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

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Keisuke Fujii, Harunobu Fujimura, Masazumi Honda, Duc Truyen Le

High Energy Physics - Lattice · Submitted 4 Nov 2026 · 113 references

Pith's verdict: Clean protocol paper with a sharp measurement-complexity claim and a straightforward demonstration; the resource estimates are preliminary but the core idea is sound.

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CitationDuc Truyen Le, Harunobu Fujimura, Keisuke Fujii, Masazumi Honda. "Quantum Computing for Entanglement Asymmetry and the Quantum Mpemba Effect in Field Theory." arXiv:2611.05987 (2026). https://pith.science/paper/MC4LF5JJ
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The paper

The paper claims

A quantum measurement protocol estimates the second Rényi entanglement asymmetry with O(ε^{-2}) measurements independent of system size, enabling studies of the quantum Mpemba effect in lattice field theories.

The reading

This paper constructs a quantum measurement protocol for estimating the second Rényi entanglement asymmetry, a quantity that measures symmetry breaking on a subsystem and can detect the quantum Mpemba effect. The protocol combines the SWAP test with a charge-resolving procedure based on randomized symmetry rotations. The key advantage is that for a fixed additive accuracy ε, the required number of measurements scales as O($ε^{{-2}}$) with no explicit dependence on the total system size, making it attractive for large quantum many-body systems and lattice quantum field theories. The authors demonstrate the protocol by classically simulating the real-time evolution of the lattice Schwinger model with a θ term, showing the quantum Mpemba effect for both benchmark spin-product states and fermion-parity-preserving initial states. They also estimate the quantum resources needed for NISQ devices, fault-tolerant quantum computers, and an early-FTQC architecture.

What carries the argument

The central mechanism is the combination of the SWAP test (a controlled-SWAP operation between two copies of the system) with a charge-resolving procedure that symmetrizes one copy by randomly applying powers of the unitary operator U_A = $e^{{2π i Q_A / 2^n}}$, where Q_A is the subsystem charge operator. This randomized symmetry twirling implements the projection onto the diagonal charge sectors of the reduced density matrix without explicit matrix multiplication, allowing the purity of the symmetrized state to be estimated using the same SWAP test circuit. The measurement complexity for each purity estimate is O($ε^{{-2}}$) because the SWAP test expectation value has a statistical error bounded by 1/√N_shot.

Extended reading notes

Core claim

The paper claims that the second Rényi entanglement asymmetry can be estimated using a protocol that combines the SWAP test with a randomized symmetry-twirling operation, achieving a measurement complexity of O($ε^{{-2}}$) for a fixed additive accuracy ε, independent of the total system size. This is accomplished by expressing the symmetrized reduced density matrix as a uniform mixture of rotated copies of the state, realized by randomly applying powers of a unitary operator to one copy before the SWAP test. The protocol is applied to the lattice Schwinger model, where classical state-vector simulations show that the second Rényi entanglement asymmetry decays faster for initial states with larger asymmetry, a signature of the quantum Mpemba effect. The paper further provides resource estimates for implementing the protocol on various quantum hardware platforms.

Load-bearing premise

The classical state-vector simulations of the lattice Schwinger model rely on a second-order Suzuki-Trotter decomposition with time step δt = 0.02/g, and the authors note that rigorous error bounds for these specific parameters (system size up to N=18, time up to tg=1) are not provided, although they believe the decomposition accurately represents the exact unitary evolution.

Editorial extensions

If this is right

  • The protocol enables measurement of the Rényi entanglement asymmetry on near-term quantum devices with a number of measurements that does not grow with the number of qubits, potentially allowing studies of symmetry restoration dynamics in systems too large for classical simulation.
  • For the lattice Schwinger model, the classical simulations demonstrate that the quantum Mpemba effect persists for fermion-parity-preserving initial states, which are physically allowed fermionic states, suggesting that the effect is a robust feature of this field theory.
  • The resource estimates indicate that small-scale implementations (e.g., N=8) are within reach of current NISQ technology with two-qubit gate fidelities around 10^{-4}, while larger systems (N~50) require fault-tolerant or early-FTQC approaches.
  • The protocol can be extended to higher Rényi orders and other symmetry groups by generalizing the symmetry-twirling operation and using multi-copy measurements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the measurement complexity truly scales as O(ε^{-2}) without any hidden system-size dependence, then the protocol circumvents the exponential measurement cost of full state tomography for estimating entanglement asymmetry, making it a practical tool for quantum simulators and future quantum computers.
  • The observation that the quantum Mpemba effect in the lattice Schwinger model depends on the initial charge-sector distribution and also on effective temperature (for parity-preserving states) suggests a richer phenomenology that could be studied systematically once larger systems become accessible via quantum hardware.
  • The resource estimates highlight a crossover: for current NISQ devices the bottleneck is gate error, while for fault-tolerant machines the T-gate count dominates. The early-FTQC approach using TE-PAI may be the most practical route for intermediate system sizes (up to ~80 qubits).
Original abstract

