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arXiv:2111.10084

On the application of Sylvester's law of inertia to QUBO formulations for systems of linear equations

Sun Woo Park, Kyungtaek Jun

Published Nov 19, 2021• 0 citations
arXivHTMLPDF

Abstract

Previous research on quantum annealing methods focused on effectively modeling systems of linear equations by utilizing quadratic unconstrained binary optimization (QUBO) formulations. These studies take part in enhancing quantum computing algorithms, which extract properties of quantum computers suitable for improving classical computational models. In this paper, we further develop the QUBO formulations of systems of linear equations by applying Sylvester's law of inertia, which explores matrix congruence of any real symmetric matrix to a diagonal matrix. We expect that the proposed algorithm can effectively implement higher dimensional systems of linear equations on a quantum computer. Further experimental verification of the proposed QUBO models as well as their comparisons to classical algorithms are also made.

DOI
10.48550/arxiv.2111.10084

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Recent studies on quantum computing algorithms focus on excavating features of quantum computers which have potential for contributing to computational model enhancements. Among various approaches, quantum annealing methods effectively parallelize quadratic unconstrained binary optimization (QUBO) formulations of systems of linear equations. In this paper, we simplify these formulations by exploiting congruence of real symmetric matrices to diagonal matrices. We further exhibit computational merits of the proposed QUBO models, which can outperform classical algorithms such as QR and SVD decomposition.
Abstract With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all of science and engineering. Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on the gate model quantum computers, was invented and several advanced variations have been developed. For a given square matrix A∈R(n×n) and a vector b∈R(n), we will find unconstrained binary optimization (QUBO) models for a vector x∈R(n) that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, We provide a python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in D-Wave system.
Quadratic Unconstrained Binary Optimization (QUBO) can be seen as a generic language for optimization problems. QUBOs attract particular attention since they can be solved with quantum hardware, like quantum annealers or quantum gate computers running QAOA. In this paper, we present two novel QUBO formulations for k-SAT and Hamiltonian Cycles that scale significantly better than existing approaches. For k-SAT we reduce the growth of the QUBO matrix from O(k) to O(log(k)). For Hamiltonian Cycles the matrix no longer grows quadratically in the number of nodes, as currently, but linearly in the number of edges and logarithmically in the number of nodes.
With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all science and engineering. The Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on gate model quantum computers, was invented and several advanced variations have been developed. For a given n by n matrix A and a vector b, we will find unconstrained binary optimization (QUBO) models for a vector x that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, we provide a Python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in the D-Wave system.
Many quantum computing algorithms are being developed with the advent of quantum computers. Solving linear systems is one of the most fundamental problems in almost all of science and engineering. HHL algorithm, a monumental quantum algorithm for solving linear systems on the gate model quantum computers, was invented and several advanced variations have been developed. However, HHL-based algorithms have a lot of limitations in spite of their importance. We address solving linear systems on a D-Wave quantum annealing device. To formulate a quadratic unconstrained binary optimization (QUBO) model for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate this QUBO model on the D-Wave system and discuss the results.
Quantum optimization shows significant potential in tackling complex combinatorial challenges, prompting our research exploration into the domain of quantum computing to address the Bin Packing Problem (BPP) through Quadratic Unconstrained Binary Optimization (QUBO) formulations. This work-in-progress paper outlines our proposal for a thorough study that encompasses classical bin packing formulations, initiating with the Mixed Integer Linear Program, followed by a comparative analysis of the corresponding QUBO representations, and adopting a structured approach of Quantum Annealing on the D-Wave machine. While our work is still in progress, continuous comparative analyses are being conducted to evaluate various QUBO formulations, derived from diverse classical bin packing models, aiming to identify the most efficient QUBO for addressing the BPP. On another front, efforts are dedicated to refining the solution process further, with a focus on enhancing the precision of the selected QUBO formulation.