Quantum Speed-ups for Semidefinite Programming
Fernando G. S. L. Brandao, Krysta Svore
TL;DR
This work introduces a quantum algorithm for solving semidefinite programs that achieves a square-root speed-up in both the matrix dimension n and the number of constraints m, by combining quantum Gibbs sampling with a matrix multiplicative weights framework based on Arora and Kale. The method reduces the SDP to a feasibility problem and uses Gibbs-based oracles and Jaynes’ principle to approximate the necessary dual updates, while sparsifying intermediate Hamiltonians to maintain tractable sparsity. It provides unconditional polynomial speed-ups (in n and m) and proves a matching quantum lower bound up to polylog factors, along with a detailed analysis of correctness and complexity. The results suggest that quantum resources could meaningfully accelerate SDP- and LP-like optimization in practice, and raise open questions about dependence on problem size parameters R and δ, robustness to noise, and potential improvements for specific SDP instances.
Abstract
We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time $n^{\frac{1}{2}} m^{\frac{1}{2}} s^2 \text{poly}(\log(n), \log(m), R, r, 1/δ)$, with $n$ and $s$ the dimension and row-sparsity of the input matrices, respectively, $m$ the number of constraints, $δ$ the accuracy of the solution, and $R, r$ a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in $n$ and $m$. We prove the algorithm cannot be substantially improved (in terms of $n$ and $m$) giving a $Ω(n^{\frac{1}{2}}+m^{\frac{1}{2}})$ quantum lower bound for solving semidefinite programs with constant $s, R, r$ and $δ$. The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.