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Robbiati, Matteo; Pedicillo, Edoardo; Pasquale, Andrea; Li, Xiaoyue; Wright, Andrew; Farias, Renato M.S.; Giang, Khanh Uyen; Son, Jeongrak; Knörzer, Johannes; Goh, Siong Thye; Khoo, Jun Yong; Ng, Nelly H.Y.; Holmes, Zoë; Carrazza, Stefano; Gluza, Marek" /> <meta name="keywords" content="CERN Document Server, WebSearch, CERN Document Server" /> <script type="text/javascript" src="https://cds.cern.ch/js/jquery.min.js"></script>  <link rel="stylesheet" href="https://cds.cern.ch/img/webnews.css" type="text/css" />  <script type="text/javascript" src="https://cds.cern.ch/js/webnews.js?v=20131009"></script> <meta property="fb:app_id" content="137353533001720"/> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: {inlineMath: [['$','$']], processEscapes: true}, showProcessingMessages: false, messageStyle: "none" }); </script> <script src="/MathJax/MathJax.js?config=TeX-AMS_CHTML" type="text/javascript"> </script>  <meta content="arXiv : Double-bracket quantum algorithms for high-fidelity ground state preparation" name="citation_title" /> <meta content="Robbiati, Matteo" name="citation_author" /> <meta content="Li, Xiaoyue" name="citation_author" /> <meta content="Holmes, Zoë" name="citation_author" /> <meta content="Giang, Khanh Uyen" name="citation_author" /> <meta content="Son, Jeongrak" name="citation_author" /> <meta content="Goh, Siong Thye" name="citation_author" /> <meta content="Carrazza, Stefano" name="citation_author" /> <meta content="Wright, Andrew" name="citation_author" /> <meta content="Ng, Nelly H.Y." name="citation_author" /> <meta content="Pasquale, Andrea" name="citation_author" /> <meta content="Knörzer, Johannes" name="citation_author" /> <meta content="Farias, Renato M.S." name="citation_author" /> <meta content="Khoo, Jun Yong" name="citation_author" /> <meta content="Gluza, Marek" name="citation_author" /> <meta content="Pedicillo, Edoardo" name="citation_author" /> <meta content="2024/08/07" name="citation_publication_date" /> <meta name="citation_online_date" content="2024/08/22"> <meta content="TIF-UNIMI-2024-6" name="citation_technical_report_number" /> <meta content="arXiv:2408.03987" name="citation_technical_report_number" /> <meta name="citation_pdf_url" content="https://cds.cern.ch/record/2907597/files/2408.03987.pdf" />  <meta content="arXiv" property="og:title" /> <meta content="Double-bracket quantum algorithms for high-fidelity ground state preparation" property="og:title" /> <meta content="website" property="og:type" /> <meta content="https://cds.cern.ch/record/2907597" property="og:url" /> <meta name="twitter:image" content="https://cds.cern.ch/record/2907597/files/diagram.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2907597/files/diagram.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2907597/files/diagram.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2907597/files/hdw_efficient_XXZ_10Q7L13S_cma_jumps.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2907597/files/hdw_efficient_XXZ_10Q7L13S_cma_jumps.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2907597/files/hdw_efficient_XXZ_10Q7L13S_cma_scatter.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2907597/files/hdw_efficient_XXZ_10Q7L13S_cma_scatter.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2907597/files/hw_preserving_XXZ_10Q3L42S_cma_jumps.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2907597/files/hw_preserving_XXZ_10Q3L42S_cma_jumps.png" /> <meta content="CERN Document Server" property="og:site_name" /> <meta content="arXiv" property="og:description" /> <meta content="Ground state preparation is a key area where quantum computers are expected to prove advantageous. Double-bracket quantum algorithms (DBQAs) have been recently proposed to diagonalize Hamiltonians and in this work we show how to use them to prepare ground states. We propose to improve an initial state preparation by adding a few steps of DBQAs. The interfaced method systematically achieves a better fidelity while significantly reducing the computational cost of the procedure. For a Heisenberg model, we compile our algorithm using CZ and single-qubit gates into circuits that match capabilities of near-term quantum devices. Moreover, we show that DBQAs can benefit from the experimental