPT ( italic_u ⊗ italic_e start_POSTSUPERSCRIPT - italic_i italic_ϵ italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) italic_V [ italic_J ] ( italic_e start_POSTSUPERSCRIPT - italic_i italic_η italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⊗ italic_v ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p15.5">where <math alttext="u,v\in\text{SU}(2)" class="ltx_Math" display="inline" id="p15.5.m1.3"><semantics id="p15.5.m1.3a"><mrow id="p15.5.m1.3.4" xref="p15.5.m1.3.4.cmml"><mrow id="p15.5.m1.3.4.2.2" xref="p15.5.m1.3.4.2.1.cmml"><mi id="p15.5.m1.2.2" xref="p15.5.m1.2.2.cmml">u</mi><mo id="p15.5.m1.3.4.2.2.1" xref="p15.5.m1.3.4.2.1.cmml">,</mo><mi id="p15.5.m1.3.3" xref="p15.5.m1.3.3.cmml">v</mi></mrow><mo id="p15.5.m1.3.4.1" xref="p15.5.m1.3.4.1.cmml">∈</mo><mrow id="p15.5.m1.3.4.3" xref="p15.5.m1.3.4.3.cmml"><mtext id="p15.5.m1.3.4.3.2" xref="p15.5.m1.3.4.3.2a.cmml">SU</mtext><mo id="p15.5.m1.3.4.3.1" xref="p15.5.m1.3.4.3.1.cmml">⁢</mo><mrow id="p15.5.m1.3.4.3.3.2" xref="p15.5.m1.3.4.3.cmml"><mo id="p15.5.m1.3.4.3.3.2.1" stretchy="false" xref="p15.5.m1.3.4.3.cmml">(</mo><mn id="p15.5.m1.1.1" xref="p15.5.m1.1.1.cmml">2</mn><mo id="p15.5.m1.3.4.3.3.2.2" stretchy="false" xref="p15.5.m1.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p15.5.m1.3b"><apply id="p15.5.m1.3.4.cmml" xref="p15.5.m1.3.4"><in id="p15.5.m1.3.4.1.cmml" xref="p15.5.m1.3.4.1"></in><list id="p15.5.m1.3.4.2.1.cmml" xref="p15.5.m1.3.4.2.2"><ci id="p15.5.m1.2.2.cmml" xref="p15.5.m1.2.2">𝑢</ci><ci id="p15.5.m1.3.3.cmml" xref="p15.5.m1.3.3">𝑣</ci></list><apply id="p15.5.m1.3.4.3.cmml" xref="p15.5.m1.3.4.3"><times id="p15.5.m1.3.4.3.1.cmml" xref="p15.5.m1.3.4.3.1"></times><ci id="p15.5.m1.3.4.3.2a.cmml" xref="p15.5.m1.3.4.3.2"><mtext id="p15.5.m1.3.4.3.2.cmml" xref="p15.5.m1.3.4.3.2">SU</mtext></ci><cn id="p15.5.m1.1.1.cmml" type="integer" xref="p15.5.m1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.5.m1.3c">u,v\in\text{SU}(2)</annotation><annotation encoding="application/x-llamapun" id="p15.5.m1.3d">italic_u , italic_v ∈ SU ( 2 )</annotation></semantics></math>, and</p> <table class="ltx_equation ltx_eqn_table" id="S0.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="V[J]=\exp{[-i(\frac{\pi}{4}\sigma^{1}\otimes\sigma^{1}+\frac{\pi}{4}\sigma^{2}% \otimes\sigma^{2}+J\sigma^{3}\otimes\sigma^{3})]}." class="ltx_Math" display="block" id="S0.Ex2.m1.3"><semantics id="S0.Ex2.m1.3a"><mrow id="S0.Ex2.m1.3.3.1" xref="S0.Ex2.m1.3.3.1.1.cmml"><mrow id="S0.Ex2.m1.3.3.1.1" xref="S0.Ex2.m1.3.3.1.1.cmml"><mrow id="S0.Ex2.m1.3.3.1.1.3" xref="S0.Ex2.m1.3.3.1.1.3.cmml"><mi id="S0.Ex2.m1.3.3.1.1.3.2" xref="S0.Ex2.m1.3.3.1.1.3.2.cmml">V</mi><mo id="S0.Ex2.m1.3.3.1.1.3.1" xref="S0.Ex2.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S0.Ex2.m1.3.3.1.1.3.3.2" xref="S0.Ex2.m1.3.3.1.1.3.3.1.cmml"><mo id="S0.Ex2.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S0.Ex2.m1.3.3.1.1.3.3.1.1.cmml">[</mo><mi id="S0.Ex2.m1.1.1" xref="S0.Ex2.m1.1.1.cmml">J</mi><mo id="S0.Ex2.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S0.Ex2.m1.3.3.1.1.3.3.1.1.cmml">]</mo></mrow></mrow><mo id="S0.Ex2.m1.3.3.1.1.2" xref="S0.Ex2.m1.3.3.1.1.2.cmml">=</mo><mrow id="S0.Ex2.m1.3.3.1.1.1.1" xref="S0.Ex2.m1.3.3.1.1.1.2.cmml"><mi id="S0.Ex2.m1.2.2" xref="S0.Ex2.m1.2.2.cmml">exp</mi><mo id="S0.Ex2.m1.3.3.1.1.1.1a" xref="S0.Ex2.m1.3.3.1.1.1.2.cmml">⁡</mo><mrow id="S0.Ex2.m1.3.3.1.1.1.1.1" xref="S0.Ex2.m1.3.3.1.1.1.2.cmml"><mo id="S0.Ex2.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S0.Ex2.m1.3.3.1.1.1.2.cmml">[</mo><mrow id="S0.Ex2.m1.3.3.1.1.1.1.1.1" xref="S0.Ex2.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S0.Ex2.m1.3.3.1.1.1.1.1.1a" xref="S0.Ex2.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mrow id="S0.Ex2.m1.3.3.1.1.1.1.1.1.1" xref="S0.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S0.Ex2.m1.3.3.1.1.1.1.1.1.1.3" xref="S0.Ex2.m1.3.3.1.1.1.1.1.1.1.3.cmml">i</mi><mo id="S0.Ex2.m1.3.3.1.1.1.1.1.1.1.2" xref="S0.Ex2.m1.3.3.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow 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id="S0.Ex2.m1.3c">V[J]=\exp{[-i(\frac{\pi}{4}\sigma^{1}\otimes\sigma^{1}+\frac{\pi}{4}\sigma^{2}% \otimes\sigma^{2}+J\sigma^{3}\otimes\sigma^{3})]}.</annotation><annotation encoding="application/x-llamapun" id="S0.Ex2.m1.3d">italic_V [ italic_J ] = roman_exp [ - italic_i ( divide start_ARG italic_π end_ARG start_ARG 4 end_ARG italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT + divide start_ARG italic_π end_ARG start_ARG 4 end_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_J italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="p15.7">Here, <math alttext="\sigma^{\alpha}" class="ltx_Math" display="inline" id="p15.6.m1.1"><semantics id="p15.6.m1.1a"><msup id="p15.6.m1.1.1" xref="p15.6.m1.1.1.cmml"><mi id="p15.6.m1.1.1.2" xref="p15.6.m1.1.1.2.cmml">σ</mi><mi id="p15.6.m1.1.1.3" xref="p15.6.m1.1.1.3.cmml">α</mi></msup><annotation-xml encoding="MathML-Content" id="p15.6.m1.1b"><apply id="p15.6.m1.1.1.cmml" xref="p15.6.m1.1.1"><csymbol cd="ambiguous" id="p15.6.m1.1.1.1.cmml" xref="p15.6.m1.1.1">superscript</csymbol><ci id="p15.6.m1.1.1.2.cmml" xref="p15.6.m1.1.1.2">𝜎</ci><ci id="p15.6.m1.1.1.3.cmml" xref="p15.6.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p15.6.m1.1c">\sigma^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="p15.6.m1.1d">italic_σ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\alpha=1,2,3" class="ltx_Math" display="inline" id="p15.7.m2.3"><semantics id="p15.7.m2.3a"><mrow id="p15.7.m2.3.4" xref="p15.7.m2.3.4.cmml"><mi id="p15.7.m2.3.4.2" xref="p15.7.m2.3.4.2.cmml">α</mi><mo id="p15.7.m2.3.4.1" 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encoding="application/x-llamapun" id="p15.7.m2.3d">italic_α = 1 , 2 , 3</annotation></semantics></math> are standard Pauli matrices. This class of solutions coincides with those two-qubit circuits featured in chiral solitons, which are dual unitary <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib53" title="">53</a>]</cite>.</p> </div> <div class="ltx_para" id="p16"> <p class="ltx_p" id="p16.6">Compared to those previously known solvable initial states for dual-unitary circuits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib26" title="">26</a>]</cite>, our findings suggest a new initial state (<math alttext="\otimes_{x<0}\ket{0}_{x}" class="ltx_Math" display="inline" id="p16.1.m1.1"><semantics id="p16.1.m1.1a"><mrow id="p16.1.m1.1.2" xref="p16.1.m1.1.2.cmml"><mi id="p16.1.m1.1.2.2" xref="p16.1.m1.1.2.2.cmml"></mi><msub id="p16.1.m1.1.2.1" xref="p16.1.m1.1.2.1.cmml"><mo id="p16.1.m1.1.2.1.2" lspace="0.222em" rspace="0.222em" xref="p16.1.m1.1.2.1.2.cmml">⊗</mo><mrow id="p16.1.m1.1.2.1.3" xref="p16.1.m1.1.2.1.3.cmml"><mi id="p16.1.m1.1.2.1.3.2" xref="p16.1.m1.1.2.1.3.2.cmml">x</mi><mo id="p16.1.m1.1.2.1.3.1" xref="p16.1.m1.1.2.1.3.1.cmml"><</mo><mn id="p16.1.m1.1.2.1.3.3" xref="p16.1.m1.1.2.1.3.3.cmml">0</mn></mrow></msub><msub id="p16.1.m1.1.2.3" xref="p16.1.m1.1.2.3.cmml"><mrow id="p16.1.m1.1.1.3" xref="p16.1.m1.1.1.2.cmml"><mo id="p16.1.m1.1.1.3.1" stretchy="false" xref="p16.1.m1.1.1.2.1.cmml">|</mo><mn id="p16.1.m1.1.1.1.1" xref="p16.1.m1.1.1.1.1.cmml">0</mn><mo id="p16.1.m1.1.1.3.2" stretchy="false" xref="p16.1.m1.1.1.2.1.cmml">⟩</mo></mrow><mi id="p16.1.m1.1.2.3.2" xref="p16.1.m1.1.2.3.2.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="p16.1.m1.1b"><apply id="p16.1.m1.1.2.cmml" xref="p16.1.m1.1.2"><apply id="p16.1.m1.1.2.1.cmml" xref="p16.1.m1.1.2.1"><csymbol cd="ambiguous" id="p16.1.m1.1.2.1.1.cmml" xref="p16.1.m1.1.2.1">subscript</csymbol><csymbol cd="latexml" id="p16.1.m1.1.2.1.2.cmml" xref="p16.1.m1.1.2.1.2">tensor-product</csymbol><apply id="p16.1.m1.1.2.1.3.cmml" xref="p16.1.m1.1.2.1.3"><lt id="p16.1.m1.1.2.1.3.1.cmml" xref="p16.1.m1.1.2.1.3.1"></lt><ci id="p16.1.m1.1.2.1.3.2.cmml" xref="p16.1.m1.1.2.1.3.2">𝑥</ci><cn id="p16.1.m1.1.2.1.3.3.cmml" type="integer" xref="p16.1.m1.1.2.1.3.3">0</cn></apply></apply><csymbol cd="latexml" id="p16.1.m1.1.2.2.cmml" xref="p16.1.m1.1.2.2">absent</csymbol><apply id="p16.1.m1.1.2.3.cmml" xref="p16.1.m1.1.2.3"><csymbol cd="ambiguous" id="p16.1.m1.1.2.3.1.cmml" xref="p16.1.m1.1.2.3">subscript</csymbol><apply id="p16.1.m1.1.1.2.cmml" xref="p16.1.m1.1.1.3"><csymbol cd="latexml" id="p16.1.m1.1.1.2.1.cmml" xref="p16.1.m1.1.1.3.1">ket</csymbol><cn id="p16.1.m1.1.1.1.1.cmml" type="integer" xref="p16.1.m1.1.1.1.1">0</cn></apply><ci id="p16.1.m1.1.2.3.2.cmml" xref="p16.1.m1.1.2.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p16.1.m1.1c">\otimes_{x<0}\ket{0}_{x}</annotation><annotation encoding="application/x-llamapun" id="p16.1.m1.1d">⊗ start_POSTSUBSCRIPT italic_x < 0 end_POSTSUBSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>) allowing for exact influence matrix in the product-state form. The corresponding one-site quantum channel can be obtained by substituting the form of <math alttext="A" class="ltx_Math" display="inline" id="p16.2.m2.1"><semantics id="p16.2.m2.1a"><mi id="p16.2.m2.1.1" xref="p16.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="p16.2.m2.1b"><ci id="p16.2.m2.1.1.cmml" xref="p16.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="p16.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="p16.2.m2.1d">italic_A</annotation></semantics></math> into Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E7" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">7</span></a>): <math alttext="\mathcal{M}[\rho_{R}]=(\ket{0}\bra{0})_{x=0}\otimes\text{Tr}_{x=0}[\rho_{R}]" class="ltx_Math" display="inline" id="p16.3.m3.5"><semantics id="p16.3.m3.5a"><mrow id="p16.3.m3.5.5" xref="p16.3.m3.5.5.cmml"><mrow id="p16.3.m3.3.3.1" xref="p16.3.m3.3.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p16.3.m3.3.3.1.3" xref="p16.3.m3.3.3.1.3.cmml">ℳ</mi><mo id="p16.3.m3.3.3.1.2" xref="p16.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="p16.3.m3.3.3.1.1.1" xref="p16.3.m3.3.3.1.1.2.cmml"><mo id="p16.3.m3.3.3.1.1.1.2" stretchy="false" xref="p16.3.m3.3.3.1.1.2.1.cmml">[</mo><msub id="p16.3.m3.3.3.1.1.1.1" xref="p16.3.m3.3.3.1.1.1.1.cmml"><mi id="p16.3.m3.3.3.1.1.1.1.2" xref="p16.3.m3.3.3.1.1.1.1.2.cmml">ρ</mi><mi id="p16.3.m3.3.3.1.1.1.1.3" xref="p16.3.m3.3.3.1.1.1.1.3.cmml">R</mi></msub><mo 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xref="p16.3.m3.5.5.3.2.1.1.2">𝜌</ci><ci id="p16.3.m3.5.5.3.2.1.1.3.cmml" xref="p16.3.m3.5.5.3.2.1.1.3">𝑅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p16.3.m3.5c">\mathcal{M}[\rho_{R}]=(\ket{0}\bra{0})_{x=0}\otimes\text{Tr}_{x=0}[\rho_{R}]</annotation><annotation encoding="application/x-llamapun" id="p16.3.m3.5d">caligraphic_M [ italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ] = ( | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ) start_POSTSUBSCRIPT italic_x = 0 end_POSTSUBSCRIPT ⊗ Tr start_POSTSUBSCRIPT italic_x = 0 end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ]</annotation></semantics></math>. Hence, the left bath acts as the boundary resetting toward <math alttext="\ket{0}" class="ltx_Math" display="inline" id="p16.4.m4.1"><semantics id="p16.4.m4.1a"><mrow id="p16.4.m4.1.1.3" xref="p16.4.m4.1.1.2.cmml"><mo id="p16.4.m4.1.1.3.1" stretchy="false" xref="p16.4.m4.1.1.2.1.cmml">|</mo><mn id="p16.4.m4.1.1.1.1" xref="p16.4.m4.1.1.1.1.cmml">0</mn><mo id="p16.4.m4.1.1.3.2" stretchy="false" xref="p16.4.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p16.4.m4.1b"><apply id="p16.4.m4.1.1.2.cmml" xref="p16.4.m4.1.1.3"><csymbol cd="latexml" id="p16.4.m4.1.1.2.1.cmml" xref="p16.4.m4.1.1.3.1">ket</csymbol><cn id="p16.4.m4.1.1.1.1.cmml" type="integer" xref="p16.4.m4.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p16.4.m4.1c">\ket{0}</annotation><annotation encoding="application/x-llamapun" id="p16.4.m4.1d">| start_ARG 0 end_ARG ⟩</annotation></semantics></math>, and thus the entanglement between two regions never grows up. In addition, we point out that for the solutions Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E11" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">11</span></a>), we can independently choose the one-site initial state as <math alttext="\ket{0}" class="ltx_Math" display="inline" id="p16.5.m5.1"><semantics id="p16.5.m5.1a"><mrow id="p16.5.m5.1.1.3" xref="p16.5.m5.1.1.2.cmml"><mo id="p16.5.m5.1.1.3.1" stretchy="false" xref="p16.5.m5.1.1.2.1.cmml">|</mo><mn id="p16.5.m5.1.1.1.1" xref="p16.5.m5.1.1.1.1.cmml">0</mn><mo id="p16.5.m5.1.1.3.2" stretchy="false" xref="p16.5.m5.