A fast and exact algorithm for stabilizer Rényi entropy via XOR-FWHT

Abstract: Quantum advantage is widely understood to rely on key quantum resources beyond entanglement, among which nonstabilizerness (quantum ``magic'') plays a central role in enabling universal quantum computation. However, a direct brute-force enumeration of all Pauli strings and the corresponding expectation values from a length-$2^N$ state vector, where $N$ is the system size, yields an overall computational cost scaling as $O(8^N)$, which quickly becomes infeasible as the system size grows. Here we reformulate the second-order stabilizer Rényi entropy in a bitstring language, expose an underlying XOR-convolution structure on $\mathbb Z_2^N$, and reduce the computation to $2^N$ fast Walsh-Hadamard transforms of length, together with pointwise operations, yielding a deterministic and exact XOR fast Walsh-Hadamard transforms algorithm with runtime scaling $O(N4^N)$ and natural parallelism. This algorithm enables high-precision, medium-scale exact calculations for generic state vectors. It provides a practical tool for probing the scaling, phase diagnostics, and dynamical fine structure of quantum magic in many-body systems.