availability of increasing circuit depths. Whenever an approximate ground state can be prepared without exhausting the available circuit depth, then DBQAs can be enlisted to algorithmically seek a higher fidelity preparation." property="og:description" />  <meta content="summary" name="twitter:card" /> <style></style> </head> <body class="CERN32Document32Server search" lang="ja">  <div id="cern-toolbar"> <h1><a href="http://cern.ch" title="CERN">CERN <span>Accelerating science</span></a></h1> <ul> <li class="cern-accountlinks"><a class="cern-account" href="https://cds.cern.ch/youraccount/login?ln=ja&referer=https%3A//cds.cern.ch/record/2907597%3Fln%3Dja" title="Sign in to your CERN account">Sign in</a></li> <li><a class="cern-directory" href="http://cern.ch/directory" title="Search CERN resources and browse the directory">Directory</a></li> </ul> </div>   <div role="banner" class="clearfix" id="header"> <div class="header-inner inner"> <hgroup class="clearfix"> <h2 id="site-name"> <a rel="home" title="Home" href="/"><span>CERN Document Server</span></a> </h2> <h3 id="site-slogan">Access articles, reports and multimedia content in HEP</h3> </hgroup> <div role="navigation" id="main-navigation" class="cdsmenu"> <h2 class="element-invisible">Main menu</h2><ul class="links inline clearfix"> <li class="menu-386 first active-trail"><a class="active-trail" href="https://cds.cern.ch/?ln=ja">検索</a></li> <li class="menu-444 "><a class="" title="" href="https://cds.cern.ch/submit?ln=ja">アップロード</a></li> <li class="menu-426 "><a class="" href="https://cds.cern.ch/help/?ln=ja">ヘルプ</a></li> <li class="leaf hassubcdsmenu"> <a hreflang="en" class="header" href="https://cds.cern.ch/youraccount/display?ln=ja">あなたのページ</a> <ul class="subsubcdsmenu"><li><a href="https://cds.cern.ch/youralerts/list?ln=ja">Your alerts</a></li><li><a href="https://cds.cern.ch/yourbaskets/display?ln=ja">Your baskets</a></li><li><a href="https://cds.cern.ch/yourcomments?ln=ja">Your comments</a></li><li><a href="https://cds.cern.ch/youralerts/display?ln=ja">Your searches</a></li></ul></li> </ul> </div> </div> </div>  <table class="navtrailbox"> <tr> <td class="navtrailboxbody"> <a href="/?ln=ja" class="navtrail">ホーム</a> > Double-bracket quantum algorithms for high-fidelity ground state preparation </td> </tr> </table> </div> <div class="pagebody"><div class="pagebodystripemiddle"> <div class="detailedrecordbox"> <div class="detailedrecordtabs"> <div> <ul class="detailedrecordtabs"><li class="on first"><a href="/record/2907597/?ln=ja">Information </a></li><li class=""><a href="/record/2907597/files?ln=ja">Files </a></li></ul> <div id="tabsSpacer" style="clear:both;height:0px"> </div></div> </div> <div class="detailedrecordboxcontent"> <div class="top-left-folded"></div> <div class="top-right-folded"></div> <div class="inside">  <abbr class="unapi-id" title="2907597"></abbr>  <link href="/img/download_and_embed_buttons.css" rel="stylesheet" type="text/css" /> <style type="text/css">  .video_talk_wrapper{ max-width: 640px; min-height: 360px; } </style> <table class="formatRecordTableFullWidth"> <tr> <td class="formatRecordHeader" style="background-image: url('/img/journals.jpg');" colspan="2">  Preprint </td> </tr> <tr><td class="formatRecordLabel"> Report number </td><td style="padding-left:5px;">TIF-UNIMI-2024-6 ; <a href="http://arxiv.org/abs/arXiv:2408.03987">arXiv:2408.03987</a></td></tr> <tr><td class="formatRecordLabel"> Title </td><td style="padding-left:5px;"><b>Double-bracket quantum algorithms for high-fidelity ground state preparation</b></td></tr> <style> .video_control_container, #video_controlbutton { display: none; }#video_detailbox_download {display: none;}#video_detailbox_embed {display: none;}</style> <div class="video_control_container"> <div style="box-sizing: inherit;position: relative;min-height: 1em;margin: 1em 0;background: #f8f8f9;padding: 1em 1.5em;line-height: 1.4285em;transition: opacity .1s ease,color .1s ease,background .1s ease,box-shadow .1s ease,-webkit-box-shadow .1s ease;border-radius: .28571429rem;font-size: 1.1em;margin-top: 0;margin-bottom: 6px;background-color: #f8ffff;color: #276f86;box-shadow: 0 0 0 1px #a9d5de inset,0 0 0 0 transparent;"> <i class="fa fa-info"></i><span style="padding-left: 9px;">If you experience any problem watching the video, click