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

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\u27 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

Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $\varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $\bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},\ldots,A_{\tau}$, we propose a procedure that gives a good approximation of $A=A_{1}+\cdots+A_{\tau}$. $\bullet$ Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020

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

We give two 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 access to an oracle to the matrices at unit cost. We show that it has run time Õ(s^2(√((mϵ)^(−10)) + √((nϵ)^(−12))), with ϵ 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 matrices are given as quantum states. We show that its run time is Õ (√m + poly(r))⋅poly(log m,log n,B,ϵ^(−1)), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically 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 ρ with rank at most r, we show we can find in time √m⋅poly(log m,log n,r,ϵ^(−1)) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements, up to error ϵ. 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 method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest

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

We give two 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 access to an oracle to the matrices at unit cost. We show that it has run time Õ(s^2(√((mϵ)^(−10)) + √((nϵ)^(−12))), with ϵ 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 matrices are given as quantum states. We show that its run time is Õ (√m + poly(r))⋅poly(log m,log n,B,ϵ^(−1)), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically 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 ρ with rank at most r, we show we can find in time √m⋅poly(log m,log n,r,ϵ^(−1)) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements, up to error ϵ. 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 method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest

Quantum algorithms for machine learning and optimization

The theories of optimization and machine learning answer foundational questions in computer science and lead to new algorithms for practical applications. While these topics have been extensively studied in the context of classical computing, their quantum counterparts are far from well-understood. In this thesis, we explore algorithms that bridge the gap between the fields of quantum computing and machine learning. First, we consider general optimization problems with only function evaluations. For two core problems, namely general convex optimization and volume estimation of convex bodies, we give quantum algorithms as well as quantum lower bounds that constitute the quantum speedups of both problems to be polynomial compared to their classical counterparts. We then consider machine learning and optimization problems with input data stored explicitly as matrices. We first look at semidefinite programs and provide quantum algorithms with polynomial speedup compared to the classical state-of-the-art. We then move to machine learning and give the optimal quantum algorithms for linear and kernel-based classifications. To complement with our quantum algorithms, we also introduce a framework for quantum-inspired classical algorithms, showing that for low-rank matrix arithmetics there can only be polynomial quantum speedup. Finally, we study statistical problems on quantum computers, with the focus on testing properties of probability distributions. We show that for testing various properties including L1-distance, L2-distance, Shannon and Renyi entropies, etc., there are polynomial quantum speedups compared to their classical counterparts. We also extend these results to testing properties of quantum states

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described