A quantum compiler design method by using linear combinations of permutations

TL;DR

This paper presents a novel quantum compiler design method that transforms matrices into permutation-based quantum circuits using linear combinations, leveraging Birkhoff's algorithm and optimization techniques.

Contribution

It introduces a new approach to express matrices as linear combinations of permutations for quantum circuit synthesis, enhancing quantum compiler design.

Findings

Matrix conversion into doubly stochastic form demonstrated

Permutation decomposition of matrices shown via Birkhoff's algorithm

Potential optimization techniques for quantum compilation discussed

Abstract

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.