Finding a Hadamard Matrix by Simulated Annealing of Spin-Vectors

TL;DR

This paper introduces a novel method using simulated annealing on an Ising model to find Hadamard matrices, demonstrating success on low order cases including those difficult to construct by traditional methods.

Contribution

The paper presents a new heuristic approach for constructing Hadamard matrices via simulated annealing on a graph-based Ising model representation.

Findings

Successfully finds low order Hadamard matrices.

Can discover matrices not trivially constructed by Sylvester method.

Demonstrates effectiveness of physics-inspired heuristics for combinatorial problems.

Abstract

Reformulation of a combinatorial problem into optimization of a statistical-mechanics system, enables finding a better solution using heuristics derived from a physical process, such as by the SA (Simulated Annealing). In this paper, we present a Hadamard matrix (H-matrix) searching method based on the SA on an Ising model. By equivalence, an H-matrix can be converted into an SH (Seminormalized Hadamard) matrix; whose first columns are unity vector and the rest ones are vectors with equal number of -1 and +1 called SH-vectors. We define SH spin-vectors to represent the SH vectors, which play a similar role to the spins on an Ising model. The topology of the lattice is generalized into a graph, whose edges represent orthogonality relationship among the SH spin-vectors. Started from a randomly generated quasi H-matrix Q, which is a matrix similar to the SH-matrix without imposing orthogonality, we perform the SA. The transitions of Q are conducted by random exchange of {+,-} spin-pair within the SH-spin vectors which follow the Metropolis update rule. Upon transition toward zero-energy, the Q-matrix is evolved following a Markov chain toward an orthogonal matrix, at which point the H-matrix is said to be found. We demonstrate the capability of the proposed method to find some low order H-matrices, including the ones that cannot trivially be constructed by the Sylvester method.