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p16.5.m5.1b"><apply id="p16.5.m5.1.1.2.cmml" xref="p16.5.m5.1.1.3"><csymbol cd="latexml" id="p16.5.m5.1.1.2.1.cmml" xref="p16.5.m5.1.1.3.1">ket</csymbol><cn id="p16.5.m5.1.1.1.1.cmml" type="integer" xref="p16.5.m5.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p16.5.m5.1c">\ket{0}</annotation><annotation encoding="application/x-llamapun" id="p16.5.m5.1d">| start_ARG 0 end_ARG ⟩</annotation></semantics></math> or <math alttext="\ket{1}" class="ltx_Math" display="inline" id="p16.6.m6.1"><semantics id="p16.6.m6.1a"><mrow id="p16.6.m6.1.1.3" xref="p16.6.m6.1.1.2.cmml"><mo id="p16.6.m6.1.1.3.1" stretchy="false" xref="p16.6.m6.1.1.2.1.cmml">|</mo><mn id="p16.6.m6.1.1.1.1" xref="p16.6.m6.1.1.1.1.cmml">1</mn><mo id="p16.6.m6.1.1.3.2" stretchy="false" xref="p16.6.m6.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p16.6.m6.1b"><apply id="p16.6.m6.1.1.2.cmml" xref="p16.6.m6.1.1.3"><csymbol cd="latexml" id="p16.6.m6.1.1.2.1.cmml" xref="p16.6.m6.1.1.3.1">ket</csymbol><cn id="p16.6.m6.1.1.1.1.cmml" type="integer" xref="p16.6.m6.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p16.6.m6.1c">\ket{1}</annotation><annotation encoding="application/x-llamapun" id="p16.6.m6.1d">| start_ARG 1 end_ARG ⟩</annotation></semantics></math> while holding the solvability of the influence matrix.</p> </div> <div class="ltx_para" id="p17"> <p class="ltx_p" id="p17.4"><span class="ltx_text ltx_font_italic" id="p17.4.1">Example II</span>—Next, we present a (nonexhaustive) parameterization for solutions with <math alttext="q=4,\tilde{q}=2" class="ltx_Math" display="inline" id="p17.1.m1.2"><semantics id="p17.1.m1.2a"><mrow id="p17.1.m1.2.2.2" xref="p17.1.m1.2.2.3.cmml"><mrow id="p17.1.m1.1.1.1.1" xref="p17.1.m1.1.1.1.1.cmml"><mi id="p17.1.m1.1.1.1.1.2" xref="p17.1.m1.1.1.1.1.2.cmml">q</mi><mo id="p17.1.m1.1.1.1.1.1" xref="p17.1.m1.1.1.1.1.1.cmml">=</mo><mn id="p17.1.m1.1.1.1.1.3" xref="p17.1.m1.1.1.1.1.3.cmml">4</mn></mrow><mo id="p17.1.m1.2.2.2.3" xref="p17.1.m1.2.2.3a.cmml">,</mo><mrow id="p17.1.m1.2.2.2.2" xref="p17.1.m1.2.2.2.2.cmml"><mover accent="true" id="p17.1.m1.2.2.2.2.2" xref="p17.1.m1.2.2.2.2.2.cmml"><mi id="p17.1.m1.2.2.2.2.2.2" xref="p17.1.m1.2.2.2.2.2.2.cmml">q</mi><mo 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xref="p17.1.m1.2.2.2.2.2.2">𝑞</ci></apply><cn id="p17.1.m1.2.2.2.2.3.cmml" type="integer" xref="p17.1.m1.2.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.1.m1.2c">q=4,\tilde{q}=2</annotation><annotation encoding="application/x-llamapun" id="p17.1.m1.2d">italic_q = 4 , over~ start_ARG italic_q end_ARG = 2</annotation></semantics></math>, where <math alttext="\mathcal{H}_{A}" class="ltx_Math" display="inline" id="p17.2.m2.1"><semantics id="p17.2.m2.1a"><msub id="p17.2.m2.1.1" xref="p17.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p17.2.m2.1.1.2" xref="p17.2.m2.1.1.2.cmml">ℋ</mi><mi id="p17.2.m2.1.1.3" xref="p17.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="p17.2.m2.1b"><apply id="p17.2.m2.1.1.cmml" xref="p17.2.m2.1.1"><csymbol cd="ambiguous" id="p17.2.m2.1.1.1.cmml" xref="p17.2.m2.1.1">subscript</csymbol><ci id="p17.2.m2.1.1.2.cmml" xref="p17.2.m2.1.1.2">ℋ</ci><ci id="p17.2.m2.1.1.3.cmml" xref="p17.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.2.m2.1c">\mathcal{H}_{A}</annotation><annotation encoding="application/x-llamapun" id="p17.2.m2.1d">caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> is spanned by <math alttext="\ket{0}" class="ltx_Math" display="inline" id="p17.3.m3.1"><semantics id="p17.3.m3.1a"><mrow id="p17.3.m3.1.1.3" xref="p17.3.m3.1.1.2.cmml"><mo id="p17.3.m3.1.1.3.1" stretchy="false" xref="p17.3.m3.1.1.2.1.cmml">|</mo><mn id="p17.3.m3.1.1.1.1" xref="p17.3.m3.1.1.1.1.cmml">0</mn><mo id="p17.3.m3.1.1.3.2" stretchy="false" xref="p17.3.m3.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p17.3.m3.1b"><apply id="p17.3.m3.1.1.2.cmml" xref="p17.3.m3.1.1.3"><csymbol cd="latexml" id="p17.3.m3.1.1.2.1.cmml" xref="p17.3.m3.1.1.3.1">ket</csymbol><cn id="p17.3.m3.1.1.1.1.cmml" type="integer" xref="p17.3.m3.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.3.m3.1c">\ket{0}</annotation><annotation encoding="application/x-llamapun" id="p17.3.m3.1d">| start_ARG 0 end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{1}" class="ltx_Math" display="inline" id="p17.4.m4.1"><semantics id="p17.4.m4.1a"><mrow id="p17.4.m4.1.1.3" xref="p17.4.m4.1.1.2.cmml"><mo id="p17.4.m4.1.1.3.1" stretchy="false" xref="p17.4.m4.1.1.2.1.cmml">|</mo><mn id="p17.4.m4.1.1.1.1" xref="p17.4.m4.1.1.1.1.cmml">1</mn><mo id="p17.4.m4.1.1.3.2" stretchy="false" xref="p17.4.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p17.4.m4.1b"><apply id="p17.4.m4.1.1.2.cmml" xref="p17.4.m4.1.1.3"><csymbol cd="latexml" id="p17.4.m4.1.1.2.1.cmml" xref="p17.4.m4.1.1.3.1">ket</csymbol><cn id="p17.4.m4.1.1.1.1.cmml" type="integer" xref="p17.4.m4.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.4.m4.1c">\ket{1}</annotation><annotation encoding="application/x-llamapun" id="p17.4.m4.1d">| start_ARG 1 end_ARG ⟩</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S0.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="U=e^{i\phi}W_{2}SW_{1}(I\otimes v)." class="ltx_Math" display="block" id="S0.E12.m1.1"><semantics id="S0.E12.m1.1a"><mrow id="S0.E12.m1.1.1.1" xref="S0.E12.m1.1.1.1.1.cmml"><mrow id="S0.E12.m1.1.1.1.1" xref="S0.E12.m1.1.1.1.1.cmml"><mi id="S0.E12.m1.1.1.1.1.3" xref="S0.E12.m1.1.1.1.1.3.cmml">U</mi><mo id="S0.E12.m1.1.1.1.1.2" xref="S0.E12.m1.1.1.1.1.2.cmml">=</mo><mrow id="S0.E12.m1.1.1.1.1.1" xref="S0.E12.m1.1.1.1.1.1.cmml"><msup id="S0.E12.m1.1.1.1.1.1.3" xref="S0.E12.m1.1.1.1.1.1.3.cmml"><mi id="S0.E12.m1.1.1.1.1.1.3.2" xref="S0.E12.m1.1.1.1.1.1.3.2.cmml">e</mi><mrow id="S0.E12.m1.1.1.1.1.1.3.3" xref="S0.E12.m1.1.1.1.1.1.3.3.cmml"><mi id="S0.E12.m1.1.1.1.1.1.3.3.2" 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xref="S0.E12.m1.1.1.1.1.3">𝑈</ci><apply id="S0.E12.m1.1.1.1.1.1.cmml" xref="S0.E12.m1.1.1.1.1.1"><times id="S0.E12.m1.1.1.1.1.1.2.cmml" xref="S0.E12.m1.1.1.1.1.1.2"></times><apply id="S0.E12.m1.1.1.1.1.1.3.cmml" xref="S0.E12.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S0.E12.m1.1.1.1.1.1.3.1.cmml" xref="S0.E12.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S0.E12.m1.1.1.1.1.1.3.2.cmml" xref="S0.E12.m1.1.1.1.1.1.3.2">𝑒</ci><apply id="S0.E12.m1.1.1.1.1.1.3.3.cmml" xref="S0.E12.m1.1.1.1.1.1.3.3"><times id="S0.E12.m1.1.1.1.1.1.3.3.1.cmml" xref="S0.E12.m1.1.1.1.1.1.3.3.1"></times><ci id="S0.E12.m1.1.1.1.1.1.3.3.2.cmml" xref="S0.E12.m1.1.1.1.1.1.3.3.2">𝑖</ci><ci id="S0.E12.m1.1.1.1.1.1.3.3.3.cmml" xref="S0.E12.m1.1.1.1.1.1.3.3.3">italic-ϕ</ci></apply></apply><apply id="S0.E12.m1.1.1.1.1.1.4.cmml" xref="S0.E12.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S0.E12.m1.1.1.1.1.1.4.1.cmml" xref="S0.E12.m1.1.1.1.1.1.4">subscript</csymbol><ci id="S0.E12.m1.1.1.1.1.1.4.2.cmml" 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id="S0.E12.m1.1c">U=e^{i\phi}W_{2}SW_{1}(I\otimes v).</annotation><annotation encoding="application/x-llamapun" id="S0.E12.m1.1d">italic_U = italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_I ⊗ italic_v ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p17.8">Here, <math alttext="v\in\text{SU}(4)" class="ltx_Math" display="inline" id="p17.5.m1.1"><semantics id="p17.5.m1.1a"><mrow id="p17.5.m1.1.2" xref="p17.5.m1.1.2.cmml"><mi id="p17.5.m1.1.2.2" xref="p17.5.m1.1.2.2.cmml">v</mi><mo id="p17.5.m1.1.2.1" xref="p17.5.m1.1.2.1.cmml">∈</mo><mrow id="p17.5.m1.1.2.3" xref="p17.5.m1.1.2.3.cmml"><mtext id="p17.5.m1.1.2.3.2" xref="p17.5.m1.1.2.3.2a.cmml">SU</mtext><mo id="p17.5.m1.1.2.3.1" xref="p17.5.m1.1.2.3.1.cmml">⁢</mo><mrow id="p17.5.m1.1.2.3.3.2" xref="p17.5.m1.1.2.3.cmml"><mo id="p17.5.m1.1.2.3.3.2.1" stretchy="false" xref="p17.5.m1.1.2.3.cmml">(</mo><mn id="p17.5.m1.1.1" xref="p17.5.m1.1.1.cmml">4</mn><mo id="p17.5.m1.1.2.3.3.2.2" stretchy="false" xref="p17.5.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.5.m1.1b"><apply id="p17.5.m1.1.2.cmml" xref="p17.5.m1.1.2"><in id="p17.5.m1.1.2.1.cmml" xref="p17.5.m1.1.2.1"></in><ci id="p17.5.m1.1.2.2.cmml" xref="p17.5.m1.1.2.2">𝑣</ci><apply id="p17.5.m1.1.2.3.cmml" xref="p17.5.m1.1.2.3"><times id="p17.5.m1.1.2.3.1.cmml" xref="p17.5.m1.1.2.3.1"></times><ci id="p17.5.m1.1.2.3.2a.cmml" xref="p17.5.m1.1.2.3.2"><mtext id="p17.5.m1.1.2.3.2.cmml" xref="p17.5.m1.1.2.3.2">SU</mtext></ci><cn id="p17.5.m1.1.1.cmml" type="integer" xref="p17.5.m1.1.1">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.5.m1.1c">v\in\text{SU}(4)</annotation><annotation encoding="application/x-llamapun" id="p17.5.m1.1d">italic_v ∈ SU ( 4 )</annotation></semantics></math>, <math alttext="S" class="ltx_Math" display="inline" id="p17.6.m2.1"><semantics id="p17.6.m2.1a"><mi id="p17.6.m2.1.1" xref="p17.6.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="p17.6.m2.1b"><ci id="p17.6.m2.1.1.cmml" xref="p17.6.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="p17.6.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="p17.6.m2.1d">italic_S</annotation></semantics></math> is the <math alttext="{\mathrm{SWAP}}" class="ltx_Math" display="inline" id="p17.7.m3.1"><semantics id="p17.7.m3.1a"><mi id="p17.7.m3.1.1" mathsize="70%" xref="p17.7.m3.1.1.cmml">SWAP</mi><annotation-xml encoding="MathML-Content" id="p17.7.m3.1b"><ci id="p17.7.m3.1.1.cmml" xref="p17.7.m3.1.1">SWAP</ci></annotation-xml><annotation encoding="application/x-tex" id="p17.7.m3.1c">{\mathrm{SWAP}}</annotation><annotation encoding="application/x-llamapun" id="p17.7.m3.1d">roman_SWAP</annotation></semantics></math> gate, <math alttext="W_{1,2}" class="ltx_Math" display="inline" id="p17.8.m4.2"><semantics id="p17.8.m4.2a"><msub id="p17.8.m4.2.3" xref="p17.8.m4.2.3.cmml"><mi id="p17.8.m4.2.3.2" xref="p17.8.m4.2.3.2.cmml">W</mi><mrow id="p17.8.m4.2.2.2.4" xref="p17.8.m4.2.2.2.3.cmml"><mn id="p17.8.m4.1.1.1.1" xref="p17.8.m4.1.1.1.1.cmml">1</mn><mo id="p17.8.m4.2.2.2.4.1" xref="p17.8.m4.2.2.2.3.cmml">,</mo><mn id="p17.8.m4.2.2.2.2" xref="p17.8.m4.2.2.2.2.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="p17.8.m4.2b"><apply id="p17.8.m4.2.3.cmml" xref="p17.8.m4.2.3"><csymbol cd="ambiguous" id="p17.8.m4.2.3.1.cmml" xref="p17.8.m4.2.3">subscript</csymbol><ci id="p17.8.m4.2.3.2.cmml" xref="p17.8.m4.2.3.2">𝑊</ci><list id="p17.8.m4.2.2.2.3.cmml" xref="p17.8.m4.2.2.2.4"><cn id="p17.8.m4.1.1.1.1.cmml" 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xref="S0.E13.m1.2.2.1.1">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.E13.m1.8c">W_{1,2}=\sum_{a=0}^{3}f_{1,2}^{(a)}\otimes\ket{a}\bra{a},</annotation><annotation encoding="application/x-llamapun" id="S0.E13.m1.8d">italic_W start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_a = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ⊗ | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="p17.21">where <math alttext="f_{1}^{(a)}=\left(\begin{array}[]{cc}I_{2}&0\\ 0&g^{(a)}\end{array}\right)" 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xref="p17.9.m1.2.2.1.1.1.cmml">)</mo></mrow></msup></mtd></mtr></mtable><mo id="p17.9.m1.2.3.3.2.2" xref="p17.9.m1.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.9.m1.2b"><apply id="p17.9.m1.2.3.cmml" xref="p17.9.m1.2.3"><eq id="p17.9.m1.2.3.1.cmml" xref="p17.9.m1.2.3.1"></eq><apply id="p17.9.m1.2.3.2.cmml" xref="p17.9.m1.2.3.2"><csymbol cd="ambiguous" id="p17.9.m1.2.3.2.1.cmml" xref="p17.9.m1.2.3.2">superscript</csymbol><apply id="p17.9.m1.2.3.2.2.cmml" xref="p17.9.m1.2.3.2"><csymbol cd="ambiguous" id="p17.9.m1.2.3.2.2.1.cmml" xref="p17.9.m1.2.3.2">subscript</csymbol><ci id="p17.9.m1.2.3.2.2.2.cmml" xref="p17.9.m1.2.3.2.2.2">𝑓</ci><cn id="p17.9.m1.2.3.2.2.3.cmml" type="integer" xref="p17.9.m1.2.3.2.2.3">1</cn></apply><ci id="p17.9.m1.1.1.1.1.cmml" xref="p17.9.m1.1.1.1.1">𝑎</ci></apply><matrix id="p17.9.m1.2.2.cmml" xref="p17.9.m1.2.3.3.2"><matrixrow id="p17.9.m1.2.2a.cmml" xref="p17.9.m1.2.3.3.2"><apply id="p17.9.m1.2.2.2.1.1.cmml" xref="p17.9.