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Quantum Computing for Entanglement Asymmetry and the Quantum Mpemba Effect in Field Theory

This paper proposes a measurement-efficient quantum protocol combining the SWAP test with randomized symmetry rotations to estimate Rényi entanglement asymmetry and investigate the quantum Mpemba effect in lattice quantum field theories, demonstrating its feasibility through simulations of the Schwinger model and resource analysis for both NISQ and fault-tolerant quantum computers.

Original author: Keisuke Fujii, Harunobu Fujimura, Masazumi Honda, Duc Truyen Le

Published 2026-11-05

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CC BY 4.0

Original author: Keisuke Fujii, Harunobu Fujimura, Masazumi Honda, Duc Truyen Le

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet, invisible world of quantum physics, particles do not always behave the way our everyday experience suggests. One of the most puzzling phenomena in this realm is how systems return to a state of balance after being disturbed. Usually, we expect a system that starts far from equilibrium to take a long time to settle down, while one that starts close to equilibrium should relax quickly. However, nature sometimes defies this intuition. In a phenomenon known as the Mpemba effect, a system that begins in a more chaotic, "hotter" state can actually reach a calm, balanced state faster than one that starts in a cooler, more ordered state. While this effect was first noticed in the freezing of water, physicists have recently discovered that it also occurs in the quantum world, where it is linked to how symmetries—fundamental rules of balance—are restored within a part of a larger system. Understanding this process is crucial for mapping the behavior of complex materials and the fundamental forces of the universe, but studying it is incredibly difficult because the mathematics involved is so vast that even the most powerful supercomputers struggle to keep up.

To tackle this challenge, a team of researchers has developed a new method for using quantum computers to measure these elusive dynamics. Instead of trying to simulate the entire universe at once, which would require impossible amounts of computing power, they devised a clever protocol that focuses on a specific, measurable quantity called entanglement asymmetry. This quantity acts like a gauge, telling scientists how much a small section of a quantum system has broken the rules of symmetry. The researchers combined two existing techniques: a method for comparing two copies of a quantum state and a process that randomly rotates the system to isolate specific types of symmetry. The result is a measurement strategy that is remarkably efficient. Unlike older methods that would require exponentially more measurements as the system grew larger, this new approach requires a number of measurements that depends only on the desired precision, not on the size of the system. This makes it possible to study complex quantum field theories, which describe how particles interact, on quantum hardware that is currently available or will be available in the near future.

The team tested their idea by simulating a specific model of physics known as the lattice Schwinger model, which is a simplified version of quantum electrodynamics, the theory describing how light and matter interact. They used classical computers to run simulations of how their new measurement protocol would work on this model. They prepared the system in different initial states, some with strong symmetry breaking and some with weak breaking, and watched how they evolved over time. The simulations confirmed that the quantum Mpemba effect does indeed occur in this setting. They observed that the states which started with the strongest initial disruption to symmetry were the ones that restored that symmetry the fastest. This counterintuitive behavior was clearly visible in the data, with the "hotter" states crossing paths with the "cooler" ones as they both moved toward equilibrium. The researchers also examined states that respected a specific rule about particle parity, ensuring the results were physically realistic, and found that the effect persisted even under these stricter conditions.

Beyond proving the concept works in a simulation, the authors carefully calculated the resources needed to run this experiment on actual quantum machines. They looked at three different types of hardware: current noisy devices, future fault-tolerant computers, and an intermediate architecture designed for early error correction. Their analysis showed that while small-scale demonstrations of this effect are within reach of today's technology, larger simulations will require significant advances in hardware reliability and error correction. For instance, running the simulation on a system with fifty qubits would require a level of gate precision that is currently at the edge of what is possible, while larger systems would need the full power of future fault-tolerant machines. The study provides a clear roadmap for how these resources scale, showing that the number of operations needed grows predictably with the size of the system and the desired accuracy.

This work represents a significant step forward in connecting abstract quantum theory with practical experimental capabilities. By providing a measurement protocol that scales efficiently, the researchers have opened a door to investigating anomalous relaxation dynamics in systems that were previously too complex to study. The findings suggest that the quantum Mpemba effect is a robust feature of quantum field theories, not just a curiosity of simple models. While the current results are based on simulations, the framework they have built is ready for implementation on real quantum hardware. As quantum computers continue to improve, this method will allow scientists to probe the real-time evolution of symmetry in the fundamental laws of physics, potentially revealing new insights into how the universe organizes itself after a disturbance. The path from a theoretical idea to a measurable reality is now clearly mapped, inviting the next generation of experiments to explore the strange, fast-relaxing side of the quantum world.

Technical Summary: Quantum Computing for Entanglement Asymmetry and the Quantum Mpemba Effect in Field Theory

Problem Statement
The paper addresses the challenge of studying non-equilibrium dynamics, specifically the "quantum Mpemba effect," in quantum field theories (QFTs). The quantum Mpemba effect describes a phenomenon where a system prepared further from equilibrium (with stronger initial symmetry breaking) relaxes to a symmetric state faster than one prepared closer to equilibrium. Quantifying this effect requires measuring the entanglement asymmetry ( $\Delta S_A$ ), a metric defined by the relative entropy between a subsystem's reduced density matrix ( $\rho_A$ ) and its symmetrized counterpart ( $\rho_{A,S}$ ).

