Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness

TL;DR

This paper presents a fast classical method to compute nonstabilizerness in quantum states, significantly reducing computational costs and enabling studies of magic in complex quantum systems.

Contribution

It introduces an efficient sampling framework combining Walsh-Hadamard transforms and partitioning, with a Monte Carlo estimator for nonstabilizerness that scales well with system size.

Findings

Sample complexity does not grow with system size in benchmarks.

T gates' nonstabilizerness approaches dilute limit with modest Clifford scrambling.

Method enables analysis of magic in highly entangled and dynamic quantum states.

Abstract

Quantum magic, quantified by nonstabilizerness, measures departures from stabilizer structure and underlies potential quantum speedups. We introduce an efficient classical framework for computing stabilizer R\'enyi entropies and stabilizer nullity of generic $N$-qubit wavefunctions. The method combines the fast Walsh-Hadamard transform with an exact partition of Pauli operators, reducing the average cost per sampled Pauli string from $\mathcal{O}(2^N)$ to $\mathcal{O}(N)$. We further develop a Monte Carlo estimator with Clifford preconditioning and find that the required number of samples shows no visible growth with $N$ in our benchmarks. Applying the method to $T$-doped random Clifford circuits, we identify the scrambling ratio $\eta$ (Clifford gates per $T$ gate) as the key parameter governing magic growth. Each $T$ gate approaches its dilute-limit nonstabilizerness power with only modest Clifford scrambling. Our approach enables quantitative studies of magic in highly entangled states and long-time nonequilibrium dynamics.