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

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.