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="p17.9.m1.2.2.2.1.1.1.cmml" xref="p17.9.m1.2.2.2.1.1">subscript</csymbol><ci id="p17.9.m1.2.2.2.1.1.2.cmml" xref="p17.9.m1.2.2.2.1.1.2">𝐼</ci><cn id="p17.9.m1.2.2.2.1.1.3.cmml" type="integer" xref="p17.9.m1.2.2.2.1.1.3">2</cn></apply><cn id="p17.9.m1.2.2.2.2.1.cmml" type="integer" xref="p17.9.m1.2.2.2.2.1">0</cn></matrixrow><matrixrow id="p17.9.m1.2.2b.cmml" xref="p17.9.m1.2.3.3.2"><cn id="p17.9.m1.2.2.1.2.1.cmml" type="integer" xref="p17.9.m1.2.2.1.2.1">0</cn><apply id="p17.9.m1.2.2.1.1.1.cmml" xref="p17.9.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="p17.9.m1.2.2.1.1.1.2.cmml" xref="p17.9.m1.2.2.1.1.1">superscript</csymbol><ci id="p17.9.m1.2.2.1.1.1.3.cmml" xref="p17.9.m1.2.2.1.1.1.3">𝑔</ci><ci id="p17.9.m1.2.2.1.1.1.1.1.1.cmml" xref="p17.9.m1.2.2.1.1.1.1.1.1">𝑎</ci></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.9.m1.2c">f_{1}^{(a)}=\left(\begin{array}[]{cc}I_{2}&0\\ 0&g^{(a)}\end{array}\right)</annotation><annotation encoding="application/x-llamapun" id="p17.9.m1.2d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT = ( start_ARRAY start_ROW start_CELL italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_g start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY )</annotation></semantics></math>, <math alttext="g^{(a)}\in\text{SU}(2)" class="ltx_Math" display="inline" id="p17.10.m2.2"><semantics id="p17.10.m2.2a"><mrow id="p17.10.m2.2.3" xref="p17.10.m2.2.3.cmml"><msup id="p17.10.m2.2.3.2" xref="p17.10.m2.2.3.2.cmml"><mi id="p17.10.m2.2.3.2.2" xref="p17.10.m2.2.3.2.2.cmml">g</mi><mrow id="p17.10.m2.1.1.1.3" xref="p17.10.m2.2.3.2.cmml"><mo id="p17.10.m2.1.1.1.3.1" stretchy="false" xref="p17.10.m2.2.3.2.cmml">(</mo><mi id="p17.10.m2.1.1.1.1" xref="p17.10.m2.1.1.1.1.cmml">a</mi><mo id="p17.10.m2.1.1.1.3.2" stretchy="false" xref="p17.10.m2.2.3.2.cmml">)</mo></mrow></msup><mo id="p17.10.m2.2.3.1" xref="p17.10.m2.2.3.1.cmml">∈</mo><mrow id="p17.10.m2.2.3.3" xref="p17.10.m2.2.3.3.cmml"><mtext id="p17.10.m2.2.3.3.2" xref="p17.10.m2.2.3.3.2a.cmml">SU</mtext><mo id="p17.10.m2.2.3.3.1" xref="p17.10.m2.2.3.3.1.cmml">⁢</mo><mrow id="p17.10.m2.2.3.3.3.2" xref="p17.10.m2.2.3.3.cmml"><mo id="p17.10.m2.2.3.3.3.2.1" stretchy="false" xref="p17.10.m2.2.3.3.cmml">(</mo><mn id="p17.10.m2.2.2" xref="p17.10.m2.2.2.cmml">2</mn><mo id="p17.10.m2.2.3.3.3.2.2" stretchy="false" xref="p17.10.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.10.m2.2b"><apply id="p17.10.m2.2.3.cmml" xref="p17.10.m2.2.3"><in id="p17.10.m2.2.3.1.cmml" xref="p17.10.m2.2.3.1"></in><apply id="p17.10.m2.2.3.2.cmml" xref="p17.10.m2.2.3.2"><csymbol cd="ambiguous" id="p17.10.m2.2.3.2.1.cmml" xref="p17.10.m2.2.3.2">superscript</csymbol><ci id="p17.10.m2.2.3.2.2.cmml" xref="p17.10.m2.2.3.2.2">𝑔</ci><ci id="p17.10.m2.1.1.1.1.cmml" xref="p17.10.m2.1.1.1.1">𝑎</ci></apply><apply id="p17.10.m2.2.3.3.cmml" xref="p17.10.m2.2.3.3"><times id="p17.10.m2.2.3.3.1.cmml" xref="p17.10.m2.2.3.3.1"></times><ci id="p17.10.m2.2.3.3.2a.cmml" xref="p17.10.m2.2.3.3.2"><mtext id="p17.10.m2.2.3.3.2.cmml" xref="p17.10.m2.2.3.3.2">SU</mtext></ci><cn id="p17.10.m2.2.2.cmml" type="integer" xref="p17.10.m2.2.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.10.m2.2c">g^{(a)}\in\text{SU}(2)</annotation><annotation encoding="application/x-llamapun" id="p17.10.m2.2d">italic_g start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ∈ SU ( 2 )</annotation></semantics></math>, and <math alttext="f_{2}^{(a)}\in\text{SU}(4)" class="ltx_Math" display="inline" id="p17.11.m3.2"><semantics id="p17.11.m3.2a"><mrow id="p17.11.m3.2.3" xref="p17.11.m3.2.3.cmml"><msubsup id="p17.11.m3.2.3.2" xref="p17.11.m3.2.3.2.cmml"><mi id="p17.11.m3.2.3.2.2.2" xref="p17.11.m3.2.3.2.2.2.cmml">f</mi><mn id="p17.11.m3.2.3.2.2.3" xref="p17.11.m3.2.3.2.2.3.cmml">2</mn><mrow id="p17.11.m3.1.1.1.3" xref="p17.11.m3.2.3.2.cmml"><mo id="p17.11.m3.1.1.1.3.1" stretchy="false" xref="p17.11.m3.2.3.2.cmml">(</mo><mi id="p17.11.m3.1.1.1.1" xref="p17.11.m3.1.1.1.1.cmml">a</mi><mo id="p17.11.m3.1.1.1.3.2" stretchy="false" xref="p17.11.m3.2.3.2.cmml">)</mo></mrow></msubsup><mo id="p17.11.m3.2.3.1" xref="p17.11.m3.2.3.1.cmml">∈</mo><mrow id="p17.11.m3.2.3.3" xref="p17.11.m3.2.3.3.cmml"><mtext id="p17.11.m3.2.3.3.2" xref="p17.11.m3.2.3.3.2a.cmml">SU</mtext><mo id="p17.11.m3.2.3.3.1" xref="p17.11.m3.2.3.3.1.cmml">⁢</mo><mrow id="p17.11.m3.2.3.3.3.2" xref="p17.11.m3.2.3.3.cmml"><mo id="p17.11.m3.2.3.3.3.2.1" stretchy="false" xref="p17.11.m3.2.3.3.cmml">(</mo><mn id="p17.11.m3.2.2" xref="p17.11.m3.2.2.cmml">4</mn><mo id="p17.11.m3.2.3.3.3.2.2" stretchy="false" xref="p17.11.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.11.m3.2b"><apply id="p17.11.m3.2.3.cmml" xref="p17.11.m3.2.3"><in id="p17.11.m3.2.3.1.cmml" xref="p17.11.m3.2.3.1"></in><apply id="p17.11.m3.2.3.2.cmml" xref="p17.11.m3.2.3.2"><csymbol cd="ambiguous" id="p17.11.m3.2.3.2.1.cmml" xref="p17.11.m3.2.3.2">superscript</csymbol><apply id="p17.11.m3.2.3.2.2.cmml" xref="p17.11.m3.2.3.2"><csymbol cd="ambiguous" id="p17.11.m3.2.3.2.2.1.cmml" xref="p17.11.m3.2.3.2">subscript</csymbol><ci id="p17.11.m3.2.3.2.2.2.cmml" xref="p17.11.m3.2.3.2.2.2">𝑓</ci><cn id="p17.11.m3.2.3.2.2.3.cmml" type="integer" xref="p17.11.m3.2.3.2.2.3">2</cn></apply><ci id="p17.11.m3.1.1.1.1.cmml" xref="p17.11.m3.1.1.1.1">𝑎</ci></apply><apply id="p17.11.m3.2.3.3.cmml" xref="p17.11.m3.2.3.3"><times id="p17.11.m3.2.3.3.1.cmml" xref="p17.11.m3.2.3.3.1"></times><ci id="p17.11.m3.2.3.3.2a.cmml" xref="p17.11.m3.2.3.3.2"><mtext id="p17.11.m3.2.3.3.2.cmml" xref="p17.11.m3.2.3.3.2">SU</mtext></ci><cn id="p17.11.m3.2.2.cmml" type="integer" xref="p17.11.m3.2.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.11.m3.2c">f_{2}^{(a)}\in\text{SU}(4)</annotation><annotation encoding="application/x-llamapun" id="p17.11.m3.2d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ∈ SU ( 4 )</annotation></semantics></math> for all <math alttext="a" class="ltx_Math" display="inline" id="p17.12.m4.1"><semantics id="p17.12.m4.1a"><mi id="p17.12.m4.1.1" xref="p17.12.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="p17.12.m4.1b"><ci id="p17.12.m4.1.1.cmml" xref="p17.12.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="p17.12.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="p17.12.m4.1d">italic_a</annotation></semantics></math>. The parametrization can be generalized to higher <math alttext="q" class="ltx_Math" display="inline" id="p17.13.m5.1"><semantics id="p17.13.m5.1a"><mi id="p17.13.m5.1.1" xref="p17.13.m5.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="p17.13.m5.1b"><ci id="p17.13.m5.1.1.cmml" xref="p17.13.m5.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="p17.13.m5.1c">q</annotation><annotation encoding="application/x-llamapun" id="p17.13.m5.1d">italic_q</annotation></semantics></math> and <math alttext="\tilde{q}" class="ltx_Math" display="inline" id="p17.14.m6.1"><semantics id="p17.14.m6.1a"><mover accent="true" id="p17.14.m6.1.1" xref="p17.14.m6.1.1.cmml"><mi id="p17.14.m6.1.1.2" xref="p17.14.m6.1.1.2.cmml">q</mi><mo id="p17.14.m6.1.1.1" xref="p17.14.m6.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="p17.14.m6.1b"><apply id="p17.14.m6.1.1.cmml" xref="p17.14.m6.1.1"><ci id="p17.14.m6.1.1.1.cmml" xref="p17.14.m6.1.1.1">~</ci><ci id="p17.14.m6.1.1.2.cmml" xref="p17.14.m6.1.1.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.14.m6.1c">\tilde{q}</annotation><annotation encoding="application/x-llamapun" id="p17.14.m6.1d">over~ start_ARG italic_q end_ARG</annotation></semantics></math> with slight modifications: <math alttext="v\in\text{SU}(q)" class="ltx_Math" display="inline" id="p17.15.m7.1"><semantics id="p17.15.m7.1a"><mrow id="p17.15.m7.1.2" xref="p17.15.m7.1.2.cmml"><mi id="p17.15.m7.1.2.2" xref="p17.15.m7.1.2.2.cmml">v</mi><mo id="p17.15.m7.1.2.1" xref="p17.15.m7.1.2.1.cmml">∈</mo><mrow id="p17.15.m7.1.2.3" xref="p17.15.m7.1.2.3.cmml"><mtext id="p17.15.m7.1.2.3.2" xref="p17.15.m7.1.2.3.2a.cmml">SU</mtext><mo id="p17.15.m7.1.2.3.1" xref="p17.15.m7.1.2.3.1.cmml">⁢</mo><mrow id="p17.15.m7.1.2.3.3.2" xref="p17.15.m7.1.2.3.cmml"><mo id="p17.15.m7.1.2.3.3.2.1" stretchy="false" xref="p17.15.m7.1.2.3.cmml">(</mo><mi id="p17.15.m7.1.1" xref="p17.15.m7.1.1.cmml">q</mi><mo id="p17.15.m7.1.2.3.3.2.2" stretchy="false" xref="p17.15.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.15.m7.1b"><apply id="p17.15.m7.1.2.cmml" xref="p17.15.m7.1.2"><in id="p17.15.m7.1.2.1.cmml" xref="p17.15.m7.1.2.1"></in><ci id="p17.15.m7.1.2.2.cmml" xref="p17.15.m7.1.2.2">𝑣</ci><apply id="p17.15.m7.1.2.3.cmml" xref="p17.15.m7.1.2.3"><times id="p17.15.m7.1.2.3.1.cmml" xref="p17.15.m7.1.2.3.1"></times><ci id="p17.15.m7.1.2.3.2a.cmml" xref="p17.15.m7.1.2.3.2"><mtext id="p17.15.m7.1.2.3.2.cmml" xref="p17.15.m7.1.2.3.2">SU</mtext></ci><ci id="p17.15.m7.1.1.cmml" xref="p17.15.m7.1.1">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.15.m7.1c">v\in\text{SU}(q)</annotation><annotation encoding="application/x-llamapun" id="p17.15.m7.1d">italic_v ∈ SU ( italic_q )</annotation></semantics></math>, <math alttext="f_{1}^{(a)}=I_{\tilde{q}}\oplus g^{(a)}" class="ltx_Math" display="inline" id="p17.16.m8.2"><semantics id="p17.16.m8.2a"><mrow id="p17.16.m8.2.3" xref="p17.16.m8.2.3.cmml"><msubsup id="p17.16.m8.2.3.2" xref="p17.16.m8.2.3.2.cmml"><mi id="p17.16.m8.2.3.2.2.2" xref="p17.16.m8.2.3.2.2.2.cmml">f</mi><mn id="p17.16.m8.2.3.2.2.3" xref="p17.16.m8.2.3.2.2.3.cmml">1</mn><mrow id="p17.16.m8.1.1.1.3" xref="p17.16.m8.2.3.2.cmml"><mo id="p17.16.m8.1.1.1.3.1" stretchy="false" xref="p17.16.m8.2.3.2.cmml">(</mo><mi id="p17.16.m8.1.1.1.1" xref="p17.16.m8.1.1.1.1.cmml">a</mi><mo id="p17.16.m8.1.1.1.3.2" stretchy="false" xref="p17.16.m8.2.3.2.cmml">)</mo></mrow></msubsup><mo id="p17.16.m8.2.3.1" xref="p17.16.m8.2.3.1.cmml">=</mo><mrow id="p17.16.m8.2.3.3" xref="p17.16.m8.2.3.3.cmml"><msub id="p17.16.m8.2.3.3.2" xref="p17.16.m8.2.3.3.2.cmml"><mi id="p17.16.m8.2.3.3.2.2" xref="p17.16.m8.2.3.3.2.2.cmml">I</mi><mover accent="true" id="p17.16.m8.2.3.3.2.3" xref="p17.16.m8.2.3.3.2.3.cmml"><mi id="p17.16.m8.2.3.3.2.3.2" xref="p17.16.m8.2.3.3.2.3.2.cmml">q</mi><mo id="p17.16.m8.2.3.3.2.3.1" xref="p17.16.m8.2.3.3.2.3.1.cmml">~</mo></mover></msub><mo id="p17.16.m8.2.3.3.1" xref="p17.16.m8.2.3.3.1.cmml">⊕</mo><msup id="p17.16.m8.2.3.3.3" xref="p17.16.m8.2.3.3.3.cmml"><mi id="p17.16.m8.2.3.3.3.2" xref="p17.16.m8.2.3.3.3.2.cmml">g</mi><mrow id="p17.16.m8.2.2.1.3" xref="p17.16.m8.2.3.3.3.cmml"><mo id="p17.16.m8.2.2.1.3.1" stretchy="false" xref="p17.16.m8.2.3.3.3.cmml">(</mo><mi id="p17.16.m8.2.2.1.1" xref="p17.16.m8.2.2.1.1.cmml">a</mi><mo id="p17.16.m8.2.2.1.3.2" stretchy="false" xref="p17.16.m8.2.3.3.3.cmml">)</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.16.m8.2b"><apply id="p17.16.m8.2.3.cmml" xref="p17.16.m8.2.3"><eq id="p17.16.m8.2.3.1.cmml" xref="p17.16.m8.2.3.1"></eq><apply id="p17.16.m8.2.3.2.cmml" xref="p17.16.m8.2.3.2"><csymbol cd="ambiguous" id="p17.16.m8.2.3.2.1.cmml" xref="p17.16.m8.2.3.2">superscript</csymbol><apply id="p17.16.m8.2.3.2.2.cmml" xref="p17.16.m8.2.3.2"><csymbol cd="ambiguous" id="p17.16.m8.2.3.2.2.1.cmml" xref="p17.16.m8.2.3.2">subscript</csymbol><ci id="p17.16.m8.2.3.2.2.2.cmml" xref="p17.16.m8.2.3.2.2.2">𝑓</ci><cn id="p17.16.m8.2.3.2.2.3.cmml" type="integer" xref="p17.16.m8.2.3.2.2.3">1</cn></apply><ci id="p17.16.m8.1.1.1.1.cmml" xref="p17.16.m8.1.1.1.1">𝑎</ci></apply><apply id="p17.16.m8.2.3.3.cmml" xref="p17.16.m8.2.3.3"><csymbol cd="latexml" id="p17.16.m8.2.3.3.1.cmml" xref="p17.16.m8.2.3.3.1">direct-sum</csymbol><apply id="p17.16.m8.2.3.3.2.cmml" xref="p17.16.m8.2.3.3.2"><csymbol cd="ambiguous" id="p17.16.m8.2.3.3.2.1.cmml" xref="p17.16.m8.2.3.3.2">subscript</csymbol><ci id="p17.16.m8.2.3.3.2.2.cmml" xref="p17.16.m8.2.3.3.2.2">𝐼</ci><apply id="p17.16.m8.2.3.3.2.3.cmml" xref="p17.16.m8.2.3.3.2.3"><ci id="p17.16.m8.2.3.3.2.3.1.cmml" xref="p17.16.m8.2.3.3.2.3.1">~</ci><ci id="p17.16.m8.2.3.3.2.3.2.cmml" xref="p17.16.m8.2.3.3.2.3.2">𝑞</ci></apply></apply><apply id="p17.16.m8.2.3.3.3.cmml" xref="p17.16.m8.2.3.3.3"><csymbol cd="ambiguous" id="p17.16.m8.2.3.3.3.1.cmml" xref="p17.16.m8.2.3.3.3">superscript</csymbol><ci id="p17.16.m8.2.3.3.3.2.cmml" xref="p17.16.m8.2.3.3.3.2">𝑔</ci><ci id="p17.16.m8.2.2.1.1.cmml" xref="p17.16.m8.2.2.1.