gates<\/p>\n\n\n\n
We\u2019ve proved that the CNOT gate along with all possible single-qubit gates<\/p>\n\n\n\n
form a universal set. But this set is still in\ufb01nite. We\u2019d like to do better: to<\/p>\n\n\n\n
get a \ufb01nite set of gates as in the classical case. Of course we must realize that<\/p>\n\n\n\n
the set of possible single qubit gates is itself in\ufb01nite as opposed to the \ufb01nite<\/p>\n\n\n\n
number of gates in classical computation. Yet it is surprising that there exist<\/p>\n\n\n\n
more rigorous theorems (e.g., the Solovay\u2013Kitaev Theorem [23]) con\ufb01rming<\/p>\n\n\n\n
the universality of a smaller set of gates, such as for example, H, CNOT,<\/p>\n\n\n\n
S =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 i<\/p>\n\n\n\n
#<\/p>\n\n\n\n
and T =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 e<\/p>\n\n\n\n
i\u03c0\/4<\/p>\n\n\n\n
#<\/p>\n\n\n\n
or the To\ufb00oli and H gates. But these sets of<\/p>\n\n\n\n
gates cannot be used to construct arbitrary gates to in\ufb01nite precision. So the<\/p>\n\n\n\n
theorems actually prove that one can approximate arbitrary unitary gates, to<\/p>\n\n\n\n
any degree of accuracy, by using a \ufb01nite set of gates. We will not dwell on<\/p>\n\n\n\n
these theorems or their proofs here.<\/p>\n","protected":false},"excerpt":{"rendered":"
gates We\u2019ve proved that the CNOT gate along with all possible single-qubit gates form a universal set. But this set is still in\ufb01nite. We\u2019d like to do better: to get a \ufb01nite set of gates as in the classical case. Of course we must realize that the set of possible single qubit gates is itself […]<\/p>\n","protected":false},"author":1,"featured_media":4041,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[496],"tags":[],"class_list":["post-4076","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-gates-and-circuits"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-2.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4076","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4076"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4076\/revisions"}],"predecessor-version":[{"id":4077,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4076\/revisions\/4077"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4041"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4076"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4076"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}