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<h2 class="hd hd-2 unit-title">Fault-tolerant non-clifford gates</h2>
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Clifford gate implementations via teleportation
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Which of the following are true? </p>
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<text>[mathjaxinline]\sqrt(S)[/mathjaxinline] is a Clifford gate because [mathjaxinline]S[/mathjaxinline] is a Clifford gate.</text>
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<text>Each copy of the state [mathjaxinline]|T\rangle[/mathjaxinline] allows us one use of a [mathjaxinline]T[/mathjaxinline] gate in computation.</text>
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<text>Let [mathjaxinline]V[/mathjaxinline] be an arbitrary quantum circuit. We can implement [mathjaxinline]V\sigma _ a V^\dagger[/mathjaxinline] using Clifford gates only for arbitrary Pauli [mathjaxinline]\sigma _ a[/mathjaxinline].</text>
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<text>The correction step of the teleportation of circuit [mathjaxinline]V[/mathjaxinline] as described in the video involves a circuit in [mathjaxinline]C_2[/mathjaxinline].</text>
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// $('#site-search-box').bind("enterKey", go_to_search);
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window.open(gurl, "MITx 8.370x subtitle source");
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<h2 class="hd hd-2 unit-title">CQ: Fault-tolerant construction of the T gate</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.2x+2T2018+type@problem+block@cq_M2L10_Fault-tolerant_construction_of_the_T_gate">
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Fault-tolerant construction of the T gate
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.2x+2T2018+type@problem+block@cq_M2L10_Fault-tolerant_construction_of_the_T_gate-problem-progress"></div>
<div class="problem">
<div>
<p>
This is a concept question, provided to check your understanding of the previous lecture video clips. </p>
<p>
Which of the following are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cq_M2L10_Fault-tolerant_construction_of_the_T_gate_2_1">
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<text>In order to prepare the encoded [mathjaxinline]T[/mathjaxinline] gate we have to project the [mathjaxinline]|0\rangle[/mathjaxinline] state onto the [mathjaxinline]+1[/mathjaxinline] eigenspace of the logical operator [mathjaxinline]TXT^ dag[/mathjaxinline] operator.</text>
</label>
</div>
<div class="field">
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<text>Implementation of a [mathjaxinline]k^{\rm th}[/mathjaxinline] level fault-tolerant [mathjaxinline]T[/mathjaxinline] gate is possible using only single qubit rotations for arbitrary [mathjaxinline]k[/mathjaxinline].</text>
</label>
</div>
<div class="field">
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<text>[mathjaxinline]TXT^ dag[/mathjaxinline] is a Pauli operator.</text>
</label>
</div>
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<text>The [mathjaxinline]T[/mathjaxinline] gate can be implemented transversally in the Steane code.</text>
</label>
</div>
<span id="answer_cq_M2L10_Fault-tolerant_construction_of_the_T_gate_2_1"/>
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<div class="indicator-container">
<span class="status unanswered" id="status_cq_M2L10_Fault-tolerant_construction_of_the_T_gate_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
</div>
<div class="action">
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<div class="submit-attempt-container">
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<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_cq_M2L10_Fault-tolerant_construction_of_the_T_gate">
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is-hidden"
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data-graded="True">
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</div>
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<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.2x+2T2018+type@vertical+block@Magic_state_distillation_-_briefly" data-course-id="course-v1:MITx+8.371.2x+2T2018" data-has-score="False" data-graded="True" data-runtime-version="1" data-init="VerticalStudentView" data-request-token="c923a22602e811efa5270afff417eba9">
<h2 class="hd hd-2 unit-title">Magic state distillation - briefly</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10e">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-block-type="video" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10e" data-course-id="course-v1:MITx+8.371.2x+2T2018" data-has-score="False" data-graded="True" data-runtime-version="1" data-init="XBlockToXModuleShim" data-request-token="c923a22602e811efa5270afff417eba9">
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<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
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Your browser does not support this video format. Try using a different browser.
