An efficient Pauli decomposition algorithm for structured matrices

Abstract: Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only $k = \mathsf{poly}(n)$ Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in $n$. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least $1 - δ$, for any $δ$. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in $n$, $k$, and $\log(1/δ)$. These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.