While the concept has been explored in quantum many-body systems, applying it to lattice quantum field theories is difficult. Conventional numerical methods, such as Monte Carlo sampling in the path-integral formulation, fail for real-time evolution due to the severe sign problem. Furthermore, existing quantum measurement techniques for entanglement asymmetry (e.g., full state tomography or generic randomized measurements) suffer from measurement costs that scale exponentially with the subsystem size, rendering them impractical for systems with many degrees of freedom.

Methodology
The authors propose a quantum measurement protocol designed to estimate the second Rényi entanglement asymmetry ( $\Delta S_A^{(2)}$ ) with favorable measurement complexity. The protocol integrates two main components:

SWAP Test: Used to estimate the purity of the reduced density matrix, $\text{Tr}_A[\rho_A^2]$ .
Charge-Resolving Procedure: A method to estimate the purity of the symmetrized state, $\text{Tr}_A[\rho_{A,S}^2]$ , without explicitly constructing the symmetrized matrix. This is achieved by applying randomized symmetry rotations (specifically, unitary operators $U_A^x = e^{2\pi i x Q_A / 2^n}$ ) to one copy of the system before performing the SWAP test. This effectively projects the state onto charge sectors and removes off-diagonal coherences.

The protocol requires two copies of the system evolved in parallel. By repeating the measurement $N_{\text{shot}}$ times, the authors demonstrate that the statistical error scales as $O(\epsilon^{-2})$ for a fixed additive accuracy $\epsilon$ , independent of the total system size. This contrasts sharply with methods requiring exponential scaling in subsystem size.

Key Contributions and Results

Protocol Construction: The paper constructs a circuit (illustrated in Fig. 6) that combines time evolution, randomized symmetry twirling, and SWAP tests. It provides a rigorous derivation showing that the measurement cost does not explicitly depend on the system size, making it suitable for lattice QFTs.
Application to the Lattice Schwinger Model: The authors apply the protocol to the (1+1)-dimensional lattice Schwinger model (quantum electrodynamics with a $\theta$ $θ$ term). Using classical state-vector simulations, they investigate the quantum Mpemba effect for two classes of initial states:
1. Spin-product states: Benchmark states (ferromagnetic and Néel) that break symmetry but do not strictly obey fermion-parity superselection rules.
2. Fermion-parity-preserving states: Physically admissible states that respect the superselection rule while still breaking the continuous $U(1)$ symmetry.
Observation of the Mpemba Effect: In both cases, the simulations show that initial states with larger initial entanglement asymmetry (stronger symmetry breaking) exhibit faster relaxation (symmetry restoration) than those with weaker initial breaking. This crossing of relaxation curves serves as the signature of the quantum Mpemba effect.
Mechanism Analysis: For spin-product states, the authors provide a qualitative explanation linking the relaxation rate to the initial charge-sector distribution. States with broader charge distributions (centered around larger charge sectors) access more energy eigenstates and oscillation frequencies, leading to faster dephasing and symmetry restoration.
Resource Estimation: The paper estimates the resources required to implement this protocol on three types of hardware:
- NISQ Devices: For small systems (e.g., $N=8$ ), the required two-qubit gate count is within the reach of current trapped-ion technology, though error accumulation remains a challenge for larger sizes.
- Fault-Tolerant Quantum Computers (FTQC): The authors estimate T-gate counts and runtime, assuming a logical T-gate throughput of 1 MHz. They show that while the gate count scales polynomially with system size, the runtime for large systems (e.g., $N=50$ ) remains feasible (seconds to minutes) in a fully fault-tolerant regime.
- Early-FTQC (STAR ver. 3): Using Time Evolution by Probabilistic Angle Interpolation (TE-PAI) and error mitigation, the authors estimate that systems up to $N \approx 79$ could be simulated with a physical error rate of $10^{-3}$ , albeit with a significant physical qubit overhead ( $\sim 10^5$ qubits for $N=50$ ).

Significance and Claims
The paper claims to provide a measurement-efficient framework for investigating anomalous symmetry-restoration dynamics in quantum many-body systems and quantum field theories. By decoupling the measurement complexity from the system size, the protocol overcomes a primary bottleneck in studying real-time dynamics in QFTs on quantum hardware.

The authors emphasize that their work connects a theoretical measurement protocol to a concrete field-theory application (the Schwinger model) and provides quantitative resource estimates for near-term and future quantum hardware. They explicitly state that their numerical results are limited to small system sizes ( $N \le 18$ ) and that the observed finite-size effects (such as residual asymmetry at late times) require further investigation in the thermodynamic limit using more scalable classical methods (like tensor networks) or quantum hardware. The paper does not claim to have solved the general sign problem for all QFTs but rather offers a specific, scalable pathway for probing non-equilibrium symmetry dynamics in lattice models.

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