Probabilistic Construction and Analysis of Seminormalized Hadamard Matrices

Let o be a 4k-length column vector whose all entries are 1s, with k a positive integer. Let V={v_i} be a set of semi-normalized Hadamard (SH)-vectors, which are 4k-length vectors whose 2k entries are -1s and the remaining 2k are 1s. We define a 4k-order QSH (Quasi SH)-matrix, Q, as a 4kx4k matrix where the first column is o and the remaining ones are distinct v_i in V. When Q is orthogonal, it becomes an SH-matrix H. Therefore, 4k-order SH-matrices can be built by enumerate all possible Q from every combination of v_i, then evaluate the orthogonality of each one of them. Since such exhaustive method requires a large amount of computing resource, we can employ probabilistic algorithms to construct H, such as, by Random Vector Selection (RVS) or the Orthogonalization by Simulated Annealing (OSA) algorithms. We demonstrate the constructions of low-order SH-matrices by using these methods. We also analyze some probabilistic aspects of the constructions, including orthogonal probability p between a pair of randomly selected SH-vectors, the existence probability p_H|Q that a randomly generated Q is in fact an SH-matrix H, and address the discrepancy of the distribution between the known number of SH-matrix with expected number derived from the probabilistic analysis

Finding Hadamard matrices by a quantum annealing machine

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.Comment: 21 pages, 4 figure