Quantum Approximate Optimization for Hard Problems in Linear Algebra

TL;DR

This paper investigates applying the Quantum Approximate Optimization Algorithm (QAOA) to solve Binary Linear Least Squares problems, comparing its performance with classical methods and discussing implementation challenges on current quantum hardware.

Contribution

It demonstrates the potential and limitations of QAOA for linear algebra problems, highlighting its performance relative to classical algorithms and addressing practical implementation issues.

Findings

Simulated Annealing outperforms QAOA at low depths for BLLS.

QAOA can produce solutions as samples directly, not just wavefunction amplitudes.

Implementation challenges on noisy quantum hardware are significant.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a quantum computational framework for solving quantum or classical optimization tasks. Here, we explore using QAOA for Binary Linear Least Squares (BLLS); a problem that can serve as a building block of several other hard problems in linear algebra, such as the Non-negative Binary Matrix Factorization (NBMF) and other variants of the Non-negative Matrix Factorization (NMF) problem. Most of the previous efforts in quantum computing for solving these problems were done using the quantum annealing paradigm. For the scope of this work, our experiments were done on noiseless quantum simulators, a simulator including a device-realistic noise-model, and two IBM Q 5-qubit machines. We highlight the possibilities of using QAOA and QAOA-like variational algorithms for solving such problems, where trial solutions can be obtained directly as samples, rather than being amplitude-encoded in the quantum wavefunction. Our numerics show that Simulated Annealing can outperform QAOA for BLLS at a QAOA depth of $p\leq3$ for the probability of sampling the ground state. Finally, we point out some of the challenges involved in current-day experimental implementations of this technique on cloud-based quantum computers.