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Peer-reviewedCC BY 4.0Physical Review Research

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

  • Xuyang Huang1
  • Han-Ze Li1,2
  • Ching Hua Lee2
  • Jian-Xin Zhong1
  1. 1Shanghai University
  2. 2National University of Singapore
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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 ( 8 N ) = O ( d 3 ) for an N -qubit state with d = 2 N . 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 ( N 4 N ) = O ( d 2 log 2 d ) 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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Cite

BibTeXRIS
BibTeX
@article{huang2026fast,
  title = {{Fast and exact approach for stabilizer Rényi entropy via the XOR-FWHT algorithm}},
  author = {Xuyang Huang and Han-Ze Li and Ching Hua Lee and Jian-Xin Zhong},
  journal = {Physical Review Research},
  year = {2026},
  month = oct,
  doi = {10.1103/f7ss-6tl1},
  url = {https://link.aps.org/doi/10.1103/f7ss-6tl1},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Fast and exact approach for stabilizer Rényi entropy via the XOR-FWHT algorithm
AU  - Xuyang Huang
AU  - Han-Ze Li
AU  - Ching Hua Lee
AU  - Jian-Xin Zhong
PY  - 2026
DA  - 2026/10/19
JO  - Physical Review Research
DO  - 10.1103/f7ss-6tl1
UR  - https://link.aps.org/doi/10.1103/f7ss-6tl1
AB  - 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 ( 8 N ) = O ( d 3 ) for an N -qubit state with d = 2 N . 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 ( N 4 N ) = O ( d 2 log 2 d ) 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.
SN  - 2643-1564
ER  -