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Evaluating Many-Body Stabilizer Rényi Entropy by Sampling Reduced Pauli Strings: Singularities, Volume Law, and Nonlocal Magic

Yi-Ming Ding1,2,3,*, Zhe Wang2,3, and Zheng Yan2,3,†

  • 1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, China
  • 2Department of Physics, School of Science and Research Center for Industries of the Future, Westlake University, Hangzhou 310030, China
  • 3Institute of Natural Sciences, Westlake Institute for Advanced Study, Hangzhou 310024, China

  • *Contact author: dingyiming@westlake.edu.cn
  • †Contact author: zhengyan@westlake.edu.cn
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PRX Quantum 6, 030328 – Published 18 August, 2025

DOI: https://doi.org/10.1103/pyzr-jmvw

Abstract

We present a novel quantum Monte Carlo method for evaluating the α-stabilizer Rényi entropy (SRE) for any integer α≥2. By interpreting the α-SRE as partition-function ratios, we eliminate the sign problem in the imaginary-time path integral by sampling reduced Pauli strings within a reduced configuration space, which enables efficient classical computations of the α-SRE and its derivatives to explore magic in previously inaccessible two- or higher-dimensional systems. We first isolate the free-energy part in 2-SRE, which is a trivial term. Notably, at quantum critical points in one-dimensional or two-dimensional transverse-field Ising (TFI) models, we reveal nontrivial singularities associated with the characteristic function contribution, directly tied to magic. Their interplay leads to complicated behaviors of 2-SRE, avoiding extrema at critical points generally. In contrast, analyzing the volume-law correction to SRE reveals a discontinuity tied to criticalities, suggesting that it is more informative than the full-state magic. For conformal critical points, we claim that it could reflect nonlocal magic residing in correlations. Finally, we verify that 2-SRE fails to characterize magic in mixed states (e.g., Gibbs states), yielding nonphysical results. This work provides a powerful tool for exploring the roles of magic in large-scale many-body systems and reveals the intrinsic relation between magic and many-body physics.

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Physics Subject Headings (PhySH)

Popular Summary

While entanglement is a fundamental feature of quantum systems, it is not sufficient to achieve quantum advantage. In particular, stabilizer states can be highly entangled, while still being amenable to efficient classical simulation. To capture the nonclassicality beyond entanglement, the concept of magic has been introduced. It quantifies how far a quantum state deviates from the set of stabilizer states. Given the central role of entanglement in probing criticality and quantum phases, it is natural to ask whether magic plays a similarly fundamental role. However, computing magic for many-body systems remains a major challenge, especially in higher dimensions, calling for the development of scalable and efficient computational tools.

In this work, we present a novel quantum Monte Carlo (QMC) algorithm for efficiently computing the stabilizer Rényi entropy (SRE), a measure of magic, and its derivatives. We uncover rich behaviors in SRE, governed by the interplay between contributions from free energy and the characteristic function, with critical signatures manifesting in the singularities of their derivatives. Moreover, we show that volume-law corrections to SRE encode essential information. In particular, we extract universal signatures related to the g factor of the underlying boundary conformal field theory.

Our work represents a significant advancement in QMC algorithms and opens new directions for exploring how many-body magic characterizes quantum phases, critical phenomena, and conformal field theory, particularly in high-dimensional systems.

Article Text

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PRX Quantum 6, 030328– Published 18 August, 2025
  • Received 26 January 2025
  • Revised 17 May 2025
  • Accepted 23 July 2025
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