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 M2 to (M3 + M2 − 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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机译:在所有具有相应阶数的二进制矩阵的集合中找到Hadamard矩阵(H矩阵)是一个难题,有可能可以通过量子计算解决。我们提出了一种寻找H矩阵问题的哈密顿量公式化的方法,并解决了它在现有的量子退火机(QAM)上的实现局限性,该量子退火机允许最多二次项,而该问题自然会引入高阶项。对于M阶H矩阵,这样的限制将变量的数量从M 2 sup>增加到(M 3 sup> + M 2 sup> − M )/ 2,这使得汉密尔顿方程的公式过于详尽,无法手工完成。我们使用符号计算技术来解决此问题。讨论了三种相关情况:(1)找到N 展开▼