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Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

Authors Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, Xiaodi Wu

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Author Details

Fernando G. S. L. Brandão
  • Institute of Quantum Information and Matter, California Institute of Technology, USA
Amir Kalev
  • Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Tongyang Li
  • Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Cedric Yen-Yu Lin
  • Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Krysta M. Svore
  • Station Q, Quantum Architectures and Computation Group, Microsoft Research, USA
Xiaodi Wu
  • Joint Center for Quantum Information and Computer Science, University of Maryland, USA

Acknowledgements

We thank Scott Aaronson, Joran van Apeldoorn, Andr{á}s Gilyén, Cupjin Huang, and anonymous reviewers for helpful discussions. We are also grateful to Joran van Apeldoorn and Andr{á}s Gilyén for sharing a working draft of [Apeldoorn and Gilyén, 2018] with us. FB was supported by NSF. CYL and AK were supported by the Department of Defense. TL was supported by NSF CCF-1526380. XW was supported by NSF grants CCF-1755800 and CCF-1816695, and also by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program.

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Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.27

Abstract

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. 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 trace-norm 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 low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Quantum query complexity
  • Theory of computation → Convex optimization
Keywords
  • quantum algorithms
  • semidefinite program
  • convex optimization

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