the download button below</span> </div> <a id="video_controlbutton_download" rel="download_button" class="video_controlbutton" href="#">Download</a> <a id="video_controlbutton_embed" class="video_controlbutton" href="#">Embed</a> </div> <div id="video_detailbox_download" style="margin:10px"> </div> <div id="video_detailbox_embed" style="margin:10px"> </div> <tr><td class="formatRecordLabel"> <span style="white-space:nowrap;">Author(s)</span> </td><td style="padding-left:5px;"> <script type="text/javascript"> function toggle_authors_visibility(){ var more = document.getElementById('more'); var link = document.getElementById('link'); var extension = document.getElementById('extension'); if (more.style.display=='none'){ more.style.display = ''; extension.style.display = 'none'; link.innerHTML = "隠す" } else { more.style.display = 'none'; extension.style.display = ''; link.innerHTML = "15 人のオーサーを表示" } link.style.color = "rgb(204,0,0);" } function set_up(){ var extension = document.getElementById('extension'); extension.innerHTML = ""; toggle_authors_visibility(); } </script> <a name="show_hide" /><a href="https://cds.cern.ch/search?f=author&p=Robbiati%2C%20Matteo&ln=ja">Robbiati, Matteo</a> (CERN ; Milan U.) ; <a href="https://cds.cern.ch/search?f=author&p=Pedicillo%2C%20Edoardo&ln=ja">Pedicillo, Edoardo</a> (Milan U. ; Technol. Innovation Inst., UAE) ; <a href="https://cds.cern.ch/search?f=author&p=Pasquale%2C%20Andrea&ln=ja">Pasquale, Andrea</a> (Milan U. ; Technol. Innovation Inst., UAE) ; <a href="https://cds.cern.ch/search?f=author&p=Li%2C%20Xiaoyue&ln=ja">Li, Xiaoyue</a> (Nanyang Technol. U.) ; <a href="https://cds.cern.ch/search?f=author&p=Wright%2C%20Andrew&ln=ja">Wright, Andrew</a> (LPHE, Lausanne) ; <a href="https://cds.cern.ch/search?f=author&p=Farias%2C%20Renato%20M.S.&ln=ja">Farias, Renato M.S.</a> (Technol. Innovation Inst., UAE ; Rio de Janeiro Federal U.) ; <a href="https://cds.cern.ch/search?f=author&p=Giang%2C%20Khanh%20Uyen&ln=ja">Giang, Khanh Uyen</a> (Nanyang Technol. U.) ; <a href="https://cds.cern.ch/search?f=author&p=Son%2C%20Jeongrak&ln=ja">Son, Jeongrak</a> (Nanyang Technol. U.) ; <a href="https://cds.cern.ch/search?f=author&p=Kn%C3%B6rzer%2C%20Johannes&ln=ja">Knörzer, Johannes</a> (Zurich, ETH) ; <a href="https://cds.cern.ch/search?f=author&p=Goh%2C%20Siong%20Thye&ln=ja">Goh, Siong Thye</a> (A-STAR, Singapore)<span id="more" style=""> ; <a href="https://cds.cern.ch/search?f=author&p=Khoo%2C%20Jun%20Yong&ln=ja">Khoo, Jun Yong</a> (A-STAR, Singapore) ; <a href="https://cds.cern.ch/search?f=author&p=Ng%2C%20Nelly%20H.Y.&ln=ja">Ng, Nelly H.Y.</a> (Nanyang Technol. U.) ; <a href="https://cds.cern.ch/search?f=author&p=Holmes%2C%20Zo%C3%AB&ln=ja">Holmes, Zoë</a> (LPHE, Lausanne) ; <a href="https://cds.cern.ch/search?f=author&p=Carrazza%2C%20Stefano&ln=ja">Carrazza, Stefano</a> (CERN ; Milan U. ; Technol. Innovation Inst., UAE ; INFN, Milan) ; <a href="https://cds.cern.ch/search?f=author&p=Gluza%2C%20Marek&ln=ja">Gluza, Marek</a> (Nanyang Technol. U.)</span> <span id="extension"></span> <small><i><a id="link" href="#" onclick="toggle_authors_visibility()" style="color:rgb(204,0,0);"></a></i></small><script type="text/javascript">set_up()</script></td></tr>  <script type="text/javascript"> /* * Changes the Imprint title to Date - Duration as requested by RQF0820114 * This is a hack not to touch the general imprint format element */ $(document).ready(function() { var collections = 'PREPRINT'; if (collections.indexOf('E-LEARNING') > -1) { $('.cern-imprint-label').text('Date - Duration'); } }) </script> <tr><td class="formatRecordLabel cern-imprint-label"> Imprint </td><td style="padding-left:5px;">2024-08-07</td></tr> <tr><td class="formatRecordLabel"> Number of pages </td><td style="padding-left:5px;">18</td></tr>  <tr><td class="formatRecordLabel"> Note </td><td style="padding-left:5px;">5 pages + appendix, 4 figures, code available at: <a href="https://github.com/qiboteam/boostvqe">https://github.com/qiboteam/boostvqe</a></td></tr> <tr><td class="formatRecordLabel"> Subject category </td><td style="padding-left:5px;">quant-ph ; General Theoretical Physics</td></tr> <tr><td class="formatRecordLabel"> Abstract </td><td style="padding-left:5px;">Ground state preparation is a key area where quantum computers are expected to prove