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arXiv:2610.16494

Scalable Quantum Monte Carlo for Shannon-Rényi Entropy: An Experiment-visitable Probe of Quantum Phases and Criticality

Dong-Xu Liu, Zhe Wang, Zenan Liu Yi-Ming Ding, Zheng Yan

Published Oct 12, 2026• 0 citations
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Abstract

The Shannon-Rényi (SR) entropy characterizes the probability distribution of many-body wave functions in a chosen basis and is directly accessible from single-shot measurement snapshots in cold-atom experiments, yet its theoretical evaluation in interacting systems carries exponential cost. Here we construct a field-theoretic spacetime manifold for the SR entropy in which $q$ replicas are glued at the imaginary-time boundaries, recasting it as a ratio of generalized partition functions amenable to large-scale quantum Monte Carlo simulation. On this basis we develop a reweight-annealing framework with two complementary strategies: a virtual-parameter scheme interpolating between unglued and fully glued limits via binomial-weighted partial gluing, and a real-parameter scheme using a physical coupling constant as the annealing variable within stochastic series expansion. Both reduce the computational cost from exponential to polynomial. Most importantly, we establish for the first time a finite-size scaling theory for the SR entropy near a quantum critical point, $S_q=a(g)L^2+S_q^0\left((g-g_c)L^{1/ν}\right)$; data collapse of its first derivative yields the correlation-length exponent $ν$ for both the O(3) and Ising universality classes, realized in the spin-$\frac{1}{2}$ columnar dimerized Heisenberg model and the bilayer Heisenberg-Ising model. The same simulations extract the Goldstone-mode count $N_G=2$ and the $Z_2$ ground-state degeneracy $\ln 2$ from subleading terms. These results elevate the SR entropy to a standalone, order-parameter-free diagnostic of quantum criticality with the same scaling status as the entanglement entropy, and close the loop between theory and quantum simulation by enabling quantitative benchmarking of experimental snapshots from state-of-the-art quantum simulators.

DOI
10.48550/arxiv.2610.16494

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Entanglement entropy (EE) plays a central role in the intersection of quantum information science and condensed matter physics. However, scanning the EE for two-dimensional and higher-dimensional systems still remains challenging. To address this challenge, we propose a quantum Monte Carlo scheme capable of extracting large-scale data of Rényi EE with high precision and low technical barrier. Its advantages lie in the following aspects: a single simulation can obtain the continuous variation curve of EE with respect to parameters, greatly reducing the computational cost; the algorithm implementation is simplified, and there is no need to alter the spacetime manifold during the simulation, making the code easily extendable to various many-body models. Additionally, we introduce a formula to calculate the derivative of EE without resorting to numerical differentiation from dense EE data. We then demonstrate the feasibility of using EE and its derivative to find phase transition points, critical exponents, and various phases. Entanglement entropy is a crucial quantity in several research fields, yet its calculation in higher-dimensional systems remains challenging. Here the authors develop a quantum Monte Carlo-based algorithm to efficiently extract large-scale entanglement entropy data and apply it to quantum critical systems.
Abstract We develop a nonequilibrium increment method in quantum Monte Carlo simulations to obtain the Rényi entanglement entropy of various quantum many-body systems with high efficiency and precision. To demonstrate its power, we show the results on a few important yet difficult (2 + 1)dquantum lattice models, ranging from the Heisenberg quantum antiferromagnet with spontaneous symmetry breaking, the quantum critical point with O(3) conformal field theory (CFT) to the toric code Z2{{\mathbb{Z}}}_{2}Z2​ Z2 topological ordered state and the Kagome Z2{{\mathbb{Z}}}_{2}Z2​ Z2 quantum spin liquid model with frustration and multi-spin interactions. In all these cases, our method either reveals the precise CFT data from the logarithmic correction or extracts the quantum dimension in topological order, from the dominant area law in finite-size scaling, with very large system sizes, controlled errorbars, and minimal computational costs. Our method, therefore, establishes a controlled and practical computation paradigm to obtain the difficult yet important universal properties in highly entangled quantum matter.
Our understanding of physical phenomena hinges on finding universal core mechanisms that unite them. The concept of universality is deeply ingrained in the study of quantum many-body systems. At zero temperature, microscopically different systems with long-range order collapse into their universal state described by a handful of universal parameters. Establishing those parameters implies identification of the universal theory that is effectively describing the system and ultimately providing the desired understanding. We take up this task with the help of Renyi entanglement entropy. Defined with respect to a system bipartition, this measure quantifies information shared between the subsystems. Understanding what insights into universality are encoded within this information-theoretic quantity as well as developing numerical tools to efficiently estimate the Renyi entanglement entropy are the subjects of this thesis. \n \nOn the computational end of this far-reaching goal, we develop a novel theoretical framework for constructing improved Renyi entanglement entropy estimators in the context of d+1 quantum Monte Carlo methods. The discovery of a connection of this methodology to the well-established Kandel-Domany formalism provides a clear path towards generalization. Additionally, we embrace a data-driven approach towards learning the ground state wavefunction. We demonstrate how a restricted Boltzmann machine can be used to reconstruct the Renyi entanglement entropy from projective measurements of a quantum ground state. Furthermore, we extend this classical generative architecture to a quantum analogue that we call the quantum Boltzmann machine. \n \nOn the theoretical side of this endeavour, we study the Renyi entanglement entropy scaling terms for two quantum lattice models embedded in two dimensional space via extensive quantum Monte Carlo simulations. For the ground state of the XY model, we provide conclusive numerical evidence for a logarithmic contribution that uniquely characterizes the continuous symmetry of the emerging order parameter. Moreover, we confirm the form of the subleading universal geometric contribution arising due to the bosonic nature of low-energy degrees of freedom in this model. For the critical ground state of the transverse field Ising model, we develop a novel scaling procedure to extract a universal number κ revealed via a cylindrical entangling bipartition in the thin-slice limit. The combined product of our work sheds new light on the entanglement-based classification of universality and brings a suite of new powerful numerical tools to continue illuminating this theoretical program in the future.
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.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.