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<p>
See this nice paper by Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave Wecker, and Matthias Troyer, on <a href="https://arxiv.org/abs/1605.03590" target="_blank">Elucidating Reaction Mechanisms on Quantum Computers</a> </p>
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Quantum simulation issues and ideas
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<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
Which of the following statements is true about quantum simulations, as discussed by Prof. Chuang in lecture? </p>
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<text>The simulation output is generally a quantum state, which collapses upon measurement, and thus how efficiently the desired result can be measured is a key issue</text>
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<text>Improved approximations may allow classical simulations to superced the efficiency of quantum simulations for certain problems</text>
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<text>Quantum simulations need not be digital; they may also involve analog deformation of one system's Hamiltonian to become the desired system's Hamiltonign</text>
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<text>A key early step in quantum simulation is ensuring that the simulating system's Hamiltonian can in principle be used to generate the desired system's Hamiltonian</text>
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console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
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<h2 class="hd hd-2 unit-title">Grover's quantum search algorithm - recap</h2>
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<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
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<div class="closed-captions"></div>
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<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
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var course_root_link = $('span.nav-item-course').find('a').attr('href');
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});
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get_search_module();
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new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
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$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
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}
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if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
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'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
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var vuid = vblock.data('usage-id').split('@');
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vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
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gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
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catch(err){
console.log(err);
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setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
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<h2 class="hd hd-2 unit-title">CQ: Grover's search algorithm as a quantum simulation</h2>
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Grover&#39;s search algorithm as a quantum simulation
</h3>
<div class="problem-progress" id="block-v1:MITx+8.370.2x+1T2018+type@problem+block@u46_cq_grover_sim-problem-progress"></div>
<div class="problem">
<div>
<p>
This is a concept question, provided to double-check your understanding of the previous video clip. </p>
<p>
What does the analysis of the Hamiltonian simulation of Grover's algorithm show? Check all of the following that apply: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_u46_cq_grover_sim_2_1">
<fieldset aria-describedby="status_u46_cq_grover_sim_2_1">
<div class="field">
<input type="checkbox" name="input_u46_cq_grover_sim_2_1[]" id="input_u46_cq_grover_sim_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="u46_cq_grover_sim_2_1-choice_0-label" for="input_u46_cq_grover_sim_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_u46_cq_grover_sim_2_1">
<text>The simulation of [mathjaxinline]H = H_1 + H_2[/mathjaxinline] can alawys be done with no error, by composing separate simulations of [mathjaxinline]H_1[/mathjaxinline] and [mathjaxinline]H_2[/mathjaxinline]</text>
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<text>Any quantum algorithm [mathjaxinline]U[/mathjaxinline] can be simulated just by obtaining its Hamiltonian [mathjaxinline]H[/mathjaxinline] from expressing [mathjaxinline]U=e^{iHt}[/mathjaxinline]</text>
</label>
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<text>The cost of Hamiltonian simulation must include steps needed to generate [mathjaxinline]H=H_1+H_2[/mathjaxinline] from [mathjaxinline]H_1[/mathjaxinline] and [mathjaxinline]H_2[/mathjaxinline], e.g. via Trotterization, and this can be expensive</text>
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<text>Oracles such as are used in Grover's algorithm to mark the search result, cannot be included in a Hamiltonian model for quantum algorithms, because they are unitary circuits</text>
</label>
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// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
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{
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console.log("3. search_module_url = ", search_module_url);
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console.log("course_root_link = ", course_root_link);
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var sterm = $('#site-search-box').val();
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new_url = search_module_url + "?q=" + sterm;
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+ " id='site-search-box'"
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if (event.which == 13) {
event.preventDefault();
go_to_search();
}
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// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
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'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
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}
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}else{
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}
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mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
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var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
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var cindex = Number(cst.data('index'));
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gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
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<p><img src="/assets/courseware/v1/aa2ce1d55ecd0adeb2315f66132c0b9b/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_Nature_Cover_Final_Full2-594x782.jpg" width="300"/></p><p>
Quantum computers have now been used to simulate simple molecules. See, for example: </p><ul class="itemize"><li><p>
IBM paper on <a href="https://www.nature.com/articles/nature23879" target="_blank">Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets</a> (Nature, v549, p242, 2017) </p></li><li><p>
Google paper on <a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.031007" target="_blank">Scalable Quantum Simulation of Molecular Energies</a> (Phys. Rev. X, v6, 031007, 2016) </p></li></ul>
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<h2 class="hd hd-2 unit-title">Quantum simulation for quantum chemistry - variational quantum eigensolver</h2>
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<p>
Quantum simulations of the small molecules H2 and BeH2 have been realized using superconducting qubits running the variational quantum eignsolver algorithm. Below is a video discussing the experiment, which was done at IBM in 2017. </p>
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<h3 class="hd hd-2">Experimental quantum simulation of small molecules using VQE</h3>
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<h2 class="hd hd-2 unit-title">CQ: Quantum chemistry simulations</h2>
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Quantum chemistry simulations
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This is a concept question, provided to double-check your understanding of the previous video clip. </p>
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Which of the following statements are true, regarding the quantum simulation of quantum chemistry problems, and the Hamiltonian [mathjaxinline]H_{QC}[/mathjaxinline] involved? </p>
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<text>[mathjaxinline]H_{QC}[/mathjaxinline] describes fermionic interactions, and thus [mathjaxinline]H_{QC}[/mathjaxinline] must first be trasnformed into bosonic form for quantum simulation, e.g. by Jordan-Wigner or a Bravyi-Kitaev transformation</text>
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<text>It is generally simple to find the ground-state energy of [mathjaxinline]H_{QC}[/mathjaxinline], but very hard to find the energies of the first and higher excited states</text>
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<text>The 'variational quantum eigensolver' algorithm is provably exponentially fast, just like the quantum phase estimation algorithm</text>
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<text>The quantum phase estimation algorithm can in principle be used to simulate [mathjaxinline]H_{QC}[/mathjaxinline] efficiently, but may require very large numbers of qubits and long execution times, for even the smallest relevant quantum chemistry problems</text>
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<text>The true run-time of the variational quantum eigensolver algorithm is unknown, but this algorithm is enticing because of its use of classical computation together witn potentially short quantumm subroutines</text>
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