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arXiv:2501.12146 · PRX Quantum

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

Yi-Ming Ding, Zhe Wang, Zheng Yan

Published Jul 23, 2025• 23 citations
arXivHTMLPDF

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 within a , 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 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.

DOI
10.1103/pyzr-jmvw

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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 within a , 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 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.
Quantum magic---the resource that separates universal quantum computation from efficiently simulable Clifford circuits---has emerged as a diagnostic of many-body quantum states, yet the stabilizer Rényi entropy (SRE) that quantifies it remains largely inaccessible in interacting fermion systems. Estimating the global SRE requires a sum over exponentially many Majorana strings, which in determinant quantum Monte Carlo can be sampled without a sign problem only for the interacting density matrix obtained after averaging the auxiliary fields: sampling strings and fields simultaneously incurs a sign problem even for sign-problem-free models, so that the known sign-free alternative is a nested Monte Carlo that does not scale. We propose instead the two-point SRE---a practical stabilizer-Rényi correlator built from rank-2 SREs of one- and two-site reduced density matrices, which are fixed exactly by single-particle Green functions and density correlations already measured in standard simulations---and show that it obeys universal finite-size scaling at fermionic quantum critical points. In the one-dimensional half-filled spinless ttt-VVV chain, the correlator distinguishes algebraic and exponential decay regimes and tracks the inverse-logarithmic finite-size drift characteristic of the Berezinskii--Kosterlitz--Thouless transition. On the honeycomb lattice, sign-problem-free quantum Monte Carlo yields Gross--Neveu--Ising scaling with finite anomalous dimension at zero temperature and two-dimensional Ising collapse at the thermal transition. Our results suggest stabilizer-Rényi microscopy as a spatially resolved, quantitatively universal probe of critical correlations in interacting fermionic matter, on par with conventional order-parameter correlators and accessible to quantum-simulator measurements.
Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or “magic”) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time O(N\chi^3) O ( N χ 3 ) for matrix product states (MPSs) of bond dimension \chi χ . We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual 2 2 -SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with O(8^{N/2}) O ( 8 N / 2 ) time and O(2^N) O ( 2 N ) memory, which we demonstrate for 24 24 qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.
Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most 444, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.
Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.