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  • Letter
  • Open Access

Fast and exact approach for stabilizer Rényi entropy via the XOR-FWHT algorithm

Xuyang Huang1,*, Han-Ze Li1,2,*, Ching Hua Lee2,†, and Jian-Xin Zhong1,‡

  • *These authors contributed equally to this work.
  • †Contact author: phylch@nus.edu.sg
  • ‡Contact author: jxzhong@shu.edu.cn
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Phys. Rev. Research 8, L042005 – Published 19 October, 2026

DOI: https://doi.org/10.1103/f7ss-6tl1

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 deterministic evaluation of the second-order stabilizer Rényi entropy for generic many-body quantum states remains computationally challenging, with brute-force methods scaling as O(8N)=O(d3) for an N-qubit state with d=2N. Here, we develop an exact algebraic reformulation and a deterministic numerical algorithm, numerically exact up to floating-point roundoff, that reduces this cost to O(N4N)=O(d2log2d) while retaining natural parallelism. This is an exponential speedup, although the scaling remains exponential in the number of qubits. This advance enables high-precision calculations for generic state vectors at medium system sizes and provides a practical tool for investigating the scaling and nonequilibrium dynamics of quantum magic in many-body systems.

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Phys. Rev. Research 8, L042005– Published 19 October, 2026
  • Received 26 March 2026
  • Accepted 2 October 2026
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