Abstract
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speedups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{10} + sqrt{n} epsilon^{12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{1}), with B an upper bound on the tracenorm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r.
We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics.
As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for lowrank Hamiltonians, given quantum states encoding these Hamiltonians, with a polylogarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.
BibTeX  Entry
@InProceedings{brando_et_al:LIPIcs:2019:10603,
author = {Fernando G. S. L. Brand{\~a}o and Amir Kalev and Tongyang Li and Cedric YenYu Lin and Krysta M. Svore and Xiaodi Wu},
title = {{Quantum SDP Solvers: Large SpeedUps, Optimality, and Applications to Quantum Learning}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {27:127:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771092},
ISSN = {18688969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10603},
URN = {urn:nbn:de:0030drops106036},
doi = {10.4230/LIPIcs.ICALP.2019.27},
annote = {Keywords: quantum algorithms, semidefinite program, convex optimization}
}
Keywords: 

quantum algorithms, semidefinite program, convex optimization 
Collection: 

46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) 
Issue Date: 

2019 
Date of publication: 

04.07.2019 