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

• URL: http://arxiv.org/abs/2301.04237v3
• Date: Thu, 11 May 2023 10:45:58 GMT
• Title: Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer
• Authors: Brandon Augustino, Giacomo Nannicini, Tam\'as Terlaky and Luis Zuluaga
• Abstract summary: We show that some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to exponentially improve the dependence of their algorithm on precision. A quantum implementation of our algorithm exhibits a worst case running time of $mathcalO left(ns + n1.5 cdot textpolylog left(n, | C |_F, frac1epsilon right)$.
• Score: 0.20999222360659603
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 Brand\~ao 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 $\mathcal{O} \left(ns + n^{1.5} \cdot \text{polylog} \left(n, \| C \|_F, \frac{1}{\epsilon} \right) \right)$.

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