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

TL;DR

This paper introduces two quantum algorithms for solving semidefinite programs (SDPs) with significant speed-ups, and applies the second to quantum state learning, achieving efficient descriptions with quantum circuit approximations.

Contribution

The paper presents novel quantum SDP solvers with optimal and poly-logarithmic complexities, and demonstrates their application to quantum state learning with improved efficiency.

Findings

First algorithm has optimal dependence on m and n, quadratic speed-up over previous methods.

Second algorithm operates efficiently with poly-logarithmic dependence on n, suitable for quantum input models.

Application to quantum state learning achieves description in time proportional to (\u221a m) with high accuracy.

Abstract

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 $\tilde{O}(s^2(\sqrt{m}\epsilon^{-10}+\sqrt{n}\epsilon^{-12}))$, 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 $m\approx n$. 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 $\tilde{O}(\sqrt{m}+\text{poly}(r))\cdot\text{poly}(\log m,\log n,B,\epsilon^{-1})$, with $B$ an upper bound on the trace-norm of all input matrices. In particular the complexity depends only 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 $\rho$ with rank at most $r$, we show we can find in time $\sqrt{m}\cdot\text{poly}(\log m,\log n,r,\epsilon^{-1})$ 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 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.