We construct a quantum measurement protocol for estimating the R\'enyi entanglement asymmetry, which quantifies symmetry breaking at the level of a subsystem and provides a probe of the quantum Mpemba effect. The protocol combines the SWAP test with a charge-resolving procedure based on randomized symmetry rotations. For a fixed additive accuracy $\epsilon$, the required number of measurements scales as $\mathcal{O}(\epsilon^{-2})$ without explicit dependence on the total system size, making the protocol particularly attractive for quantum many-body systems and lattice quantum field theories. As a representative application, we study the quantum Mpemba effect in the lattice Schwinger model with a $\theta$ term using classical state-vector simulations. We further estimate the resources required to implement the protocol on noisy intermediate-scale quantum (NISQ) devices, fault-tolerant quantum computers (FTQCs), and a representative early-FTQC architecture. Our results provide a measurement-efficient framework for investigating anomalous symmetry-restoration dynamics in quantum many-body systems and quantum field theories.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-05: 1 request, 5,028 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.04 · states quantitative result 0.83 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We construct a quantum measurement protocol for estimating the R\'enyi entanglement asymmetry, which quantifies symmetry breaking at the level of a subsystem and provides a probe of the quantum Mpemba effect. The protocol combines the SWAP test with a charge-resolving procedure based on randomized symmetry rotations. For a fixed additive accuracy $\epsilon$, the required number of measurements scales as $\mathcal{O}(\epsilon^{-2})$ without explicit dependence on the total system size, making the protocol particularly attractive for quantum many-body systems and lattice quantum field theories. As a representative application, we study the quantum Mpemba effect in the lattice Schwinger model with a $\theta$ term using classical state-vector simulations. We further estimate the resources required to implement the protocol on noisy intermediate-scale quantum (NISQ) devices, fault-tolerant quantum computers (FTQCs), and a representative early-FTQC architecture. Our results provide a measurement-efficient framework for investigating anomalous symmetry-restoration dynamics in quantum many-body systems and quantum field theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We construct a quantum measurement protocol for estimating the R\'enyi entanglement asymmetry, which quantifies symmetry breaking at the level of a subsystem and provides a probe of the quantum Mpemba effect. contribution 0.97 0.95 0.19 asserted 1.00
s1 The protocol combines the SWAP test with a charge-resolving procedure based on randomized symmetry rotations. method 1.00 0.86 0.04 asserted 1.00
s2 For a fixed additive accuracy $\epsilon$, the required number of measurements scales as $\mathcal{O}(\epsilon^{-2})$ without explicit dependence on the total system size, making the protocol particularly attractive for quantum many-body systems and lattice quantum field theories. result 0.83 0.88 0.19 asserted 1.00
s3 As a representative application, we study the quantum Mpemba effect in the lattice Schwinger model with a $\theta$ term using classical state-vector simulations. contribution 0.85 0.95 0.06 asserted 0.99
s4 We further estimate the resources required to implement the protocol on noisy intermediate-scale quantum (NISQ) devices, fault-tolerant quantum computers (FTQCs), and a representative early-FTQC architecture. contribution 0.88 0.93 0.07 asserted 0.97
s5 Our results provide a measurement-efficient framework for investigating anomalous symmetry-restoration dynamics in quantum many-body systems and quantum field theories. interpretation 0.93 0.81 0.75 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. A schematic illustration of a one-dimensional quantum system divided into two subsys￾tems, A and B. Each dot represents a quantum degree of freedom. The subsystem A is indicated by red dots, while the subsystem B is indicated by black dots. 2 Quantum Measurement Protocol for Entanglement Asymmetry In this section, we review the notion of entanglement asymmetry [5] and construct a quantum measurement protocol for estimating the R´enyi entanglement asymmetry. Combining the SWAP test [60] with a charge-resolving procedure based on randomized symmetry rotations [66], we show that the…
