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A quantum compiler design method by using linear combinations of permutations
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A matrix can be converted into a doubly stochastic matrix by using two diagonal matrices. And a doubly stochastic matrix can be written as a sum of permutation matrices. In this paper, we describe a method to write a given generic matrix in terms of quantum gates based on the block encoding. In particular, we first show how to convert a matrix into doubly stochastic matrices and by using Birkhoff's algorithm, we express that matrix in terms of a linear combination of permutations which can be mapped to quantum circuits. We then discuss a few optimization techniques that can be applied in a possibly future quantum compiler software based on the method described here.
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Error analysis of quantum operators written as a linear combination of permutations
Eigenvalue perturbation under bit- and phase-flips is studied for matrices written as linear combinations of permutations, with exact invariance of the dominant eigenvalue for positive coefficients.
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