A fast and exact approach for stabilizer R\'enyi entropy via the XOR-FWHT algorithm

TL;DR

This paper introduces a fast, exact algorithm based on the XOR-FWHT algorithm to efficiently compute the stabilizer Rènyi entropy, a key measure of quantum magic, for many-body quantum states, enabling new insights into quantum resource scaling.

Contribution

The authors develop a deterministic, exact algorithm that reduces computational complexity for stabilizer Rènyi entropy evaluation from exponential to polynomial times, facilitating high-precision analysis of quantum magic.

Findings

Allows exact calculations of quantum magic for larger systems

Enables study of phase transitions and dynamics in many-body quantum systems

Provides a practical computational tool for quantum resource analysis

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, the exact evaluation of the second-order stabilizer R\'enyi entropy for generic many-body quantum states remains computationally challenging, with brute-force methods scaling as $\mathcal O(8^N)$ for an $N$-qubit state. Here we develop a deterministic and exact algorithm that reduces this cost to $\mathcal{O}(N4^N)$ while retaining natural parallelism. This advance enables high-precision exact calculations for generic state vectors at medium system sizes, and provides a practical tool for investigating the scaling, phase structure, and nonequilibrium dynamics of quantum magic in many-body systems.