Quantum speed-ups for solving semidefinite relaxations of polynomial optimization

TL;DR

This paper introduces quantum algorithms that significantly accelerate the approximation of Lasserre hierarchy values in polynomial optimization, achieving super-quadratic speedups over classical methods under certain conditions.

Contribution

The authors develop a quantum algorithm based on matrix multiplicative weights that improves runtime for approximating Lasserre relaxations, surpassing classical approaches.

Findings

Quantum algorithms approximate Lasserre hierarchy values faster than classical methods.

Achieves super-quadratic speedup in problem dimension for polynomial optimization.

Provides specific runtime bounds for quantum algorithms in various problem settings.

Abstract

We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x: g_i(x)\ge 0\}$. Let $\lambda_k$ be the value of the order-$k$ Lasserre relaxation. Assume either (i) $f^\star=\lambda_k$ and the optimum is attained in the $\ell_1$-ball of radius $1/2$, or (ii) $S(g)$ lies in the simplex $\{x\ge 0: \sum_j x_j\le 1/2\}$, and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates $\lambda_k$ to accuracy $\varepsilon>0$ with runtime, for fixed $k$, \[ O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right), \] where $s_g$ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as $O(n^{3k}\mathrm{poly}(1/\varepsilon))$ even in the unconstrained case. As an example, we obtain an $O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})$ quantum algorithm for portfolio optimization, improving over the classical $O(n^{\omega+1}\log(1/\varepsilon))$ bound with $\omega\approx2.373$. Our approach builds on and sharpens the analysis of Apeldoorn and Gily\'en for the SDPs arising in polynomial optimization. We also show how to implement the required block encodings without QRAM. Under the stated assumptions, our method achieves a super-quadratic speedup in the problem dimension for computing Lasserre relaxations.