[Paper Review] Many-body magic via Pauli-Markov chains -- from criticality to gauge theories

Poetri Sonya Tarabunga, Emanuele Tirrito+2 more|arXiv (Cornell University)|May 29, 2023

Theoretical and Computational Physics5 citations

TL;DR

This paper introduces a Pauli-Markov chain method to efficiently compute many-body magic in quantum systems, enabling unbiased estimation of stabilizer Renyi entropies (SREs) with O(log N) scaling via tree tensor networks. It reveals that long-range magic—previously inaccessible—sharply signals quantum criticality in 1D systems and identifies the confinement-deconfinement transition in 2D Z₂ gauge theories with critical scaling.

ABSTRACT

We introduce a method to measure many-body magic in quantum systems based on a statistical exploration of Pauli strings via Markov chains. We demonstrate that sampling such Pauli-Markov chains gives ample flexibility in terms of partitions where to sample from: in particular, it enables to efficiently extract the magic contained in the correlations between widely-separated subsystems, which characterizes the nonlocality of magic. Our method can be implemented in a variety of situations. We describe an efficient sampling procedure using Tree Tensor Networks, that exploits their hierarchical structure leading to a modest $O(\log N)$ computational scaling with system size. To showcase the applicability and efficiency of our method, we demonstrate the importance of magic in many-body systems via the following discoveries: (a) for one dimensional systems, we show that long-range magic displays strong signatures of conformal quantum criticality (Ising, Potts, and Gaussian), overcoming the limitations of full state magic; (b) in two-dimensional $\mathbb{Z}_2$ lattice gauge theories, we provide conclusive evidence that magic is able to identify the confinement-deconfinement transition, and displays critical scaling behavior even at relatively modest volumes. Finally, we discuss an experimental implementation of the method, which only relies on measurements of Pauli observables.

Motivation & Objective

• To address the lack of scalable, efficient methods for computing magic in many-body quantum systems, especially non-local or long-range contributions.
• To develop a stochastic sampling framework based on Markov chains over Pauli strings that provides unbiased estimators of stabilizer Renyi entropies (SREs) of all orders.
• To enable the computation of long-range magic, which is UV-finite and captures nonlocal quantum correlations, overcoming limitations of full-state magic measures.
• To demonstrate the method’s efficiency and accuracy in detecting quantum criticality and phase transitions in both 1D and 2D systems using tree tensor networks (TTN).
• To provide a practical, experimentally feasible protocol relying only on Pauli observable measurements for measuring magic in quantum systems.

Proposed method

• The method employs a Markov chain over Pauli strings to stochastically sample the wavefunction, enabling efficient estimation of stabilizer Renyi entropies (SREs) for arbitrary partitions.
• The Markov chain is constructed to ensure detailed balance and ergodicity, yielding an unbiased estimator of SREs of all orders, even for large or non-local subsystems.
• The algorithm is implemented using tree tensor networks (TTN), which exploit hierarchical structure to achieve O(log N) computational scaling with system size N.
• Autocorrelation times and statistical errors are monitored to ensure convergence and sampling efficiency, with results validated via convergence checks across bond dimensions.
• The method is applied to compute both full-state magic and long-range magic, with the latter defined as a mutual information-like measure of nonlocal magic.
• Duality transformations (e.g., between 2D transverse-field Ising and Z₂ gauge theories) are used to map results across dual models, though long-range magic is not preserved under such maps.

Experimental results

Research questions

• RQ1 Can long-range magic—defined as the magic in correlations between widely separated subsystems—serve as a robust, UV-finite indicator of quantum criticality in 1D systems?
• RQ2 Does the proposed Pauli-Markov chain method enable efficient and accurate computation of stabilizer Renyi entropies (SREs) in large many-body systems with controlled statistical errors?
• RQ3 Can long-range magic detect the confinement-deconfinement transition in 2D Z₂ lattice gauge theories, and does it exhibit critical scaling behavior at finite system sizes?
• RQ4 How does the performance of long-range magic compare to traditional order parameters like the Binder cumulant in detecting quantum critical points, especially when using low-bond-dimension TTN approximations?
• RQ5 To what extent is long-range magic preserved under duality transformations, and how does its distribution differ across dual theories such as the 2D Ising model and Z₂ gauge theory?

Key findings

• Long-range magic displays sharp signatures of conformal quantum criticality in 1D systems (Ising, Potts, and Gaussian models), while full-state magic fails to detect criticality in the spin-1 XXZ model.
• The Pauli-Markov chain method achieves O(log N) scaling with system size when implemented via tree tensor networks, enabling efficient computation of SREs even for large systems.
• In 2D Z₂ lattice gauge theories, long-range magic identifies the confinement-deconfinement transition and exhibits critical scaling behavior at relatively modest system sizes, providing conclusive evidence of its utility.
• The standard deviation of the SRE estimator scales as σ ∝ N^(-0.503) for m₁, indicating reliable statistical convergence with increasing system size.
• The method outperforms the Binder cumulant in detecting the critical point in the 2D transverse-field Ising model when using low-bond-dimension TTN (χ=30), as the latter fails to show clear crossing behavior due to poor ground state representation.
• While SREs are preserved under duality transformations (e.g., between 2D Ising and Z₂ gauge theories), long-range magic is not invariant, reflecting its non-local nature and differing distribution across dual models.