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<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
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if (typeof String.prototype.startsWith != 'function') {
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String.prototype.startsWith = function (str){
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};
}
if(typeof(String.prototype.trim) === "undefined")
{
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if ($(this).text().trim().startsWith("Search this course")){
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console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
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if (event.which == 13) {
event.preventDefault();
go_to_search();
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});
// $('#site-search-box').bind("enterKey", go_to_search);
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return eval(x);
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'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
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}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + 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;
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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/";
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setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
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catch(err){
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<h2 class="hd hd-2 unit-title">The toric code - perspective</h2>
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<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M2L10f"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
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<div class="focus_grabber last"></div>
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<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
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jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
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var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
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var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.2x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.2x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + 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/";
}else{
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();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
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</div>
</div>
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<h2 class="hd hd-2 unit-title">The toric code - definition</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-block-type="video" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g" data-course-id="course-v1:MITx+8.371.2x+2T2018" data-has-score="False" data-graded="True" data-runtime-version="1" data-init="XBlockToXModuleShim" data-request-token="c923a22602e811efa5270afff417eba9">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
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<h3 class="hd hd-2">The toric code - definition</h3>
<div
id="video_M2L10g"
class="video closed"
data-metadata='{"transcriptLanguages": {"en": "English"}, "captionDataDir": null, "autoAdvance": false, "poster": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.371.2x+2T2018/xblock/block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g/handler/publish_completion", "transcriptLanguage": "en", "showCaptions": "true", "completionEnabled": false, "prioritizeHls": false, "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.2x+2T2018/xblock/block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g/handler/transcript/available_translations", "ytApiUrl": "https://www.youtube.com/iframe_api", "autohideHtml5": false, "start": 0.0, "generalSpeed": 1.0, "recordedYoutubeIsAvailable": true, "speed": null, "end": 0.0, "ytMetadataEndpoint": "", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "autoplay": false, "saveStateUrl": "/courses/course-v1:MITx+8.371.2x+2T2018/xblock/block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g/handler/xmodule_handler/save_user_state", "saveStateEnabled": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.2x+2T2018/xblock/block-v1:MITx+8.371.2x+2T2018+type@video+block@M2L10g/handler/transcript/translation/__lang__", "streams": "1.00:PHekwqXTVq0", "completionPercentage": 0.95, "duration": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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data-poster='null'
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<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M2L10g"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.2x+2T2018+type@html+block@site_search_box7">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.2x+2T2018+type@html+block@site_search_box7" data-course-id="course-v1:MITx+8.371.2x+2T2018" data-has-score="False" data-graded="True" data-runtime-version="1" data-init="XBlockToXModuleShim" data-request-token="c923a22602e811efa5270afff417eba9">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.2x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.2x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript">
var add_site_search = function(){
course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', '');
jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
};
}
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.2x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.2x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + 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/";
}else{
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();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.371.2x+2T2018+type@vertical+block@CQ_Toric_codes" data-course-id="course-v1:MITx+8.371.2x+2T2018" data-has-score="False" data-graded="True" data-runtime-version="1" data-init="VerticalStudentView" data-request-token="c923a22602e811efa5270afff417eba9">
<h2 class="hd hd-2 unit-title">CQ: Toric codes</h2>
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Toric codes
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<text>A torus can be viewed as a two dimensional lattice with two boundary conditions.</text>
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<text>A toric code with [mathjaxinline]n[/mathjaxinline] qubits has [mathjaxinline]n[/mathjaxinline] faces.</text>
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<text>In the toric code each vertex of a torus is taken as a qubit.</text>
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<text>The stabilizers of a toric code are characterized by faces and vertices of a two-dimensional lattice.</text>
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<h2 class="hd hd-2 unit-title">CQ: Toric-code properties</h2>
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Toric-code properties
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.2x+2T2018+type@problem+block@cq_M2L10_Toric_code_properties-problem-progress"></div>
<div class="problem">
<div>
<p>
This is a concept question, provided to check your understanding of the previous lecture video clips. </p>
<p>
Which of the following are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cq_M2L10_Toric_code_properties_2_1">
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<text>The stabilizer generators of a toric code commute because [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Z[/mathjaxinline] generators always overlap on even numbers of qubits.</text>
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<text>[mathjaxinline]\partial _2[/mathjaxinline] can be identified by a square matrix.</text>
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<text>The operator [mathjaxinline]\partial _1[/mathjaxinline] takes one face and maps it onto four edges.</text>
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<text>[mathjaxinline]\partial _1[/mathjaxinline] and [mathjaxinline]\partial _2[/mathjaxinline] are unitary matrices.</text>
</label>
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<span id="answer_cq_M2L10_Toric_code_properties_2_1"/>
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Toric code stabilizers
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This is a concept question, provided to check your understanding of the previous lecture video clips. </p>
<p>
Which of the following are true? </p>
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<text>We can view the [mathjaxinline]\mathbb {Z}_2[/mathjaxinline] linear subspace corresponding to the stabilizers of the toric code as the direct sum of the images of its boundary operators [mathjaxinline]\partial _1[/mathjaxinline] and [mathjaxinline]\partial _2[/mathjaxinline].</text>
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<text>A toric code with n qubits with genus [mathjaxinline]1[/mathjaxinline] corresponds to a two dimensional logical subspace.</text>
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<text>The product of [mathjaxinline]Z[/mathjaxinline] operators on a closed curve in a torus results in a normalizer of a toric code.</text>
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<text>Not every loop on a torus may be contracted to a point.</text>
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