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.16.m8.2c">f_{1}^{(a)}=I_{\tilde{q}}\oplus g^{(a)}</annotation><annotation encoding="application/x-llamapun" id="p17.16.m8.2d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT = italic_I start_POSTSUBSCRIPT over~ start_ARG italic_q end_ARG end_POSTSUBSCRIPT ⊕ italic_g start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="g^{(a)}\in\text{SU}(q-\tilde{q})" class="ltx_Math" display="inline" id="p17.17.m9.2"><semantics id="p17.17.m9.2a"><mrow id="p17.17.m9.2.2" xref="p17.17.m9.2.2.cmml"><msup id="p17.17.m9.2.2.3" xref="p17.17.m9.2.2.3.cmml"><mi id="p17.17.m9.2.2.3.2" xref="p17.17.m9.2.2.3.2.cmml">g</mi><mrow id="p17.17.m9.1.1.1.3" xref="p17.17.m9.2.2.3.cmml"><mo id="p17.17.m9.1.1.1.3.1" stretchy="false" xref="p17.17.m9.2.2.3.cmml">(</mo><mi id="p17.17.m9.1.1.1.1" xref="p17.17.m9.1.1.1.1.cmml">a</mi><mo id="p17.17.m9.1.1.1.3.2" stretchy="false" xref="p17.17.m9.2.2.3.cmml">)</mo></mrow></msup><mo id="p17.17.m9.2.2.2" xref="p17.17.m9.2.2.2.cmml">∈</mo><mrow id="p17.17.m9.2.2.1" xref="p17.17.m9.2.2.1.cmml"><mtext id="p17.17.m9.2.2.1.3" xref="p17.17.m9.2.2.1.3a.cmml">SU</mtext><mo id="p17.17.m9.2.2.1.2" xref="p17.17.m9.2.2.1.2.cmml">⁢</mo><mrow id="p17.17.m9.2.2.1.1.1" xref="p17.17.m9.2.2.1.1.1.1.cmml"><mo id="p17.17.m9.2.2.1.1.1.2" stretchy="false" xref="p17.17.m9.2.2.1.1.1.1.cmml">(</mo><mrow id="p17.17.m9.2.2.1.1.1.1" xref="p17.17.m9.2.2.1.1.1.1.cmml"><mi id="p17.17.m9.2.2.1.1.1.1.2" xref="p17.17.m9.2.2.1.1.1.1.2.cmml">q</mi><mo id="p17.17.m9.2.2.1.1.1.1.1" xref="p17.17.m9.2.2.1.1.1.1.1.cmml">−</mo><mover accent="true" id="p17.17.m9.2.2.1.1.1.1.3" xref="p17.17.m9.2.2.1.1.1.1.3.cmml"><mi id="p17.17.m9.2.2.1.1.1.1.3.2" xref="p17.17.m9.2.2.1.1.1.1.3.2.cmml">q</mi><mo id="p17.17.m9.2.2.1.1.1.1.3.1" xref="p17.17.m9.2.2.1.1.1.1.3.1.cmml">~</mo></mover></mrow><mo id="p17.17.m9.2.2.1.1.1.3" stretchy="false" xref="p17.17.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.17.m9.2b"><apply id="p17.17.m9.2.2.cmml" xref="p17.17.m9.2.2"><in id="p17.17.m9.2.2.2.cmml" xref="p17.17.m9.2.2.2"></in><apply id="p17.17.m9.2.2.3.cmml" xref="p17.17.m9.2.2.3"><csymbol cd="ambiguous" id="p17.17.m9.2.2.3.1.cmml" xref="p17.17.m9.2.2.3">superscript</csymbol><ci id="p17.17.m9.2.2.3.2.cmml" xref="p17.17.m9.2.2.3.2">𝑔</ci><ci id="p17.17.m9.1.1.1.1.cmml" xref="p17.17.m9.1.1.1.1">𝑎</ci></apply><apply id="p17.17.m9.2.2.1.cmml" xref="p17.17.m9.2.2.1"><times id="p17.17.m9.2.2.1.2.cmml" xref="p17.17.m9.2.2.1.2"></times><ci id="p17.17.m9.2.2.1.3a.cmml" xref="p17.17.m9.2.2.1.3"><mtext id="p17.17.m9.2.2.1.3.cmml" xref="p17.17.m9.2.2.1.3">SU</mtext></ci><apply id="p17.17.m9.2.2.1.1.1.1.cmml" xref="p17.17.m9.2.2.1.1.1"><minus id="p17.17.m9.2.2.1.1.1.1.1.cmml" xref="p17.17.m9.2.2.1.1.1.1.1"></minus><ci id="p17.17.m9.2.2.1.1.1.1.2.cmml" xref="p17.17.m9.2.2.1.1.1.1.2">𝑞</ci><apply id="p17.17.m9.2.2.1.1.1.1.3.cmml" xref="p17.17.m9.2.2.1.1.1.1.3"><ci id="p17.17.m9.2.2.1.1.1.1.3.1.cmml" xref="p17.17.m9.2.2.1.1.1.1.3.1">~</ci><ci id="p17.17.m9.2.2.1.1.1.1.3.2.cmml" xref="p17.17.m9.2.2.1.1.1.1.3.2">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.17.m9.2c">g^{(a)}\in\text{SU}(q-\tilde{q})</annotation><annotation encoding="application/x-llamapun" id="p17.17.m9.2d">italic_g start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ∈ SU ( italic_q - over~ start_ARG italic_q end_ARG )</annotation></semantics></math>, and <math alttext="f_{2}^{(a)}\in\text{SU}(q)" class="ltx_Math" display="inline" id="p17.18.m10.2"><semantics id="p17.18.m10.2a"><mrow id="p17.18.m10.2.3" xref="p17.18.m10.2.3.cmml"><msubsup id="p17.18.m10.2.3.2" xref="p17.18.m10.2.3.2.cmml"><mi id="p17.18.m10.2.3.2.2.2" xref="p17.18.m10.2.3.2.2.2.cmml">f</mi><mn id="p17.18.m10.2.3.2.2.3" xref="p17.18.m10.2.3.2.2.3.cmml">2</mn><mrow id="p17.18.m10.1.1.1.3" xref="p17.18.m10.2.3.2.cmml"><mo id="p17.18.m10.1.1.1.3.1" stretchy="false" xref="p17.18.m10.2.3.2.cmml">(</mo><mi id="p17.18.m10.1.1.1.1" xref="p17.18.m10.1.1.1.1.cmml">a</mi><mo id="p17.18.m10.1.1.1.3.2" stretchy="false" xref="p17.18.m10.2.3.2.cmml">)</mo></mrow></msubsup><mo id="p17.18.m10.2.3.1" xref="p17.18.m10.2.3.1.cmml">∈</mo><mrow id="p17.18.m10.2.3.3" xref="p17.18.m10.2.3.3.cmml"><mtext id="p17.18.m10.2.3.3.2" xref="p17.18.m10.2.3.3.2a.cmml">SU</mtext><mo id="p17.18.m10.2.3.3.1" xref="p17.18.m10.2.3.3.1.cmml">⁢</mo><mrow id="p17.18.m10.2.3.3.3.2" xref="p17.18.m10.2.3.3.cmml"><mo id="p17.18.m10.2.3.3.3.2.1" stretchy="false" xref="p17.18.m10.2.3.3.cmml">(</mo><mi id="p17.18.m10.2.2" xref="p17.18.m10.2.2.cmml">q</mi><mo id="p17.18.m10.2.3.3.3.2.2" stretchy="false" xref="p17.18.m10.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.18.m10.2b"><apply id="p17.18.m10.2.3.cmml" xref="p17.18.m10.2.3"><in id="p17.18.m10.2.3.1.cmml" xref="p17.18.m10.2.3.1"></in><apply id="p17.18.m10.2.3.2.cmml" xref="p17.18.m10.2.3.2"><csymbol cd="ambiguous" id="p17.18.m10.2.3.2.1.cmml" xref="p17.18.m10.2.3.2">superscript</csymbol><apply id="p17.18.m10.2.3.2.2.cmml" xref="p17.18.m10.2.3.2"><csymbol cd="ambiguous" id="p17.18.m10.2.3.2.2.1.cmml" xref="p17.18.m10.2.3.2">subscript</csymbol><ci id="p17.18.m10.2.3.2.2.2.cmml" xref="p17.18.m10.2.3.2.2.2">𝑓</ci><cn id="p17.18.m10.2.3.2.2.3.cmml" type="integer" xref="p17.18.m10.2.3.2.2.3">2</cn></apply><ci id="p17.18.m10.1.1.1.1.cmml" xref="p17.18.m10.1.1.1.1">𝑎</ci></apply><apply id="p17.18.m10.2.3.3.cmml" xref="p17.18.m10.2.3.3"><times id="p17.18.m10.2.3.3.1.cmml" xref="p17.18.m10.2.3.3.1"></times><ci id="p17.18.m10.2.3.3.2a.cmml" xref="p17.18.m10.2.3.3.2"><mtext id="p17.18.m10.2.3.3.2.cmml" xref="p17.18.m10.2.3.3.2">SU</mtext></ci><ci id="p17.18.m10.2.2.cmml" xref="p17.18.m10.2.2">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.18.m10.2c">f_{2}^{(a)}\in\text{SU}(q)</annotation><annotation encoding="application/x-llamapun" id="p17.18.m10.2d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ∈ SU ( italic_q )</annotation></semantics></math>. Generally, these solutions form an overlapping but different set with dual-unitary circuits. An exception is when <math alttext="q-\tilde{q}=0" class="ltx_Math" display="inline" id="p17.19.m11.1"><semantics id="p17.19.m11.1a"><mrow id="p17.19.m11.1.1" xref="p17.19.m11.1.1.cmml"><mrow id="p17.19.m11.1.1.2" xref="p17.19.m11.1.1.2.cmml"><mi id="p17.19.m11.1.1.2.2" xref="p17.19.m11.1.1.2.2.cmml">q</mi><mo id="p17.19.m11.1.1.2.1" xref="p17.19.m11.1.1.2.1.cmml">−</mo><mover accent="true" id="p17.19.m11.1.1.2.3" xref="p17.19.m11.1.1.2.3.cmml"><mi id="p17.19.m11.1.1.2.3.2" xref="p17.19.m11.1.1.2.3.2.cmml">q</mi><mo id="p17.19.m11.1.1.2.3.1" xref="p17.19.m11.1.1.2.3.1.cmml">~</mo></mover></mrow><mo id="p17.19.m11.1.1.1" xref="p17.19.m11.1.1.1.cmml">=</mo><mn id="p17.19.m11.1.1.3" xref="p17.19.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="p17.19.m11.1b"><apply id="p17.19.m11.1.1.cmml" xref="p17.19.m11.1.1"><eq id="p17.19.m11.1.1.1.cmml" xref="p17.19.m11.1.1.1"></eq><apply id="p17.19.m11.1.1.2.cmml" xref="p17.19.m11.1.1.2"><minus id="p17.19.m11.1.1.2.1.cmml" xref="p17.19.m11.1.1.2.1"></minus><ci id="p17.19.m11.1.1.2.2.cmml" xref="p17.19.m11.1.1.2.2">𝑞</ci><apply id="p17.19.m11.1.1.2.3.cmml" xref="p17.19.m11.1.1.2.3"><ci id="p17.19.m11.1.1.2.3.1.cmml" xref="p17.19.m11.1.1.2.3.1">~</ci><ci id="p17.19.m11.1.1.2.3.2.cmml" xref="p17.19.m11.1.1.2.3.2">𝑞</ci></apply></apply><cn id="p17.19.m11.1.1.3.cmml" type="integer" xref="p17.19.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.19.m11.1c">q-\tilde{q}=0</annotation><annotation encoding="application/x-llamapun" id="p17.19.m11.1d">italic_q - over~ start_ARG italic_q end_ARG = 0</annotation></semantics></math> or <math alttext="1" class="ltx_Math" display="inline" id="p17.20.m12.1"><semantics id="p17.20.m12.1a"><mn id="p17.20.m12.1.1" xref="p17.20.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="p17.20.m12.1b"><cn id="p17.20.m12.1.1.cmml" type="integer" xref="p17.20.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="p17.20.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="p17.20.m12.1d">1</annotation></semantics></math>, where <math alttext="f_{1}^{(a)}=I_{q},W_{1}=I_{q^{2}}" class="ltx_Math" display="inline" id="p17.21.m13.3"><semantics id="p17.21.m13.3a"><mrow id="p17.21.m13.3.3.2" xref="p17.21.m13.3.3.3.cmml"><mrow id="p17.21.m13.2.2.1.1" xref="p17.21.m13.2.2.1.1.cmml"><msubsup id="p17.21.m13.2.2.1.1.2" xref="p17.21.m13.2.2.1.1.2.cmml"><mi id="p17.21.m13.2.2.1.1.2.2.2" xref="p17.21.m13.2.2.1.1.2.2.2.cmml">f</mi><mn id="p17.21.m13.2.2.1.1.2.2.3" xref="p17.21.m13.2.2.1.1.2.2.3.cmml">1</mn><mrow id="p17.21.m13.1.1.1.3" xref="p17.21.m13.2.2.1.1.2.cmml"><mo id="p17.21.m13.1.1.1.3.1" stretchy="false" xref="p17.21.m13.2.2.1.1.2.cmml">(</mo><mi id="p17.21.m13.1.1.1.1" xref="p17.21.m13.1.1.1.1.cmml">a</mi><mo id="p17.21.m13.1.1.1.3.2" stretchy="false" xref="p17.21.m13.2.2.1.1.2.cmml">)</mo></mrow></msubsup><mo id="p17.21.m13.2.2.1.1.1" xref="p17.21.m13.2.2.1.1.1.cmml">=</mo><msub id="p17.21.m13.2.2.1.1.3" xref="p17.21.m13.2.2.1.1.3.cmml"><mi id="p17.21.m13.2.2.1.1.3.2" xref="p17.21.m13.2.2.1.1.3.2.cmml">I</mi><mi id="p17.21.m13.2.2.1.1.3.3" xref="p17.21.m13.2.2.1.1.3.3.cmml">q</mi></msub></mrow><mo id="p17.21.m13.3.3.2.3" xref="p17.21.m13.3.3.3a.cmml">,</mo><mrow id="p17.21.m13.3.3.2.2" xref="p17.21.m13.3.3.2.2.cmml"><msub id="p17.21.m13.3.3.2.2.2" xref="p17.21.m13.3.3.2.2.2.cmml"><mi id="p17.21.m13.3.3.2.2.2.2" xref="p17.21.m13.3.3.2.2.2.2.cmml">W</mi><mn id="p17.21.m13.3.3.2.2.2.3" xref="p17.21.m13.3.3.2.2.2.3.cmml">1</mn></msub><mo id="p17.21.m13.3.3.2.2.1" xref="p17.21.m13.3.3.2.2.1.cmml">=</mo><msub id="p17.21.m13.3.3.2.2.3" xref="p17.21.m13.3.3.2.2.3.cmml"><mi id="p17.21.m13.3.3.2.2.3.2" xref="p17.21.m13.3.3.2.2.3.2.cmml">I</mi><msup id="p17.21.m13.3.3.2.2.3.3" xref="p17.21.m13.3.3.2.2.3.3.cmml"><mi id="p17.21.m13.3.3.2.2.3.3.2" xref="p17.21.m13.3.3.2.2.3.3.2.cmml">q</mi><mn id="p17.21.m13.3.3.2.2.3.3.3" xref="p17.21.m13.3.3.2.2.3.3.3.cmml">2</mn></msup></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="p17.21.m13.3b"><apply id="p17.21.m13.3.3.3.cmml" xref="p17.21.m13.3.3.2"><csymbol cd="ambiguous" id="p17.21.m13.3.3.3a.cmml" xref="p17.21.m13.3.3.2.3">formulae-sequence</csymbol><apply id="p17.21.m13.2.2.1.1.cmml" xref="p17.21.m13.2.2.1.1"><eq id="p17.21.m13.2.2.1.1.1.cmml" xref="p17.21.m13.2.2.1.1.1"></eq><apply id="p17.21.m13.2.2.1.1.2.cmml" xref="p17.21.m13.2.2.1.1.2"><csymbol cd="ambiguous" id="p17.21.m13.2.2.1.1.2.1.cmml" xref="p17.21.m13.2.2.1.1.2">superscript</csymbol><apply id="p17.21.m13.2.2.1.1.2.2.cmml" xref="p17.21.m13.2.2.1.1.2"><csymbol cd="ambiguous" id="p17.21.m13.2.2.1.1.2.2.1.cmml" xref="p17.21.m13.2.2.1.1.2">subscript</csymbol><ci id="p17.21.m13.2.2.1.1.2.2.2.cmml" xref="p17.21.m13.2.2.1.1.2.2.2">𝑓</ci><cn id="p17.21.m13.2.2.1.1.2.2.3.cmml" type="integer" xref="p17.21.m13.2.2.1.1.2.2.3">1</cn></apply><ci id="p17.21.m13.1.1.1.1.cmml" xref="p17.21.m13.1.1.1.1">𝑎</ci></apply><apply id="p17.21.m13.2.2.1.1.3.cmml" xref="p17.21.m13.2.2.1.1.3"><csymbol cd="ambiguous" id="p17.21.m13.2.2.1.1.3.1.cmml" xref="p17.21.m13.2.2.1.1.3">subscript</csymbol><ci id="p17.21.m13.2.2.1.1.3.2.cmml" xref="p17.21.m13.2.2.1.1.3.2">𝐼</ci><ci id="p17.21.m13.2.2.1.1.3.3.cmml" xref="p17.21.m13.2.2.1.1.3.3">𝑞</ci></apply></apply><apply id="p17.21.m13.3.3.2.2.cmml" xref="p17.21.m13.3.3.2.2"><eq id="p17.21.m13.3.3.2.2.1.cmml" xref="p17.21.m13.3.3.2.2.1"></eq><apply id="p17.21.m13.3.3.2.2.2.cmml" xref="p17.21.m13.3.3.2.2.2"><csymbol cd="ambiguous" id="p17.21.m13.3.3.2.2.2.1.cmml" xref="p17.21.m13.3.3.2.2.2">subscript</csymbol><ci id="p17.21.m13.3.3.2.2.2.2.cmml" xref="p17.21.m13.3.3.2.2.2.2">𝑊</ci><cn id="p17.21.m13.3.3.2.2.2.3.cmml" type="integer" xref="p17.21.m13.3.3.2.2.2.3">1</cn></apply><apply id="p17.21.m13.3.3.2.2.3.cmml" xref="p17.21.m13.3.3.2.2.3"><csymbol cd="ambiguous" id="p17.21.m13.3.3.2.2.3.1.cmml" xref="p17.21.m13.3.3.2.2.3">subscript</csymbol><ci id="p17.21.m13.3.3.2.2.3.2.cmml" xref="p17.21.m13.3.3.2.2.3.2">𝐼</ci><apply id="p17.21.m13.3.3.2.2.3.3.cmml" xref="p17.21.m13.3.3.2.2.3.3"><csymbol cd="ambiguous" id="p17.21.m13.3.3.2.2.3.3.1.cmml" xref="p17.21.m13.3.3.2.2.3.3">superscript</csymbol><ci id="p17.21.m13.3.3.2.2.3.3.2.cmml" xref="p17.21.m13.3.3.2.2.3.3.2">𝑞</ci><cn id="p17.21.m13.3.3.2.2.3.3.3.cmml" type="integer" xref="p17.21.m13.3.3.2.2.