Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer

Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the dimension $n$ and number of constraints $m$. While their dependence on other parameters suggests no overall speedup over classical methodologies, some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to our advantage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brandão et al. (Quantum 6, 625 (2022)) to exponentially improve the dependence of their algorithm on precision. As a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained Binary Optimization problems (QUBOs) in matrix multiplication time. Provided access to a quantum read/classical write random access memory (QRAM), a quantum implementation of our algorithm exhibits a worst case running time of $\mathcal{O} \left(ns + n^{1.5} \cdot \text{polylog} \left(n, \| C \|_F, \frac{1}ε \right) \right)$.