Finding a Hadamard Matrix by Simulated Annealing of Spin-Vectors

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.