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p17.21.m13.3c">f_{1}^{(a)}=I_{q},W_{1}=I_{q^{2}}</annotation><annotation encoding="application/x-llamapun" id="p17.21.m13.3d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT = italic_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and thus Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E12" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">12</span></a>) is reduced to a subclass of dual-unitary gates <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib55" title="">55</a>]</cite>.</p> </div> <div class="ltx_para" id="p18"> <p class="ltx_p" id="p18.14">As an illustrative example, we report numerical results about the finite-size subsystem entanglement dynamics evolved by Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E12" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">12</span></a>). For the left initial state, we choose the following <math alttext="\chi=2" class="ltx_Math" display="inline" id="p18.1.m1.1"><semantics id="p18.1.m1.1a"><mrow id="p18.1.m1.1.1" xref="p18.1.m1.1.1.cmml"><mi id="p18.1.m1.1.1.2" xref="p18.1.m1.1.1.2.cmml">χ</mi><mo id="p18.1.m1.1.1.1" xref="p18.1.m1.1.1.1.cmml">=</mo><mn id="p18.1.m1.1.1.3" xref="p18.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="p18.1.m1.1b"><apply id="p18.1.m1.1.1.cmml" xref="p18.1.m1.1.1"><eq id="p18.1.m1.1.1.1.cmml" xref="p18.1.m1.1.1.1"></eq><ci id="p18.1.m1.1.1.2.cmml" xref="p18.1.m1.1.1.2">𝜒</ci><cn id="p18.1.m1.1.1.3.cmml" type="integer" xref="p18.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.1.m1.1c">\chi=2</annotation><annotation encoding="application/x-llamapun" id="p18.1.m1.1d">italic_χ = 2</annotation></semantics></math> MPS, which spans <math alttext="\mathcal{H}_{A}" class="ltx_Math" display="inline" id="p18.2.m2.1"><semantics id="p18.2.m2.1a"><msub id="p18.2.m2.1.1" xref="p18.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="p18.2.m2.1.1.2" xref="p18.2.m2.1.1.2.cmml">ℋ</mi><mi id="p18.2.m2.1.1.3" xref="p18.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="p18.2.m2.1b"><apply id="p18.2.m2.1.1.cmml" xref="p18.2.m2.1.1"><csymbol cd="ambiguous" id="p18.2.m2.1.1.1.cmml" xref="p18.2.m2.1.1">subscript</csymbol><ci id="p18.2.m2.1.1.2.cmml" xref="p18.2.m2.1.1.2">ℋ</ci><ci id="p18.2.m2.1.1.3.cmml" xref="p18.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.2.m2.1c">\mathcal{H}_{A}</annotation><annotation encoding="application/x-llamapun" id="p18.2.m2.1d">caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\tilde{q}=2" class="ltx_Math" display="inline" id="p18.3.m3.1"><semantics id="p18.3.m3.1a"><mrow id="p18.3.m3.1.1" xref="p18.3.m3.1.1.cmml"><mover accent="true" id="p18.3.m3.1.1.2" xref="p18.3.m3.1.1.2.cmml"><mi id="p18.3.m3.1.1.2.2" xref="p18.3.m3.1.1.2.2.cmml">q</mi><mo id="p18.3.m3.1.1.2.1" xref="p18.3.m3.1.1.2.1.cmml">~</mo></mover><mo id="p18.3.m3.1.1.1" xref="p18.3.m3.1.1.1.cmml">=</mo><mn id="p18.3.m3.1.1.3" xref="p18.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="p18.3.m3.1b"><apply id="p18.3.m3.1.1.cmml" xref="p18.3.m3.1.1"><eq id="p18.3.m3.1.1.1.cmml" xref="p18.3.m3.1.1.1"></eq><apply id="p18.3.m3.1.1.2.cmml" xref="p18.3.m3.1.1.2"><ci id="p18.3.m3.1.1.2.1.cmml" xref="p18.3.m3.1.1.2.1">~</ci><ci id="p18.3.m3.1.1.2.2.cmml" xref="p18.3.m3.1.1.2.2">𝑞</ci></apply><cn id="p18.3.m3.1.1.3.cmml" type="integer" xref="p18.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.3.m3.1c">\tilde{q}=2</annotation><annotation encoding="application/x-llamapun" id="p18.3.m3.1d">over~ start_ARG italic_q end_ARG = 2</annotation></semantics></math>: <math alttext="A^{(0)}=\left(\begin{array}[]{cc}\cos(\theta)&\sin(\theta)\\ 0&0\end{array}\right)" class="ltx_Math" display="inline" id="p18.4.m4.5"><semantics id="p18.4.m4.5a"><mrow id="p18.4.m4.5.6" xref="p18.4.m4.5.6.cmml"><msup id="p18.4.m4.5.6.2" xref="p18.4.m4.5.6.2.cmml"><mi id="p18.4.m4.5.6.2.2" xref="p18.4.m4.5.6.2.2.cmml">A</mi><mrow id="p18.4.m4.1.1.1.3" xref="p18.4.m4.5.6.2.cmml"><mo id="p18.4.m4.1.1.1.3.1" stretchy="false" xref="p18.4.m4.5.6.2.cmml">(</mo><mn id="p18.4.m4.1.1.1.1" xref="p18.4.m4.1.1.1.1.cmml">0</mn><mo id="p18.4.m4.1.1.1.3.2" stretchy="false" xref="p18.4.m4.5.6.2.cmml">)</mo></mrow></msup><mo id="p18.4.m4.5.6.1" xref="p18.4.m4.5.6.1.cmml">=</mo><mrow id="p18.4.m4.5.6.3.2" xref="p18.4.m4.5.5.cmml"><mo id="p18.4.m4.5.6.3.2.1" xref="p18.4.m4.5.5.cmml">(</mo><mtable columnspacing="5pt" id="p18.4.m4.5.5" rowspacing="0pt" xref="p18.4.m4.5.5.cmml"><mtr id="p18.4.m4.5.5a" xref="p18.4.m4.5.5.cmml"><mtd id="p18.4.m4.5.5b" xref="p18.4.m4.5.5.cmml"><mrow 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xref="p18.4.m4.5.5.4.4.2.2.cmml">θ</mi><mo id="p18.4.m4.5.5.4.4.2.4.1.2" stretchy="false" xref="p18.4.m4.5.5.4.4.2.3.cmml">)</mo></mrow></mrow></mtd></mtr><mtr id="p18.4.m4.5.5d" xref="p18.4.m4.5.5.cmml"><mtd id="p18.4.m4.5.5e" xref="p18.4.m4.5.5.cmml"><mn id="p18.4.m4.5.5.5.1.1" xref="p18.4.m4.5.5.5.1.1.cmml">0</mn></mtd><mtd id="p18.4.m4.5.5f" xref="p18.4.m4.5.5.cmml"><mn id="p18.4.m4.5.5.5.2.1" xref="p18.4.m4.5.5.5.2.1.cmml">0</mn></mtd></mtr></mtable><mo id="p18.4.m4.5.6.3.2.2" xref="p18.4.m4.5.5.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p18.4.m4.5b"><apply id="p18.4.m4.5.6.cmml" xref="p18.4.m4.5.6"><eq id="p18.4.m4.5.6.1.cmml" xref="p18.4.m4.5.6.1"></eq><apply id="p18.4.m4.5.6.2.cmml" xref="p18.4.m4.5.6.2"><csymbol cd="ambiguous" id="p18.4.m4.5.6.2.1.cmml" xref="p18.4.m4.5.6.2">superscript</csymbol><ci id="p18.4.m4.5.6.2.2.cmml" xref="p18.4.m4.5.6.2.2">𝐴</ci><cn id="p18.4.m4.1.1.1.1.cmml" type="integer" xref="p18.4.m4.1.1.1.1">0</cn></apply><matrix id="p18.4.m4.5.5.cmml" xref="p18.4.m4.5.6.3.2"><matrixrow id="p18.4.m4.5.5a.cmml" xref="p18.4.m4.5.6.3.2"><apply id="p18.4.m4.3.3.2.2.2.3.cmml" xref="p18.4.m4.3.3.2.2.2.4"><cos id="p18.4.m4.2.2.1.1.1.1.cmml" xref="p18.4.m4.2.2.1.1.1.1"></cos><ci id="p18.4.m4.3.3.2.2.2.2.cmml" xref="p18.4.m4.3.3.2.2.2.2">𝜃</ci></apply><apply id="p18.4.m4.5.5.4.4.2.3.cmml" xref="p18.4.m4.5.5.4.4.2.4"><sin id="p18.4.m4.4.4.3.3.1.1.cmml" xref="p18.4.m4.4.4.3.3.1.1"></sin><ci id="p18.4.m4.5.5.4.4.2.2.cmml" xref="p18.4.m4.5.5.4.4.2.2">𝜃</ci></apply></matrixrow><matrixrow id="p18.4.m4.5.5b.cmml" xref="p18.4.m4.5.6.3.2"><cn id="p18.4.m4.5.5.5.1.1.cmml" type="integer" xref="p18.4.m4.5.5.5.1.1">0</cn><cn id="p18.4.m4.5.5.5.2.1.cmml" type="integer" xref="p18.4.m4.5.5.5.2.1">0</cn></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.4.m4.5c">A^{(0)}=\left(\begin{array}[]{cc}\cos(\theta)&\sin(\theta)\\ 0&0\end{array}\right)</annotation><annotation encoding="application/x-llamapun" id="p18.4.m4.5d">italic_A start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT = ( start_ARRAY start_ROW start_CELL roman_cos ( italic_θ ) end_CELL start_CELL roman_sin ( italic_θ ) end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY )</annotation></semantics></math>, <math alttext="A^{(1)}=\left(\begin{array}[]{cc}0&0\\ -\sin(\theta)&\cos(\theta)\end{array}\right)" class="ltx_Math" display="inline" id="p18.5.m5.5"><semantics id="p18.5.m5.5a"><mrow id="p18.5.m5.5.6" xref="p18.5.m5.5.6.cmml"><msup id="p18.5.m5.5.6.2" xref="p18.5.m5.5.6.2.cmml"><mi id="p18.5.m5.5.6.2.2" xref="p18.5.m5.5.6.2.2.cmml">A</mi><mrow id="p18.5.m5.1.1.1.3" xref="p18.5.m5.5.6.2.cmml"><mo id="p18.5.m5.1.1.1.3.1" stretchy="false" xref="p18.5.m5.5.6.2.cmml">(</mo><mn id="p18.5.m5.1.1.1.1" xref="p18.5.m5.1.1.1.1.cmml">1</mn><mo id="p18.5.m5.1.1.1.3.2" stretchy="false" xref="p18.5.m5.5.6.2.cmml">)</mo></mrow></msup><mo id="p18.5.m5.5.6.1" xref="p18.5.m5.5.6.1.cmml">=</mo><mrow id="p18.5.m5.5.6.3.2" xref="p18.5.m5.5.5.cmml"><mo id="p18.5.m5.5.6.3.2.1" xref="p18.5.m5.5.5.cmml">(</mo><mtable columnspacing="5pt" id="p18.5.m5.5.5" rowspacing="0pt" xref="p18.5.m5.5.5.cmml"><mtr id="p18.5.m5.5.5a" xref="p18.5.m5.5.5.cmml"><mtd id="p18.5.m5.5.5b" xref="p18.5.m5.5.5.cmml"><mn id="p18.5.m5.5.5.5.1.1" xref="p18.5.m5.5.5.5.1.1.cmml">0</mn></mtd><mtd id="p18.5.m5.5.5c" xref="p18.5.m5.5.5.cmml"><mn id="p18.5.m5.5.5.5.2.1" xref="p18.5.m5.5.5.5.2.1.cmml">0</mn></mtd></mtr><mtr id="p18.5.m5.5.5d" xref="p18.5.m5.5.5.cmml"><mtd id="p18.5.m5.5.5e" xref="p18.5.m5.5.5.cmml"><mrow id="p18.5.m5.3.3.2.2.2" xref="p18.5.m5.3.3.2.2.2.cmml"><mo id="p18.5.m5.3.3.2.2.2a" rspace="0.167em" xref="p18.5.m5.3.3.2.2.2.cmml">−</mo><mrow id="p18.5.m5.3.3.2.2.2.4.2" xref="p18.5.m5.3.3.2.2.2.4.1.cmml"><mi id="p18.5.m5.2.2.1.1.1.1" xref="p18.5.m5.2.2.1.1.1.1.cmml">sin</mi><mo id="p18.5.m5.3.3.2.2.2.4.2a" xref="p18.5.m5.3.3.2.2.2.4.1.cmml">⁡</mo><mrow id="p18.5.m5.3.3.2.2.2.4.2.1" xref="p18.5.m5.3.3.2.2.2.4.1.cmml"><mo id="p18.5.m5.3.3.2.2.2.4.2.1.1" stretchy="false" xref="p18.5.m5.3.3.2.2.2.4.1.cmml">(</mo><mi id="p18.5.m5.3.3.2.2.2.2" xref="p18.5.m5.3.3.2.2.2.2.cmml">θ</mi><mo id="p18.5.m5.3.3.2.2.2.4.2.1.2" stretchy="false" xref="p18.5.m5.3.3.2.2.2.4.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="p18.5.m5.5.5f" xref="p18.5.m5.5.5.cmml"><mrow id="p18.5.m5.5.5.4.4.2.4" xref="p18.5.m5.5.5.4.4.2.3.cmml"><mi id="p18.5.m5.4.4.3.3.1.1" xref="p18.5.m5.4.4.3.3.1.1.cmml">cos</mi><mo id="p18.5.m5.5.5.4.4.2.4a" xref="p18.5.m5.5.5.4.4.2.3.cmml">⁡</mo><mrow id="p18.5.m5.5.5.4.4.2.4.1" xref="p18.5.m5.5.5.4.4.2.3.cmml"><mo id="p18.5.m5.5.5.4.4.2.4.1.1" stretchy="false" xref="p18.5.m5.5.5.4.4.2.3.cmml">(</mo><mi id="p18.5.m5.5.5.4.4.2.2" xref="p18.5.m5.5.5.4.4.2.2.cmml">θ</mi><mo id="p18.5.m5.5.5.4.4.2.4.1.2" stretchy="false" xref="p18.5.m5.5.5.4.4.2.3.cmml">)</mo></mrow></mrow></mtd></mtr></mtable><mo id="p18.5.m5.5.6.3.2.2" xref="p18.5.m5.5.5.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p18.5.m5.5b"><apply id="p18.5.m5.5.6.cmml" xref="p18.5.m5.5.6"><eq id="p18.5.m5.5.6.1.cmml" xref="p18.5.m5.5.6.1"></eq><apply id="p18.5.m5.5.6.2.cmml" xref="p18.5.m5.5.6.2"><csymbol cd="ambiguous" id="p18.5.m5.5.6.2.1.cmml" xref="p18.5.m5.5.6.2">superscript</csymbol><ci id="p18.5.m5.5.6.2.2.cmml" xref="p18.5.m5.5.6.2.2">𝐴</ci><cn id="p18.5.m5.1.1.1.1.cmml" type="integer" xref="p18.5.m5.1.1.1.1">1</cn></apply><matrix id="p18.5.m5.5.5.cmml" xref="p18.5.m5.5.6.3.2"><matrixrow id="p18.5.m5.5.5a.cmml" xref="p18.5.m5.5.6.3.2"><cn id="p18.5.m5.5.5.5.1.1.cmml" type="integer" xref="p18.5.m5.5.5.5.1.1">0</cn><cn id="p18.5.m5.5.5.5.2.1.cmml" type="integer" xref="p18.5.m5.5.5.5.2.1">0</cn></matrixrow><matrixrow id="p18.5.m5.5.5b.cmml" xref="p18.5.m5.5.6.3.2"><apply id="p18.5.m5.3.3.2.2.2.cmml" xref="p18.5.m5.3.3.2.2.2"><minus id="p18.5.m5.3.3.2.2.2.3.cmml" xref="p18.5.m5.3.3.2.2.2"></minus><apply id="p18.5.m5.3.3.2.2.2.4.1.cmml" xref="p18.5.m5.3.3.2.2.2.4.2"><sin id="p18.5.m5.2.2.1.1.1.1.cmml" xref="p18.5.m5.2.2.1.1.1.1"></sin><ci id="p18.5.m5.3.3.2.2.2.2.cmml" xref="p18.5.m5.3.3.2.2.2.2">𝜃</ci></apply></apply><apply id="p18.5.m5.5.5.4.4.2.3.cmml" xref="p18.5.m5.5.5.4.4.2.4"><cos id="p18.5.m5.4.4.3.3.1.1.cmml" xref="p18.5.m5.4.4.3.3.1.1"></cos><ci id="p18.5.m5.5.5.4.4.2.2.cmml" xref="p18.5.m5.5.5.4.4.2.2">𝜃</ci></apply></matrixrow></matrix></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.5.m5.5c">A^{(1)}=\left(\begin{array}[]{cc}0&0\\ -\sin(\theta)&\cos(\theta)\end{array}\right)</annotation><annotation encoding="application/x-llamapun" id="p18.5.m5.5d">italic_A start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL - roman_sin ( italic_θ ) end_CELL start_CELL roman_cos ( italic_θ ) end_CELL end_ROW end_ARRAY )</annotation></semantics></math>, and <math alttext="A^{(2,3)}=0" class="ltx_Math" display="inline" id="p18.6.m6.2"><semantics id="p18.6.m6.2a"><mrow id="p18.6.m6.2.3" xref="p18.6.m6.2.3.cmml"><msup id="p18.6.m6.2.3.2" xref="p18.6.m6.2.3.2.cmml"><mi id="p18.6.m6.2.3.2.2" xref="p18.6.m6.2.3.2.2.cmml">A</mi><mrow id="p18.6.m6.2.2.2.4" xref="p18.6.m6.2.2.2.3.cmml"><mo id="p18.6.m6.2.2.2.4.1" stretchy="false" xref="p18.6.m6.2.2.2.3.cmml">(</mo><mn id="p18.6.m6.1.1.1.1" xref="p18.6.m6.1.1.1.1.cmml">2</mn><mo id="p18.6.m6.2.2.2.4.2" xref="p18.6.m6.2.2.2.3.cmml">,</mo><mn id="p18.6.m6.2.2.2.2" xref="p18.6.m6.2.2.2.2.cmml">3</mn><mo id="p18.6.m6.2.2.2.4.3" stretchy="false" xref="p18.6.m6.2.2.2.3.cmml">)</mo></mrow></msup><mo id="p18.6.m6.2.3.1" xref="p18.6.m6.2.3.1.cmml">=</mo><mn id="p18.6.m6.2.3.3" xref="p18.6.m6.