Finding Hadamard matrices by a quantum annealing machine

TL;DR

This paper presents a quantum annealing approach to find Hadamard matrices, addressing implementation challenges due to higher-order terms by using symbolic computing, and demonstrates solutions on simulated and real quantum hardware.

Contribution

It formulates the Hamiltonian for finding Hadamard matrices suitable for quantum annealing, overcoming hardware limitations with symbolic computing techniques.

Findings

Successfully formulated the Hamiltonian for H-matrix finding.

Demonstrated solutions using both simulated and actual quantum annealing hardware.

Addressed the challenge of higher-order terms in quantum annealing implementations.

Abstract

Finding a Hadamard matrix (H-matrix) among the set of all binary matrices of corresponding order is a hard problem, which potentially can be solved by quantum computing. We propose a method to formulate the Hamiltonian of finding H-matrix problem and address its implementation limitation on existing quantum annealing machine (QAM) that allows up to quadratic terms, whereas the problem naturally introduces higher order ones. For an M-order H-matrix, such a limitation increases the number of variables from M^2 to (M^3 +M^2-M)/2, which makes the formulation of the Hamiltonian too exhaustive to do by hand. We use symbolic computing techniques to manage this problem. Three related cases are discussed: (1) finding N < M orthogonal binary vectors, (2) finding M-orthogonal binary vectors, which is equivalent to finding a H-matrix, and (3) finding N-deleted vectors of an M-order H-matrix. Solutions of the problems by a 2-body simulated annealing software and by an actual quantum annealing hardware are also discussed.

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