  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.