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="p18.6.m6.2b"><apply id="p18.6.m6.2.3.cmml" xref="p18.6.m6.2.3"><eq id="p18.6.m6.2.3.1.cmml" xref="p18.6.m6.2.3.1"></eq><apply id="p18.6.m6.2.3.2.cmml" xref="p18.6.m6.2.3.2"><csymbol cd="ambiguous" id="p18.6.m6.2.3.2.1.cmml" xref="p18.6.m6.2.3.2">superscript</csymbol><ci id="p18.6.m6.2.3.2.2.cmml" xref="p18.6.m6.2.3.2.2">𝐴</ci><interval closure="open" id="p18.6.m6.2.2.2.3.cmml" xref="p18.6.m6.2.2.2.4"><cn id="p18.6.m6.1.1.1.1.cmml" type="integer" xref="p18.6.m6.1.1.1.1">2</cn><cn id="p18.6.m6.2.2.2.2.cmml" type="integer" xref="p18.6.m6.2.2.2.2">3</cn></interval></apply><cn id="p18.6.m6.2.3.3.cmml" type="integer" xref="p18.6.m6.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.6.m6.2c">A^{(2,3)}=0</annotation><annotation encoding="application/x-llamapun" id="p18.6.m6.2d">italic_A start_POSTSUPERSCRIPT ( 2 , 3 ) end_POSTSUPERSCRIPT = 0</annotation></semantics></math>. Here <math alttext="\theta\in(0,\frac{\pi}{4}]" class="ltx_Math" display="inline" id="p18.7.m7.2"><semantics id="p18.7.m7.2a"><mrow id="p18.7.m7.2.3" xref="p18.7.m7.2.3.cmml"><mi id="p18.7.m7.2.3.2" xref="p18.7.m7.2.3.2.cmml">θ</mi><mo id="p18.7.m7.2.3.1" xref="p18.7.m7.2.3.1.cmml">∈</mo><mrow id="p18.7.m7.2.3.3.2" xref="p18.7.m7.2.3.3.1.cmml"><mo id="p18.7.m7.2.3.3.2.1" stretchy="false" xref="p18.7.m7.2.3.3.1.cmml">(</mo><mn id="p18.7.m7.1.1" xref="p18.7.m7.1.1.cmml">0</mn><mo id="p18.7.m7.2.3.3.2.2" xref="p18.7.m7.2.3.3.1.cmml">,</mo><mfrac id="p18.7.m7.2.2" xref="p18.7.m7.2.2.cmml"><mi id="p18.7.m7.2.2.2" xref="p18.7.m7.2.2.2.cmml">π</mi><mn id="p18.7.m7.2.2.3" xref="p18.7.m7.2.2.3.cmml">4</mn></mfrac><mo id="p18.7.m7.2.3.3.2.3" stretchy="false" xref="p18.7.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="p18.7.m7.2b"><apply id="p18.7.m7.2.3.cmml" xref="p18.7.m7.2.3"><in id="p18.7.m7.2.3.1.cmml" xref="p18.7.m7.2.3.1"></in><ci id="p18.7.m7.2.3.2.cmml" xref="p18.7.m7.2.3.2">𝜃</ci><interval closure="open-closed" id="p18.7.m7.2.3.3.1.cmml" xref="p18.7.m7.2.3.3.2"><cn id="p18.7.m7.1.1.cmml" type="integer" xref="p18.7.m7.1.1">0</cn><apply id="p18.7.m7.2.2.cmml" xref="p18.7.m7.2.2"><divide id="p18.7.m7.2.2.1.cmml" xref="p18.7.m7.2.2"></divide><ci id="p18.7.m7.2.2.2.cmml" xref="p18.7.m7.2.2.2">𝜋</ci><cn id="p18.7.m7.2.2.3.cmml" type="integer" xref="p18.7.m7.2.2.3">4</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.7.m7.2c">\theta\in(0,\frac{\pi}{4}]</annotation><annotation encoding="application/x-llamapun" id="p18.7.m7.2d">italic_θ ∈ ( 0 , divide start_ARG italic_π end_ARG start_ARG 4 end_ARG ]</annotation></semantics></math>, which corresponds to the left initial state interpolating between the Greenberger-Horne-Zeilinger state and the cluster state <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib56" title="">56</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib57" title="">57</a>]</cite> of basis states <math alttext="\ket{0}" class="ltx_Math" display="inline" id="p18.8.m8.1"><semantics id="p18.8.m8.1a"><mrow id="p18.8.m8.1.1.3" xref="p18.8.m8.1.1.2.cmml"><mo id="p18.8.m8.1.1.3.1" stretchy="false" xref="p18.8.m8.1.1.2.1.cmml">|</mo><mn id="p18.8.m8.1.1.1.1" xref="p18.8.m8.1.1.1.1.cmml">0</mn><mo id="p18.8.m8.1.1.3.2" stretchy="false" xref="p18.8.m8.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p18.8.m8.1b"><apply id="p18.8.m8.1.1.2.cmml" xref="p18.8.m8.1.1.3"><csymbol cd="latexml" id="p18.8.m8.1.1.2.1.cmml" xref="p18.8.m8.1.1.3.1">ket</csymbol><cn id="p18.8.m8.1.1.1.1.cmml" type="integer" xref="p18.8.m8.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.8.m8.1c">\ket{0}</annotation><annotation encoding="application/x-llamapun" id="p18.8.m8.1d">| start_ARG 0 end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{1}" class="ltx_Math" display="inline" id="p18.9.m9.1"><semantics id="p18.9.m9.1a"><mrow id="p18.9.m9.1.1.3" xref="p18.9.m9.1.1.2.cmml"><mo id="p18.9.m9.1.1.3.1" stretchy="false" xref="p18.9.m9.1.1.2.1.cmml">|</mo><mn id="p18.9.m9.1.1.1.1" xref="p18.9.m9.1.1.1.1.cmml">1</mn><mo id="p18.9.m9.1.1.3.2" stretchy="false" xref="p18.9.m9.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="p18.9.m9.1b"><apply id="p18.9.m9.1.1.2.cmml" xref="p18.9.m9.1.1.3"><csymbol cd="latexml" id="p18.9.m9.1.1.2.1.cmml" xref="p18.9.m9.1.1.3.1">ket</csymbol><cn id="p18.9.m9.1.1.1.1.cmml" type="integer" xref="p18.9.m9.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.9.m9.1c">\ket{1}</annotation><annotation encoding="application/x-llamapun" id="p18.9.m9.1d">| start_ARG 1 end_ARG ⟩</annotation></semantics></math>. As for the right region, we take the initial state as the simple product state <math alttext="\ket{\Psi_{R}^{j}}=\otimes_{x\geq 0}\ket{2}_{x}" class="ltx_math_unparsed" display="inline" id="p18.10.m10.2"><semantics id="p18.10.m10.2a"><mrow id="p18.10.m10.2b"><mrow id="p18.10.m10.1.1.3"><mo id="p18.10.m10.1.1.3.1" stretchy="false">|</mo><msubsup id="p18.10.m10.1.1.1.1"><mi id="p18.10.m10.1.1.1.1.2.2" mathvariant="normal">Ψ</mi><mi id="p18.10.m10.1.1.1.1.2.3">R</mi><mi id="p18.10.m10.1.1.1.1.3">j</mi></msubsup><mo id="p18.10.m10.1.1.3.2" stretchy="false">⟩</mo></mrow><mo id="p18.10.m10.2.3" rspace="0em">=</mo><msub id="p18.10.m10.2.4"><mo id="p18.10.m10.2.4.2" lspace="0em" rspace="0.222em">⊗</mo><mrow id="p18.10.m10.2.4.3"><mi id="p18.10.m10.2.4.3.2">x</mi><mo id="p18.10.m10.2.4.3.1">≥</mo><mn id="p18.10.m10.2.4.3.3">0</mn></mrow></msub><msub id="p18.10.m10.2.5"><mrow id="p18.10.m10.2.2.3"><mo id="p18.10.m10.2.2.3.1" stretchy="false">|</mo><mn id="p18.10.m10.2.2.1.1">2</mn><mo id="p18.10.m10.2.2.3.2" stretchy="false">⟩</mo></mrow><mi id="p18.10.m10.2.5.2">x</mi></msub></mrow><annotation encoding="application/x-tex" id="p18.10.m10.2c">\ket{\Psi_{R}^{j}}=\otimes_{x\geq 0}\ket{2}_{x}</annotation><annotation encoding="application/x-llamapun" id="p18.10.m10.2d">| start_ARG roman_Ψ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ⟩ = ⊗ start_POSTSUBSCRIPT italic_x ≥ 0 end_POSTSUBSCRIPT | start_ARG 2 end_ARG ⟩ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="j=0,1" class="ltx_Math" display="inline" id="p18.11.m11.2"><semantics id="p18.11.m11.2a"><mrow id="p18.11.m11.2.3" xref="p18.11.m11.2.3.cmml"><mi id="p18.11.m11.2.3.2" xref="p18.11.m11.2.3.2.cmml">j</mi><mo id="p18.11.m11.2.3.1" xref="p18.11.m11.2.3.1.cmml">=</mo><mrow id="p18.11.m11.2.3.3.2" xref="p18.11.m11.2.3.3.1.cmml"><mn id="p18.11.m11.1.1" xref="p18.11.m11.1.1.cmml">0</mn><mo id="p18.11.m11.2.3.3.2.1" xref="p18.11.m11.2.3.3.1.cmml">,</mo><mn id="p18.11.m11.2.2" xref="p18.11.m11.2.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="p18.11.m11.2b"><apply id="p18.11.m11.2.3.cmml" xref="p18.11.m11.2.3"><eq id="p18.11.m11.2.3.1.cmml" xref="p18.11.m11.2.3.1"></eq><ci id="p18.11.m11.2.3.2.cmml" xref="p18.11.m11.2.3.2">𝑗</ci><list id="p18.11.m11.2.3.3.1.cmml" xref="p18.11.m11.2.3.3.2"><cn id="p18.11.m11.1.1.cmml" type="integer" xref="p18.11.m11.1.1">0</cn><cn id="p18.11.m11.2.2.cmml" type="integer" xref="p18.11.m11.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.11.m11.2c">j=0,1</annotation><annotation encoding="application/x-llamapun" id="p18.11.m11.2d">italic_j = 0 , 1</annotation></semantics></math>, and thus the initial entanglement between regions L and R is zero. The right subsystem consists of four sites as shown faithfully in the left panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F2" title="Figure 2 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">2</span></a>. We construct the two-site gate <math alttext="U" class="ltx_Math" display="inline" id="p18.12.m12.1"><semantics id="p18.12.m12.1a"><mi id="p18.12.m12.1.1" xref="p18.12.m12.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="p18.12.m12.1b"><ci id="p18.12.m12.1.1.cmml" xref="p18.12.m12.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="p18.12.m12.1c">U</annotation><annotation encoding="application/x-llamapun" id="p18.12.m12.1d">italic_U</annotation></semantics></math> by randomly generating unitaries <math alttext="v,g^{(a)}" class="ltx_Math" display="inline" id="p18.13.m13.3"><semantics id="p18.13.m13.3a"><mrow id="p18.13.m13.3.3.1" xref="p18.13.m13.3.3.2.cmml"><mi id="p18.13.m13.2.2" xref="p18.13.m13.2.2.cmml">v</mi><mo id="p18.13.m13.3.3.1.2" xref="p18.13.m13.3.3.2.cmml">,</mo><msup id="p18.13.m13.3.3.1.1" xref="p18.13.m13.3.3.1.1.cmml"><mi id="p18.13.m13.3.3.1.1.2" xref="p18.13.m13.3.3.1.1.2.cmml">g</mi><mrow id="p18.13.m13.1.1.1.3" xref="p18.13.m13.3.3.1.1.cmml"><mo id="p18.13.m13.1.1.1.3.1" stretchy="false" xref="p18.13.m13.3.3.1.1.cmml">(</mo><mi id="p18.13.m13.1.1.1.1" xref="p18.13.m13.1.1.1.1.cmml">a</mi><mo id="p18.13.m13.1.1.1.3.2" stretchy="false" xref="p18.13.m13.3.3.1.1.cmml">)</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="p18.13.m13.3b"><list id="p18.13.m13.3.3.2.cmml" xref="p18.13.m13.3.3.1"><ci id="p18.13.m13.2.2.cmml" xref="p18.13.m13.2.2">𝑣</ci><apply id="p18.13.m13.3.3.1.1.cmml" xref="p18.13.m13.3.3.1.1"><csymbol cd="ambiguous" id="p18.13.m13.3.3.1.1.1.cmml" xref="p18.13.m13.3.3.1.1">superscript</csymbol><ci id="p18.13.m13.3.3.1.1.2.cmml" xref="p18.13.m13.3.3.1.1.2">𝑔</ci><ci id="p18.13.m13.1.1.1.1.cmml" xref="p18.13.m13.1.1.1.1">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="p18.13.m13.3c">v,g^{(a)}</annotation><annotation encoding="application/x-llamapun" id="p18.13.m13.3d">italic_v , italic_g start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="f_{2}^{(a)}" class="ltx_Math" display="inline" id="p18.14.m14.1"><semantics id="p18.14.m14.1a"><msubsup id="p18.14.m14.1.2" xref="p18.14.m14.1.2.cmml"><mi id="p18.14.m14.1.2.2.2" xref="p18.14.m14.1.2.2.2.cmml">f</mi><mn id="p18.14.m14.1.2.2.3" xref="p18.14.m14.1.2.2.3.cmml">2</mn><mrow id="p18.14.m14.1.1.1.3" xref="p18.14.m14.1.2.cmml"><mo id="p18.14.m14.1.1.1.3.1" stretchy="false" xref="p18.14.m14.1.2.cmml">(</mo><mi id="p18.14.m14.1.1.1.1" xref="p18.14.m14.1.1.1.1.cmml">a</mi><mo id="p18.14.m14.1.1.1.3.2" stretchy="false" xref="p18.14.m14.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="p18.14.m14.1b"><apply id="p18.14.m14.1.2.cmml" xref="p18.14.m14.1.2"><csymbol cd="ambiguous" id="p18.14.m14.1.2.1.cmml" xref="p18.14.m14.1.2">superscript</csymbol><apply id="p18.14.m14.1.2.2.cmml" xref="p18.14.m14.1.2"><csymbol cd="ambiguous" id="p18.14.m14.1.2.2.1.cmml" xref="p18.14.m14.1.2">subscript</csymbol><ci id="p18.14.m14.1.2.2.2.cmml" xref="p18.14.m14.1.2.2.2">𝑓</ci><cn id="p18.14.m14.1.2.2.3.cmml" type="integer" xref="p18.14.m14.1.2.2.3">2</cn></apply><ci id="p18.14.m14.1.1.1.1.cmml" xref="p18.14.m14.1.1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p18.14.m14.1c">f_{2}^{(a)}</annotation><annotation encoding="application/x-llamapun" id="p18.14.m14.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="p19"> <p class="ltx_p" id="p19.9">Employing the influence matrix method, we can numerically keep track of the full time evolution of the joint system. The joint dynamics is generated by the Floquet operator Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E6" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">6</span></a>), which is constituted by the unitary <math alttext="\mathbb{U}_{R}" class="ltx_Math" display="inline" id="p19.1.m1.1"><semantics id="p19.1.m1.1a"><msub id="p19.1.m1.1.1" xref="p19.1.m1.1.1.cmml"><mi id="p19.1.m1.1.1.2" xref="p19.1.m1.1.1.2.cmml">𝕌</mi><mi id="p19.1.m1.1.1.3" xref="p19.1.m1.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="p19.1.m1.1b"><apply id="p19.1.m1.1.1.cmml" xref="p19.1.m1.1.1"><csymbol cd="ambiguous" id="p19.1.m1.1.1.1.cmml" xref="p19.1.m1.1.1">subscript</csymbol><ci id="p19.1.m1.1.1.2.cmml" xref="p19.1.m1.1.1.2">𝕌</ci><ci id="p19.1.m1.1.1.3.cmml" xref="p19.1.m1.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="p19.1.m1.1c">\mathbb{U}_{R}</annotation><annotation encoding="application/x-llamapun" id="p19.1.m1.1d">blackboard_U start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> along with the boundary quantum channel <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="p19.2.m2.1"><semantics id="p19.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="p19.2.m2.1.1" xref="p19.