{"id":364,"date":"2022-12-14T20:01:25","date_gmt":"2022-12-14T20:01:25","guid":{"rendered":"https:\/\/live-optics-wp.pantheonsite.io\/pjessen\/?p=364"},"modified":"2022-12-14T20:59:05","modified_gmt":"2022-12-14T20:59:05","slug":"30_unitary","status":"publish","type":"post","link":"https:\/\/wp.optics.arizona.edu\/pjessen\/2022\/12\/14\/30_unitary\/","title":{"rendered":"30_unitary"},"content":{"rendered":"\n<p class=\"has-text-align-center\" style=\"font-size:30px\"><strong>Constructing general unitary maps from state preparations<\/strong><\/p>\n\n\n\n<p class=\"center\" style=\"text-align: center\">\u00a0<\/p>\n<hr \/>\n<p class=\"center\" style=\"text-align: center\"><strong>Merkel, Seth T., Brennen, Gavin, Jessen, Poul S., Deutsch, Ivan H.<br \/><\/strong>\u00a0<\/p>\n<p class=\"center\" style=\"text-align: center\">We present an efficient algorithm for generating unitary maps on a d-dimensional Hilbert space from a time-dependent Hamiltonian through a combination of stochastic searches and geometric construction. The protocol is based on the eigendecomposition of the map. A unitary matrix can be implemented by sequentially mapping each eigenvector to a fiducial state, imprinting the eigenphase on that state, and mapping it back to the eigenvector. This requires the design of only d state-to-state maps generated by control wave forms that are efficiently found by a gradient search with computational resources that scale polynomially in d. In contrast, the complexity of a stochastic search for a single wave form that simultaneously acts as desired on all eigenvectors scales exponentially in d. We extend this construction to design maps on an n-dimensional subspace of the Hilbert space using only n stochastic searches. Additionally, we show how these techniques can be used to control atomic spins in the ground-electronic hyperfine manifold of alkali metal atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit.<br \/>\u00a0<\/p>\n<p class=\"center\" style=\"text-align: center\"><a href=\"https:\/\/wp.optics.arizona.edu\/pjessen\/publications\/\">Back<\/a> | <a href=\"http:\/\/wp.optics.arizona.edu\/pjessen\/wp-content\/uploads\/sites\/116\/2022\/12\/30_unitary.pdf\">Full Text<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Constructing general unitary maps from state preparations<\/p>\n","protected":false},"author":245,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[35,1],"tags":[],"class_list":["post-364","post","type-post","status-publish","format-standard","hentry","category-35","category-abstracts"],"_links":{"self":[{"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/posts\/364","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/users\/245"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/comments?post=364"}],"version-history":[{"count":1,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/posts\/364\/revisions"}],"predecessor-version":[{"id":365,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/posts\/364\/revisions\/365"}],"wp:attachment":[{"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/media?parent=364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/categories?post=364"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.optics.arizona.edu\/pjessen\/wp-json\/wp\/v2\/tags?post=364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}