[Paper Review] Data and Code for "Quantum Mpemba Effect in Random Circuits"

Xhek Turkeshi|arXiv (Cornell University)|May 23, 2024

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates the quantum Mpemba effect in random unitary circuits with U(1) symmetry, demonstrating that initial states with higher entanglement asymmetry—specifically tilted ferromagnets—relax faster to the grand-canonical ensemble than less asymmetric states. Using exact numerics and analytical mappings to classical stochastic processes, the authors show the effect arises from long-time operator spreading tails, establishing its robustness in generic chaotic quantum systems beyond integrable models.

ABSTRACT

The essence of the Mpemba effect is that non-equilibrium systems may relax faster the further they are from their equilibrium configuration. In the quantum realm, this phenomenon arises in the dynamics of closed systems, where it is witnessed by fundamental features such as symmetry and entanglement. Here, we study the quantum Mpemba effect in charge-preserving random circuits on qudits combining extensive numerical simulations and analytical arguments. We show that the more asymmetric certain classes of initial states (tilted ferromagnets) are, the faster they restore symmetry and reach the grand-canonical ensemble. Conversely, other classes of states (tilted antiferromagnets) do not show the Mpemba effect. We provide a simple and general mechanism underlying the effect, based on the spreading of nonconserved operators in terms of conserved densities. Our analysis is based on minimal principles -- locality, unitarity, and symmetry. Consequently, our results represent a significant advancement in clarifying the emergence of Mpemba physics in chaotic systems.

Motivation & Objective

• To clarify the emergence of the quantum Mpemba effect in generic chaotic quantum systems beyond integrable models.
• To investigate whether the Mpemba effect persists in non-integrable, charge-conserving random circuits with local, unitary dynamics.
• To identify the role of initial state asymmetry and operator spreading in driving anomalous relaxation dynamics.
• To establish a minimal framework based on locality, unitarity, and symmetry to isolate the origin of the quantum Mpemba effect.
• To analyze the connection between entanglement asymmetry and relaxation to the grand-canonical ensemble in finite-size systems.

Proposed method

• Numerical simulations using exact diagonalization and tensor network methods for small qudit systems (q=1,2) to compute entanglement asymmetry dynamics.
• Analytical mapping of the dynamics to two coupled symmetric exclusion processes and macroscopic fluctuation theory for large-q limits.
• Use of coarse-grained effective models to describe operator spreading and asymmetry evolution at times t ≪ N_A and t ∼ N_A.
• Derivation of effective expressions for second Rényi entropy asymmetry ΔS̃_A^(2) via stochastic processes and mean-field approximations.
• Evaluation of the asymmetry contribution from non-conserved strings using path integrals and average over operator configurations.
• Application of large-time asymptotics and scaling assumptions to extract the Mpemba effect from long-time tails of operator spreading.

Figure 1: Numerical results for spin- $1/2$ circuit ( $q=1$ ) with $N=20$ , $N_{A}=3$ , averaging over $\mathcal{N}=10^{3}$ circuits. (a) Von Neumann entanglement asymmetry starting from the tilted ferromagnetic (TF) state at different $\theta$ . The crossings highlight the QME: the higher the initi

Experimental results

Research questions

• RQ1 Does the quantum Mpemba effect emerge in generic chaotic quantum circuits with U(1) symmetry?
• RQ2 How does the initial state's entanglement asymmetry influence the relaxation rate to the grand-canonical ensemble?
• RQ3 What is the role of long-time operator spreading tails in generating the Mpemba effect?
• RQ4 Can the Mpemba effect be explained without integrability, relying only on locality, unitarity, and symmetry?
• RQ5 Why do certain initial states (e.g., tilted antiferromagnets) fail to exhibit the Mpemba effect?

Key findings

• The quantum Mpemba effect is observed in tilted ferromagnetic initial states, where higher initial asymmetry leads to faster relaxation to the grand-canonical ensemble.
• Tilted antiferromagnetic states do not exhibit the Mpemba effect, indicating its dependence on initial state structure.
• The effect arises from the long-time tail of operator spreading, not from early-time dynamics, which are symmetric and non-Mpemba-like.
• At large times, the entanglement asymmetry decays as ΔS̃_A^(2) ∼ ΔS̃_A,0^(2) + cos²(θ) ΔS̃_A,1^(2), with the subleading term suppressed for larger θ (higher asymmetry).
• The leading-order decay ΔS̃_A,0^(2) decays exponentially as d^{N_A s_2,eq - v_2 t}, independent of initial state, confirming that initial state dependence is encoded in the asymmetry correction.
• The analytical mapping to symmetric exclusion processes and macroscopic fluctuation theory confirms the Mpemba effect is robust in the large-q limit and arises from statistical properties of operator trajectories.

Figure 2: Numerical results for $q=2$ . (a) Quench (solid lines) and annealed averaged (crosses) Rényi 2 entanglement asymmetry starting from the TF state, with $N=12$ and $N_{A}=3$ , and averaging over $\mathcal{N}=10^{3}$ circuits. The data shows that there is QME even at $q=2$ . (b) Annealed aver