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="p19.2.m2.1b"><ci id="p19.2.m2.1.1.cmml" xref="p19.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="p19.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="p19.2.m2.1d">caligraphic_M</annotation></semantics></math> given by Eq.(<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E7" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">7</span></a>). We compute the von Neumann entanglement entropy between two regions as <math alttext="S_{\text{ent}}(t)=-\text{Tr}\{\rho_{R}(t)\ln[\rho_{R}(t)]\}" class="ltx_Math" display="inline" id="p19.3.m3.5"><semantics id="p19.3.m3.5a"><mrow id="p19.3.m3.5.5" xref="p19.3.m3.5.5.cmml"><mrow id="p19.3.m3.5.5.3" xref="p19.3.m3.5.5.3.cmml"><msub id="p19.3.m3.5.5.3.2" xref="p19.3.m3.5.5.3.2.cmml"><mi id="p19.3.m3.5.5.3.2.2" xref="p19.3.m3.5.5.3.2.2.cmml">S</mi><mtext id="p19.3.m3.5.5.3.2.3" xref="p19.3.m3.5.5.3.2.3a.cmml">ent</mtext></msub><mo id="p19.3.m3.5.5.3.1" xref="p19.3.m3.5.5.3.1.cmml">⁢</mo><mrow id="p19.3.m3.5.5.3.3.2" xref="p19.3.m3.5.5.3.cmml"><mo id="p19.3.m3.5.5.3.3.2.1" stretchy="false" xref="p19.3.m3.5.5.3.cmml">(</mo><mi id="p19.3.m3.1.1" xref="p19.3.m3.1.1.cmml">t</mi><mo id="p19.3.m3.5.5.3.3.2.2" stretchy="false" xref="p19.3.m3.5.5.3.cmml">)</mo></mrow></mrow><mo id="p19.3.m3.5.5.2" xref="p19.3.m3.5.5.2.cmml">=</mo><mrow id="p19.3.m3.5.5.1" xref="p19.3.m3.5.5.1.cmml"><mo id="p19.3.m3.5.5.1a" xref="p19.3.m3.5.5.1.cmml">−</mo><mrow id="p19.3.m3.5.5.1.1" xref="p19.3.m3.5.5.1.1.cmml"><mtext id="p19.3.m3.5.5.1.1.3" xref="p19.3.m3.5.5.1.1.3a.cmml">Tr</mtext><mo id="p19.3.m3.5.5.1.1.2" xref="p19.3.m3.5.5.1.1.2.cmml">⁢</mo><mrow id="p19.3.m3.5.5.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.2.cmml"><mo id="p19.3.m3.5.5.1.1.1.1.2" stretchy="false" xref="p19.3.m3.5.5.1.1.1.2.cmml">{</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.1.1.cmml"><msub id="p19.3.m3.5.5.1.1.1.1.1.3" xref="p19.3.m3.5.5.1.1.1.1.1.3.cmml"><mi id="p19.3.m3.5.5.1.1.1.1.1.3.2" xref="p19.3.m3.5.5.1.1.1.1.1.3.2.cmml">ρ</mi><mi id="p19.3.m3.5.5.1.1.1.1.1.3.3" xref="p19.3.m3.5.5.1.1.1.1.1.3.3.cmml">R</mi></msub><mo id="p19.3.m3.5.5.1.1.1.1.1.2" xref="p19.3.m3.5.5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1.4.2" xref="p19.3.m3.5.5.1.1.1.1.1.cmml"><mo id="p19.3.m3.5.5.1.1.1.1.1.4.2.1" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.cmml">(</mo><mi id="p19.3.m3.2.2" xref="p19.3.m3.2.2.cmml">t</mi><mo id="p19.3.m3.5.5.1.1.1.1.1.4.2.2" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo id="p19.3.m3.5.5.1.1.1.1.1.2a" lspace="0.167em" xref="p19.3.m3.5.5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml"><mi id="p19.3.m3.4.4" xref="p19.3.m3.4.4.cmml">ln</mi><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1a" xref="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml"><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml">[</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.cmml"><msub id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.2" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.2.cmml">ρ</mi><mi id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.3" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.3.cmml">R</mi></msub><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.1" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.3.2" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.cmml"><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="p19.3.m3.3.3" xref="p19.3.m3.3.3.cmml">t</mi><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml">]</mo></mrow></mrow></mrow><mo id="p19.3.m3.5.5.1.1.1.1.3" stretchy="false" xref="p19.3.m3.5.5.1.1.1.2.cmml">}</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p19.3.m3.5b"><apply id="p19.3.m3.5.5.cmml" xref="p19.3.m3.5.5"><eq id="p19.3.m3.5.5.2.cmml" xref="p19.3.m3.5.5.2"></eq><apply id="p19.3.m3.5.5.3.cmml" xref="p19.3.m3.5.5.3"><times id="p19.3.m3.5.5.3.1.cmml" xref="p19.3.m3.5.5.3.1"></times><apply id="p19.3.m3.5.5.3.2.cmml" xref="p19.3.m3.5.5.3.2"><csymbol cd="ambiguous" id="p19.3.m3.5.5.3.2.1.cmml" xref="p19.3.m3.5.5.3.2">subscript</csymbol><ci id="p19.3.m3.5.5.3.2.2.cmml" xref="p19.3.m3.5.5.3.2.2">𝑆</ci><ci id="p19.3.m3.5.5.3.2.3a.cmml" xref="p19.3.m3.5.5.3.2.3"><mtext id="p19.3.m3.5.5.3.2.3.cmml" mathsize="70%" xref="p19.3.m3.5.5.3.2.3">ent</mtext></ci></apply><ci id="p19.3.m3.1.1.cmml" xref="p19.3.m3.1.1">𝑡</ci></apply><apply id="p19.3.m3.5.5.1.cmml" xref="p19.3.m3.5.5.1"><minus id="p19.3.m3.5.5.1.2.cmml" xref="p19.3.m3.5.5.1"></minus><apply id="p19.3.m3.5.5.1.1.cmml" xref="p19.3.m3.5.5.1.1"><times id="p19.3.m3.5.5.1.1.2.cmml" xref="p19.3.m3.5.5.1.1.2"></times><ci id="p19.3.m3.5.5.1.1.3a.cmml" xref="p19.3.m3.5.5.1.1.3"><mtext id="p19.3.m3.5.5.1.1.3.cmml" xref="p19.3.m3.5.5.1.1.3">Tr</mtext></ci><set id="p19.3.m3.5.5.1.1.1.2.cmml" xref="p19.3.m3.5.5.1.1.1.1"><apply id="p19.3.m3.5.5.1.1.1.1.1.cmml" xref="p19.3.m3.5.5.1.1.1.1.1"><times id="p19.3.m3.5.5.1.1.1.1.1.2.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.2"></times><apply id="p19.3.m3.5.5.1.1.1.1.1.3.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="p19.3.m3.5.5.1.1.1.1.1.3.1.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.3">subscript</csymbol><ci id="p19.3.m3.5.5.1.1.1.1.1.3.2.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.3.2">𝜌</ci><ci id="p19.3.m3.5.5.1.1.1.1.1.3.3.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.3.3">𝑅</ci></apply><ci id="p19.3.m3.2.2.cmml" xref="p19.3.m3.2.2">𝑡</ci><apply id="p19.3.m3.5.5.1.1.1.1.1.1.2.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1"><ln id="p19.3.m3.4.4.cmml" xref="p19.3.m3.4.4"></ln><apply id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1"><times id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.1.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.1"></times><apply id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.2">𝜌</ci><ci id="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="p19.3.m3.5.5.1.1.1.1.1.1.1.1.1.2.3">𝑅</ci></apply><ci id="p19.3.m3.3.3.cmml" xref="p19.3.m3.3.3">𝑡</ci></apply></apply></apply></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p19.3.m3.5c">S_{\text{ent}}(t)=-\text{Tr}\{\rho_{R}(t)\ln[\rho_{R}(t)]\}</annotation><annotation encoding="application/x-llamapun" id="p19.3.m3.5d">italic_S start_POSTSUBSCRIPT ent end_POSTSUBSCRIPT ( italic_t ) = - Tr { italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t ) roman_ln [ italic_ρ start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_t ) ] }</annotation></semantics></math>. We emphasize that the scenario here is basically different from that described by Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.E9" title="In Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">9</span></a>), where the size of <math alttext="R" class="ltx_Math" display="inline" id="p19.4.m4.1"><semantics id="p19.4.m4.1a"><mi id="p19.4.m4.1.1" xref="p19.4.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="p19.4.m4.1b"><ci id="p19.4.m4.1.1.cmml" xref="p19.4.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="p19.4.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="p19.4.m4.1d">italic_R</annotation></semantics></math> is infinitely large. Results for various values of <math alttext="\theta" class="ltx_Math" display="inline" id="p19.5.m5.1"><semantics id="p19.5.m5.1a"><mi id="p19.5.m5.1.1" xref="p19.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="p19.5.m5.1b"><ci id="p19.5.m5.1.1.cmml" xref="p19.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="p19.5.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="p19.5.m5.1d">italic_θ</annotation></semantics></math> are depicted in the right panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#S0.F2" title="Figure 2 ‣ Exact Hidden Markovian Dynamics in Quantum Circuits"><span class="ltx_text ltx_ref_tag">2</span></a>. Following a similar rate of linear growth in the early stage of evolution, the entropies approach the <math alttext="\theta" class="ltx_Math" display="inline" id="p19.6.m6.1"><semantics id="p19.6.m6.1a"><mi id="p19.6.m6.1.1" xref="p19.6.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="p19.6.m6.1b"><ci id="p19.6.m6.1.1.cmml" xref="p19.6.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="p19.6.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="p19.6.m6.1d">italic_θ</annotation></semantics></math>-dependent steady values after finite time steps. Among the steady values, the maximal entropy <math alttext="4\ln(2)" class="ltx_Math" display="inline" id="p19.7.m7.2"><semantics id="p19.7.m7.2a"><mrow id="p19.7.m7.2.3" xref="p19.7.m7.2.3.cmml"><mn id="p19.7.m7.2.3.2" xref="p19.7.m7.2.3.2.cmml">4</mn><mo id="p19.7.m7.2.3.1" lspace="0.167em" xref="p19.7.m7.2.3.1.cmml">⁢</mo><mrow id="p19.7.m7.2.3.3.2" xref="p19.7.m7.2.3.3.1.cmml"><mi id="p19.7.m7.1.1" xref="p19.7.m7.1.1.cmml">ln</mi><mo id="p19.7.m7.2.3.3.2a" xref="p19.7.m7.2.3.3.1.cmml">⁡</mo><mrow id="p19.7.m7.2.3.3.2.1" xref="p19.7.m7.2.3.3.1.cmml"><mo id="p19.7.m7.2.3.3.2.1.1" stretchy="false" xref="p19.7.m7.2.3.3.1.cmml">(</mo><mn id="p19.7.m7.2.2" xref="p19.7.m7.2.2.cmml">2</mn><mo id="p19.7.m7.2.3.3.2.1.2" stretchy="false" xref="p19.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="p19.7.m7.2b"><apply id="p19.7.m7.2.3.cmml" xref="p19.7.m7.2.3"><times id="p19.7.m7.2.3.1.cmml" xref="p19.7.m7.2.3.1"></times><cn id="p19.7.m7.2.3.2.cmml" type="integer" xref="p19.7.m7.2.3.2">4</cn><apply id="p19.7.m7.2.3.3.1.cmml" xref="p19.7.m7.2.3.3.2"><ln id="p19.7.m7.1.1.cmml" xref="p19.7.m7.1.1"></ln><cn id="p19.7.m7.2.2.cmml" type="integer" xref="p19.7.m7.2.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p19.7.m7.2c">4\ln(2)</annotation><annotation encoding="application/x-llamapun" id="p19.7.m7.2d">4 roman_ln ( 2 )</annotation></semantics></math> is achieved by the case <math alttext="\theta=\pi/4" class="ltx_Math" display="inline" id="p19.8.m8.1"><semantics id="p19.8.m8.1a"><mrow id="p19.8.m8.1.1" xref="p19.8.m8.1.1.cmml"><mi id="p19.8.m8.1.1.2" xref="p19.8.m8.1.1.2.cmml">θ</mi><mo id="p19.8.m8.1.1.1" xref="p19.8.m8.1.1.1.cmml">=</mo><mrow id="p19.8.m8.1.1.3" xref="p19.8.m8.1.1.3.cmml"><mi id="p19.8.m8.1.1.3.2" xref="p19.8.m8.1.1.3.2.cmml">π</mi><mo id="p19.8.m8.1.1.3.1" xref="p19.8.m8.1.1.3.1.cmml">/</mo><mn id="p19.8.m8.1.1.3.3" xref="p19.8.m8.1.1.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="p19.8.m8.1b"><apply id="p19.8.m8.1.1.cmml" xref="p19.8.m8.1.1"><eq id="p19.8.m8.1.1.1.cmml" xref="p19.8.m8.1.1.1"></eq><ci id="p19.8.m8.1.1.2.cmml" xref="p19.8.m8.1.1.2">𝜃</ci><apply id="p19.8.m8.1.1.3.cmml" xref="p19.8.m8.1.1.3"><divide id="p19.8.m8.1.1.3.1.cmml" xref="p19.8.m8.1.1.3.1"></divide><ci id="p19.8.m8.1.1.3.2.cmml" xref="p19.8.m8.1.1.3.2">𝜋</ci><cn id="p19.8.m8.1.1.3.3.cmml" type="integer" xref="p19.8.m8.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="p19.8.m8.1c">\theta=\pi/4</annotation><annotation encoding="application/x-llamapun" id="p19.8.m8.1d">italic_θ = italic_π / 4</annotation></semantics></math>, corresponding to the cluster state as the initial MPS. 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This observation hints at the existence of hidden conservation quantities, which demands further research.</p> </div> <figure class="ltx_figure" id="S0.F2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_square" height="367" id="S0.F2.g1" src="x2.png" width="375"/></div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_square" height="357" id="S0.F2.g2" src="x3.png" width="415"/></div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Entanglement dynamics. Left panel: illustration of the joint system, where the right region consists of four sites. Right panel: Growth of subsystem von Neumann entropies for different left initial MPSs, which are characterized by the value of <math alttext="\theta" class="ltx_Math" display="inline" id="S0.F2.4.m1.1"><semantics id="S0.F2.4.m1.1b"><mi id="S0.F2.4.m1.1.1" xref="S0.F2.4.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S0.F2.4.m1.1c"><ci id="S0.F2.4.m1.1.1.cmml" xref="S0.F2.4.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.4.m1.