Quantum approximate optimization for hard problems in linear algebra

Ajinkya Borle,Samuel J Lomonaco,Vincent Elfving

doi:10.21468/scipostphyscore.4.4.031

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 even for a small number of steps, simulated annealing can outperform QAOA for BLLS at a QAOA depth of p\leq3p≤3 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.

Highlights

Background and related work2.1 Background2.1.1 The binary linear least squares (BLLS) problemGiven a matrix A ∈ m×n, an unknown column vector of variables x ∈ {0, 1}n and a column vector b ∈ m (Where m > n)
Our work is certainly not the first in applying quantum approximate optimization algorithm (QAOA) to various relevant computational problems, and we refer the reader to a small list of examples [81,82,83]; in this work, we make an attempt to highlight some of the salient features and challenges of QAOA in the context of problems applicable to linear algebra and numerical analysis
Before we study the results of QAOA for BLLS with shot-noise, it is important to evaluate the theoretical performance of the same without any noise at all

Summary

Introduction

The application of quantum computing to hard optimization problems is a candidate where quantum computing may eventually outperform classical computation [1,2,3,4,5,6]. The technique seems to scale well in optimizing the expectation value as the problem size n increases (for a fixed QAOA circuit depth p). This corresponds to getting ‘good solutions’ fast but not necessarily the most optimal one (refer Section 3.4.2). When compared against older quantum devices, the newer ones seem to be getting better at approximating the expectation value for a given set of QAOA parameters. The results they produce are still very noisy for QAOA to function effectively (refer Sections 3.4.1 and 3.4.7). We have Appendices to complement and support the information in the main paper

Background

Implicit filtering optimization

Related work

Findings

QAOA for BLLS

Conclusion