advantageous. Double-bracket quantum algorithms (DBQAs) have been recently proposed to diagonalize Hamiltonians and in this work we show how to use them to prepare ground states. We propose to improve an initial state preparation by adding a few steps of DBQAs. The interfaced method systematically achieves a better fidelity while significantly reducing the computational cost of the procedure. For a Heisenberg model, we compile our algorithm using CZ and single-qubit gates into circuits that match capabilities of near-term quantum devices. Moreover, we show that DBQAs can benefit from the experimental availability of increasing circuit depths. Whenever an approximate ground state can be prepared without exhausting the available circuit depth, then DBQAs can be enlisted to algorithmically seek a higher fidelity preparation.</td></tr> <tr><td class="formatRecordLabel"> Other source </td><td style="padding-left:5px;"><a href="http://inspirehep.net/record/2816404">Inspire</a></td></tr> <tr><td class="formatRecordLabel"> Copyright/License </td><td style="padding-left:5px;">preprint: (License: <a href="http://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>)</td></tr> </table> <br /><div style="max-width:1024px;margin:auto"><div style="overflow-x:auto;display:inline;width:100%;"><a href="/record/2907597/plots#0"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2907597/files/diagram.png" title=" We propose a two-stage ground state preparation protocol: first, apply a relatively short-depth warm-start circuit; second, apply a DBQA circuit to further the ground state preparation fidelity." width="200px"/></a> <a href="/record/2907597/plots#1"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2907597/files/hw_preserving_XXZ_10Q3L42S_cma_jumps.png" title=" Visualization of the impact of \VQExDBQA~on cost function for a single VQE random seed, see Tab.~\ref{tab:xxz_results} for statistical analysis. \emph{(left)}~Training of VQE (blue lines) for 3, 4, and 5 layers of Hamming-weight preserving ansatz (hues of blue) achieves within 500 training epochs ground state energy residue $\Delta E\approx 1\%$. For more epochs, the initially rapid decrease in the cost function saturates and shows marginal improvement afterward. We initialize DBQA with VQE for selected epochs $\in \{100, 200, 500, 1000, 2000\}$ and optimize DBQA parameters with CMA-ES~\cite{cma}. In the bottom panel we show the relative difference value between the achieved energy $\tilde E_0$ and the true ground state energy $E_0$. \emph{(right)} Token cost estimates of \VQExDBQA~by counting the total number of CZ gates required for the complete protocol: training the VQE until a target epoch and the optimization of DBQA." width="200px"/></a> <a href="/record/2907597/plots#2"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2907597/files/hdw_efficient_XXZ_10Q7L13S_cma_jumps.png" title=" One example of \VQExDBQA~for $\XXZ$~ using a hardware efficient ansatz and obtained fixing the simulation random seed; the image is intended to provide qualitative information about the impact of \VQExDBQA. A more robust study of the performance is presented in Tab.~\ref{tab:xxz_results}.\emph{(left)} Training of VQE (blue lines) for 7, 8, and 9 layers (hues of blue) achieve within 500 training epochs ground state energy residue of about 1\%. We initialize DBQA with VQE for selected epochs $\in [1000, 2000, 3000, 4000, 5000]$ where we apply a DBQA optimized in its parameters with CMA-ES~\cite{cma}. \emph{(right)} Token cost estimates of \VQExDBQA~by counting the total number of two-qubit gates required to execute the complete protocol: training the VQE until a target epoch and then optimizing and applying the DBQA." width="200px"/></a> <a href='/record/2907597/plots'>Show more plots</a></div></div><br /> <br/><div><div style="clear: both;"> </div></div> <script type="text/javascript"> // Initially hide: $(".longCaption").hide(); // Allow to toggle visibility: $(".toggleLongCaption").toggle(function(){ $(this).siblings(".longCaption").show('fast'); var thisElem = $(this); thisElem.text(thisElem.text() === "さらに" ? "less" : "さらに"); },function(){ $(this).siblings(".longCaption").hide('fast'); var thisElem = $(this); thisElem.text(thisElem.text() === "さらに" ? 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