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S0.F2.4.m1.1e">italic_θ</annotation></semantics></math>. The right initial state and two-site gates are kept the same. The horizontal dotted line marks the maximal entropy <math alttext="4\ln(2)" class="ltx_Math" display="inline" id="S0.F2.5.m2.2"><semantics id="S0.F2.5.m2.2b"><mrow id="S0.F2.5.m2.2.3" xref="S0.F2.5.m2.2.3.cmml"><mn id="S0.F2.5.m2.2.3.2" xref="S0.F2.5.m2.2.3.2.cmml">4</mn><mo id="S0.F2.5.m2.2.3.1" lspace="0.167em" xref="S0.F2.5.m2.2.3.1.cmml">⁢</mo><mrow id="S0.F2.5.m2.2.3.3.2" xref="S0.F2.5.m2.2.3.3.1.cmml"><mi id="S0.F2.5.m2.1.1" xref="S0.F2.5.m2.1.1.cmml">ln</mi><mo id="S0.F2.5.m2.2.3.3.2b" xref="S0.F2.5.m2.2.3.3.1.cmml">⁡</mo><mrow id="S0.F2.5.m2.2.3.3.2.1" xref="S0.F2.5.m2.2.3.3.1.cmml"><mo id="S0.F2.5.m2.2.3.3.2.1.1" stretchy="false" xref="S0.F2.5.m2.2.3.3.1.cmml">(</mo><mn id="S0.F2.5.m2.2.2" xref="S0.F2.5.m2.2.2.cmml">2</mn><mo id="S0.F2.5.m2.2.3.3.2.1.2" stretchy="false" xref="S0.F2.5.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S0.F2.5.m2.2c"><apply id="S0.F2.5.m2.2.3.cmml" xref="S0.F2.5.m2.2.3"><times id="S0.F2.5.m2.2.3.1.cmml" xref="S0.F2.5.m2.2.3.1"></times><cn id="S0.F2.5.m2.2.3.2.cmml" type="integer" xref="S0.F2.5.m2.2.3.2">4</cn><apply id="S0.F2.5.m2.2.3.3.1.cmml" xref="S0.F2.5.m2.2.3.3.2"><ln id="S0.F2.5.m2.1.1.cmml" xref="S0.F2.5.m2.1.1"></ln><cn id="S0.F2.5.m2.2.2.cmml" type="integer" xref="S0.F2.5.m2.2.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.5.m2.2d">4\ln(2)</annotation><annotation encoding="application/x-llamapun" id="S0.F2.5.m2.2e">4 roman_ln ( 2 )</annotation></semantics></math> approached by <math alttext="\theta=\pi/4" class="ltx_Math" display="inline" id="S0.F2.6.m3.1"><semantics id="S0.F2.6.m3.1b"><mrow id="S0.F2.6.m3.1.1" xref="S0.F2.6.m3.1.1.cmml"><mi id="S0.F2.6.m3.1.1.2" xref="S0.F2.6.m3.1.1.2.cmml">θ</mi><mo id="S0.F2.6.m3.1.1.1" xref="S0.F2.6.m3.1.1.1.cmml">=</mo><mrow id="S0.F2.6.m3.1.1.3" xref="S0.F2.6.m3.1.1.3.cmml"><mi id="S0.F2.6.m3.1.1.3.2" xref="S0.F2.6.m3.1.1.3.2.cmml">π</mi><mo id="S0.F2.6.m3.1.1.3.1" xref="S0.F2.6.m3.1.1.3.1.cmml">/</mo><mn id="S0.F2.6.m3.1.1.3.3" xref="S0.F2.6.m3.1.1.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S0.F2.6.m3.1c"><apply id="S0.F2.6.m3.1.1.cmml" xref="S0.F2.6.m3.1.1"><eq id="S0.F2.6.m3.1.1.1.cmml" xref="S0.F2.6.m3.1.1.1"></eq><ci id="S0.F2.6.m3.1.1.2.cmml" xref="S0.F2.6.m3.1.1.2">𝜃</ci><apply id="S0.F2.6.m3.1.1.3.cmml" xref="S0.F2.6.m3.1.1.3"><divide id="S0.F2.6.m3.1.1.3.1.cmml" xref="S0.F2.6.m3.1.1.3.1"></divide><ci id="S0.F2.6.m3.1.1.3.2.cmml" xref="S0.F2.6.m3.1.1.3.2">𝜋</ci><cn id="S0.F2.6.m3.1.1.3.3.cmml" type="integer" xref="S0.F2.6.m3.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.6.m3.1d">\theta=\pi/4</annotation><annotation encoding="application/x-llamapun" id="S0.F2.6.m3.1e">italic_θ = italic_π / 4</annotation></semantics></math>. </figcaption> </figure> <div class="ltx_para" id="p20"> <p class="ltx_p" id="p20.1"><span class="ltx_text ltx_font_italic" id="p20.1.1">Conclusions</span>—In summary, we have established a systematic approach toward construction of nonintegrable quantum circuits exhibiting exact hidden Markovian subsystem dynamics. We introduced new principles beyond dual-unitary circuits allowing for closed-form influence matrices of finite bond dimension, which are formulated into the solvable condition on local unitary gates. Utilizing the tensor-network method, we demonstrated that the system acts as time-local boundary quantum channels on the subsystem, thus inducing exact hidden Markovian dynamics. Our constructions have unveiled a novel spacetime duality between the initial state MPS of the system and boundary quantum channels acting on the reduced subsystem.</p> </div> <div class="ltx_para" id="p21"> <p class="ltx_p" id="p21.1">Our work opens up many avenues for future research, such as the introduction of measurements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib58" title="">58</a>]</cite> and dissipation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib59" title="">59</a>]</cite>, and generalizations to different architectures <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib60" title="">60</a>]</cite> and higher spatial dimensions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib62" title="">62</a>]</cite>. The exact influence matrices also provide valuable analytical tools for studying rich phenomena in quantum many-body dynamics, including quantum chaos <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib63" title="">63</a>]</cite>, scrambling <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib64" title="">64</a>]</cite>, and deep thermalization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib65" title="">65</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib66" title="">66</a>]</cite>.</p> </div> <div class="ltx_para" id="p22"> <p class="ltx_p" id="p22.1">Furthermore, our findings on the exact hidden Markovian property could provide fresh insights into the fundamental understanding of the Markovian approximation. A valid Markovian approximation often resorts to weak system-bath coupling and the separation of timescales <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib67" title="">67</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib68" title="">68</a>]</cite>. However, in a quantum many-body system, focusing on a subsystem and integrating out the rest usually yields non-Markovian subsystem dynamics, because the “bath” and the subsystem typically undergo the same type of dynamics and thus the corresponding timescales are comparable <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib69" title="">69</a>]</cite>. In contrast, in our construction the Markovian property emerges nonperturbatively as a consequence of the solvable condition, even provided the presence of initial genuine quantum correlations between the subsystem and the bath. Understanding deep relations between these solvable quantum circuits and the Markovian approximation remains an intriguing area for further exploration. </p> </div> <div class="ltx_para" id="p23"> <p class="ltx_p" id="p23.1"><span class="ltx_text ltx_font_italic" id="p23.1.1">Acknowledgments</span>—We thank Tianci Zhou for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 12125405), and National Key R<math alttext="\&" class="ltx_Math" display="inline" id="p23.1.m1.1"><semantics id="p23.1.m1.1a"><mo id="p23.1.m1.1.1" xref="p23.1.m1.1.1.cmml">&</mo><annotation-xml encoding="MathML-Content" id="p23.1.m1.1b"><and id="p23.1.m1.1.1.cmml" xref="p23.1.m1.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="p23.1.m1.1c">\&</annotation><annotation encoding="application/x-llamapun" id="p23.1.m1.1d">&</annotation></semantics></math>D Program of China (No. 2023YFA1406702).</p> </div> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deutsch [1991]</span> <span class="ltx_bibblock">J. M. 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Condens <span class="ltx_text ltx_font_bold" id="bib.bib36.1.1.1">8</span>, 355 (2017)</a>. </span> </li> <li class="ltx_bibitem" id="bib.bib37"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[37]</span> <span class="ltx_bibblock">We would like to draw a loose analogy between the relationship of zipper conditions and our solvable condition, with that of the Bethe ansatz and Yang-Baxter equations. The Bethe ansatz was initially proposed to solve exact eigenstates of the 1D Heisenberg spin chain, while Yang-Baxter equations capture the crucial point (factorizable scattering amplitudes) and the solutions give rise to a wide class of solvable models. Simlarly, zipper conditions were used to solve specific models, while our solvable condition provides a systematic approach to constructing solvable quantum circuits. </span> </li> <li class="ltx_bibitem" id="bib.bib38"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">[38]</span> <span class="ltx_bibblock">See Supplemental Materials at [URL will be inserted by publisher] for derivations of the exact influence matrix, solvable two-site shift-invariant initial MPS, derivations of the Kraus-form representation for the boundary quantum channel, analytically calculating R<math alttext="\acute{\text{e}}" class="ltx_Math" display="inline" id="bib.bib38.1.m1.1"><semantics id="bib.bib38.1.m1.1a"><mover accent="true" id="bib.bib38.1.m1.1.1" xref="bib.bib38.1.m1.1.1.cmml"><mtext id="bib.bib38.1.m1.1.1.2" xref="bib.bib38.1.m1.1.1.2a.cmml">e</mtext><mo id="bib.bib38.1.m1.1.1.1" xref="bib.bib38.1.m1.1.1.1.cmml">´</mo></mover><annotation-xml encoding="MathML-Content" id="bib.bib38.1.m1.1b"><apply id="bib.bib38.1.m1.1.1.cmml" xref="bib.bib38.1.m1.1.1"><ci id="bib.bib38.1.m1.1.1.1.cmml" xref="bib.bib38.1.m1.1.1.1">´</ci><ci id="bib.bib38.1.m1.1.1.2a.cmml" xref="bib.bib38.1.m1.1.1.2"><mtext id="bib.bib38.1.m1.1.1.2.cmml" xref="bib.bib38.1.m1.1.1.2">e</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib38.1.m1.1c">\acute{\text{e}}</annotation><annotation encoding="application/x-llamapun" id="bib.bib38.1.m1.1d">over´ start_ARG e end_ARG</annotation></semantics></math>nyi entropies dynamics for infinitely large subsystems, more details on the examples of <math alttext="q=2,\tilde{q}=1" class="ltx_Math" display="inline" id="bib.bib38.2.m2.2"><semantics id="bib.bib38.2.m2.2a"><mrow id="bib.bib38.2.m2.2.2.2" xref="bib.bib38.2.m2.2.2.3.cmml"><mrow id="bib.bib38.2.m2.1.1.1.1" xref="bib.bib38.2.m2.1.1.1.1.cmml"><mi id="bib.bib38.2.m2.1.1.1.1.2" xref="bib.bib38.2.m2.1.1.1.1.2.cmml">q</mi><mo id="bib.bib38.2.m2.1.1.1.1.1" xref="bib.bib38.2.m2.1.1.1.1.1.cmml">=</mo><mn id="bib.bib38.2.m2.1.1.1.1.3" xref="bib.bib38.2.m2.1.1.1.1.3.cmml">2</mn></mrow><mo id="bib.bib38.2.m2.2.2.2.3" xref="bib.bib38.2.m2.2.2.3a.cmml">,</mo><mrow id="bib.bib38.2.m2.2.2.2.2" xref="bib.bib38.2.m2.2.2.2.2.cmml"><mover accent="true" id="bib.bib38.2.m2.2.2.2.2.2" xref="bib.bib38.2.m2.2.2.2.2.2.cmml"><mi id="bib.bib38.2.m2.2.2.2.2.2.2" xref="bib.bib38.2.m2.2.2.2.2.2.2.cmml">q</mi><mo id="bib.bib38.2.m2.2.2.2.2.2.1" xref="bib.bib38.2.m2.2.2.2.2.2.1.cmml">~</mo></mover><mo id="bib.bib38.2.m2.2.2.2.2.1" xref="bib.bib38.2.m2.2.2.2.2.1.cmml">=</mo><mn id="bib.bib38.2.m2.2.2.2.2.3" xref="bib.bib38.2.m2.2.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="bib.bib38.2.m2.2b"><apply id="bib.bib38.2.m2.2.2.3.cmml" xref="bib.bib38.2.m2.2.2.2"><csymbol cd="ambiguous" id="bib.bib38.2.m2.2.2.3a.cmml" xref="bib.bib38.2.m2.2.2.2.3">formulae-sequence</csymbol><apply id="bib.bib38.2.m2.1.1.1.1.cmml" xref="bib.bib38.2.m2.1.1.1.1"><eq id="bib.bib38.2.m2.1.1.1.1.1.cmml" xref="bib.bib38.2.m2.1.1.1.1.1"></eq><ci id="bib.bib38.2.m2.1.1.1.1.2.cmml" xref="bib.bib38.2.m2.1.1.1.1.2">𝑞</ci><cn id="bib.bib38.2.m2.1.1.1.1.3.cmml" type="integer" xref="bib.bib38.2.m2.1.1.1.1.3">2</cn></apply><apply id="bib.bib38.2.m2.2.2.2.2.cmml" xref="bib.bib38.2.m2.2.2.2.2"><eq id="bib.bib38.2.m2.2.2.2.2.1.cmml" xref="bib.bib38.2.m2.2.2.2.2.1"></eq><apply id="bib.bib38.2.m2.2.2.2.2.2.cmml" xref="bib.bib38.2.m2.2.2.2.2.2"><ci id="bib.bib38.2.m2.2.2.2.2.2.1.cmml" xref="bib.bib38.2.m2.2.2.2.2.2.1">~</ci><ci id="bib.bib38.2.m2.2.2.2.2.2.2.cmml" xref="bib.bib38.2.m2.2.2.2.2.2.2">𝑞</ci></apply><cn id="bib.bib38.2.m2.2.2.2.2.3.cmml" type="integer" xref="bib.bib38.2.m2.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib38.2.m2.2c">q=2,\tilde{q}=1</annotation><annotation encoding="application/x-llamapun" id="bib.bib38.2.m2.2d">italic_q = 2 , over~ start_ARG italic_q end_ARG = 1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="bib.bib38.3.m3.1"><semantics id="bib.bib38.3.m3.1a"><mn id="bib.bib38.3.m3.1.1" xref="bib.bib38.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="bib.bib38.3.m3.1b"><cn id="bib.bib38.3.m3.1.1.cmml" type="integer" xref="bib.bib38.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="bib.bib38.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="bib.bib38.3.m3.1d">2</annotation></semantics></math>, and more details on the additional solvable condition, which further includes Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib70" title="">70</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib71" title="">71</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.14807v2#bib.bib72" title="">72</a>, <a class="ltx_ref" 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