Pauli Decomposition via the Fast Walsh-Hadamard Transform

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 a new 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 O(N²log N) time and using O(1) additional memory, for an N× N matrix. A numerical implementation of our equation outperforms currently available solutions.