Pauli Decomposition via the Fast Walsh-Hadamard Transform

Abstract: The decomposition of a square matrix into a sum of Pauli strings is a classical pre-processing step required to realize many quantum algorithms. Such a decomposition requires significant computational resources for large matrices. We present an exact and explicit formula for the Pauli string coefficients which inspires an efficient algorithm to compute them. More specifically, we show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. This allows one to use the Fast Walsh-Hadamard transform and calculate all Pauli decomposition coefficients in $\mathcal{O}(N^2\log N)$ time and using $\mathcal{O}(1)$ additional memory, for an $N\times N$ matrix. A numerical implementation of our equation outperforms currently available solutions.