![\"\"](\"https:\/\/blogs.kcl.ac.uk\/kclip\/files\/2023\/05\/Prog_Channel_-Simulation.png\")
Fig. 1: Fig. 1. Time-varying quantum channel \u03b5^t (top) and its simulation \u03b5_(\u03c0^t ) via a programmable quantum processor Q controlled by the time-varying program state \u03c0^t (bottom).<\/p><\/div>\n
<\/p>\n
In our recent work<\/a>, presented at IEEE ITW 2023<\/a>, we study the more challenging setting illustrated in Fig. 1, in which the channel to be simulated varies over time. We adopt a worst-case formulation in which the channel variation is arbitrary and chosen by \u201cnature\u201d in a possibly adversarial way. To study this setting, we propose to adopt the framework of online convex optimization [4], which provides tools to track the optimal solution of time-varying convex problems. We specifically develop and analyze an online mirror descent algorithm over the space of positive definite matrices, yielding a matrix exponentiated gradient descent (MEGD) [5]. We prove that the regret of MEGD with respect to an optimized fixed program state is sublinear in time.<\/p>\n <\/p>\n Fig. 2: Generalized teleportation processor as a programmable processor Q We conduct experiments by adopting the generalized teleportation processor (GTP), shown in Fig. 2, as the programmable quantum processor. GTP can simulate exactly the class of teleportation-covariant channels, modeling Pauli and erasure channels, and is operated here in an adversarial setting with time varying dephasing channels. Fig.3 plots the normalized regret as a function of time T. We observe that, MEGD is able to obtain a normalized regret that decreases sublinearly with T, hence approaching the performance of the reference program that would have been optimal in hindsight.<\/p>\n <\/p>\n Fig. 3: Normalized regret as a function of time T for MEGD when simulating a time-varying dephasing channel with dephasing probabilities drawn independently and uniformly at each time in the interval [0.2,p_max) (setting p_max=0.2 models a constant channel).<\/p><\/div>\n [1] O. Simeone, \u201cAn introduction to quantum machine learning for engineers\u201d, [2] M. A. Nielsen and I. L. Chuang, \u201cProgrammable quantum gate arrays\u201d, Phys. Rev. Lett., vol. 79, pp. 321\u2013324, Jul 1997.<\/p>\n [3] L. Banchi, J. Pereira, S. Lloyd, and S. Pirandola, \u201cConvex optimization of programmable quantum computers\u201d, npj Quantum Information, vol. 6, no. 1, pp. 1\u201310, 2020.<\/p>\n [4] F. Orabona, \u201cA modern introduction to online learning\u201d, CoRR, vol. abs\/1912.13213, 2019.<\/p>\n [5] K. Tsuda, G. Ratsch, and M. K. Warmuth, \u201cMatrix exponentiated gradient updates for on-line learning and Bregman projection\u201d, Journal of Machine Learning Research, vol. 6, no. 34, pp. 995\u20131018, 2005.<\/p>\n","protected":false},"excerpt":{"rendered":" A quantum computer can be programmed to carry out a given functionality in different ways, including the direct engineering of pulse sequences, the design of parametric quantum circuits via quantum machine learning (QML) [1], the use of adaptive measurements on cluster states, and the optimization of a program state operating on a fixed quantum processor. […]<\/p>\n","protected":false},"author":1257,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-777","post","type-post","status-publish","format-standard","hentry","category-quantum-machine-learning","post-preview"],"_links":{"self":[{"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/posts\/777","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/users\/1257"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/comments?post=777"}],"version-history":[{"count":5,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/posts\/777\/revisions"}],"predecessor-version":[{"id":782,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/posts\/777\/revisions\/782"}],"wp:attachment":[{"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/media?parent=777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/categories?post=777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.kcl.ac.uk\/kclip\/wp-json\/wp\/v2\/tags?post=777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Experiments<\/h2>\n
![\"\"](\"https:\/\/blogs.kcl.ac.uk\/kclip\/files\/2023\/05\/Teleportation_processor.png\")
operating on one input qubit (n=1) and on a two-qubit program state \u03c0
(n_\u03c0=2).<\/p><\/div>\n![\"\"](\"https:\/\/blogs.kcl.ac.uk\/kclip\/files\/2023\/05\/uni_normalized_regret.png\")
\nFoundations and Trends in Signal Processing, vol. 16, no. 1-2, pp. 1\u2013223, 2022.<\/p>\n