Decomposing dense matrices into dense Pauli tensors

TL;DR

This paper introduces a memory-efficient, parallelizable algorithm for decomposing dense matrices into Pauli tensors, significantly speeding up the process compared to previous methods, especially for dense, arbitrary matrices.

Contribution

We develop a fixed-memory, branchless algorithm leveraging Gray code for efficient Pauli tensor decomposition, outperforming existing methods in speed and memory usage.

Findings

Achieves 1.5x to 5x speedup over state-of-the-art for N<8

Operates with fixed memory, suitable for dense matrices

Does not rely on matrix sparsity or special properties

Abstract

Decomposing a matrix into a weighted sum of Pauli strings is a common chore of the quantum computer scientist, whom is not easily discouraged by exponential scaling. But beware, a naive decomposition can be cubically more expensive than necessary! In this manuscript, we derive a fixed-memory, branchless algorithm to compute the inner product between a 2^N-by-2^N complex matrix and an N-term Pauli tensor in O(2^N) time, by leveraging the Gray code. Our scheme permits the embarrassingly parallel decomposition of a matrix into a weighted sum of Pauli strings in O(8^N) time. We implement our algorithm in Python, hosted open-source on Github, and benchmark against a recent state-of-the-art method called the "PauliComposer" which has an exponentially growing memory overhead, achieving speedups in the range of 1.5x to 5x for N < 8. Note that our scheme does not leverage sparsity, diagonality, Hermitivity or other properties of the input matrix which might otherwise enable optimised treatment in other methods. As such, our algorithm is well-suited to decomposition of dense, arbitrary, complex matrices which are expected dense in the Pauli basis, or for which the decomposed Pauli tensors are a priori unknown.