Figure 2. Schematic illustration of the reduced density matrix ρA (left) and the symmetrized reduced density matrix ρA,S (right), shown in the eigenbasis of symmetry charge QA. The shaded boxes represent the block-diagonal elements and the symbol ∗ denotes nonzero off-diagonal elements. The presence of nonzero off-diagonal charge blocks in￾dicates symmetry breaking in subsystem A. In contrast, ρA,S is defined by eliminating the off-diagonal charge blocks and is therefore always block-diagonal. When the symmetry is preserved on the subsystem A, the reduced density matrix ρA com￾mutes with the…
Figure 3. Quantum circuit for the SWAP test used to estimate the purity TrA
Figure 4. Realization of the symmetrized state ρA,S in a quantum circuit. We apply the unitary (UA) x where UA is given by (2.19) and an integer x ∈ {0, 1, · · · , 2 n − 1} is chosen uniformly at random for each experimental run. For a finite subsystem, the symmetrized reduced density matrix ρA,S can be written as ρA,S = q Xmax qA=qmin ΠqA ρAΠqA , (2.17) where qmin and qmax are minimal and maximal charges of the subsystem A, respectively. Importantly, one can rewrite this as5 ρA,S = 1 2 n 2 Xn−1 x=0 (UA) x ρA  U † A x , (2.18) where UA is a unitary operator defined as UA = e 2πi QA 2n ,…
Figure 5. Quantum circuit for estimating TrA[ρ 2 A,S], obtained by combining the quantum circuit in
Figure 6. Quantum circuit for investigating the quantum Mpemba effect. The symmetrized purity TrA h ρ 2 A,Si is obtained from the expectation value of measurements on the ancilla qubit. To evaluate the purity TrA
Figure 7. Initial second R´enyi entanglement asymmetry ∆S (2) A (t = 0) as a function of ϕ for the spin-product initial states (3.13). The subsystem size is fixed to NA = 4. The asymmetry vanishes at ϕ = 0 and ϕ = π and reaches its maximum at ϕ = π/2, reflecting the degree of symmetry breaking in the initial state. The dashed vertical lines indicate ϕ = 0.3π, 0.4π, and 0.5π, which are used in the time-evolution analysis. Because these states allow a simple characterization of the initial charge-sector distribution, they provide a useful setting in which to illustrate that the second R´enyi…
Figure 8. Time evolution of the second R´enyi entanglement asymmetry ∆S (2) A (t) for the spin￾product ferromagnetic initial state (3.14). We compare the results for ϕ = 0.3π, 0.4π, and 0.5π. The parameters are fixed to N = 18, NA = 4, θ = π/3, m/g = 4, and ag = 1/4. For the Suzuki–Trotter decomposition, we use the time step δt = 0.02/g. demonstrates the application of our measurement protocol to quantum Mpemba dynamics in a lattice quantum field theory. The initial distribution of the conserved charge provides a qualitative interpretation of this behavior [38,85]. Since the Hamiltonian commut
Figure 9. Charge-sector distributions for the spin-product ferromagnetic state |ψ1(ϕ)⟩F with N = 18. The horizontal axis denotes the total charge q, while the vertical axis denotes the probability p F q (ϕ). The left, middle, and right panels correspond to ϕ = 0.3π, 0.4π, and 0.5π, respectively. The dimension of the charge-q sector is N q  and is maximal at q = N/2. The color shading schematically represents the dimension of the corresponding charge sector. is maximal. A larger charge-sector dimension does not, by itself, guarantee faster dephasing. The relaxation rate also depends on the…
Figure 10. Charge-sector distributions for the spin-product N´eel initial state |ψ1(ϕ)⟩N with N = 18. The horizontal axis denotes the total charge q, while the vertical axis denotes the probability p N q (ϕ). The left, middle, and right panels correspond to ϕ = 0.3π, 0.4π, and 0.5π, respectively. All three distributions are centered at q = N/2, while their widths increase with ϕ. The probability weight of each charge sector is given by [85] p N q (ϕ) = min( X q,N/2) k=max(0,q−N/2)  N/2 k  N/2 q − k  cosN+2q−4k  ϕ 2  sinN−2q+4k  ϕ 2  . (3.25) The expectation value and variance of the…
Figure 11. Time evolution of the second R´enyi entanglement asymmetry ∆S (2) A (t) for the spin￾product N´eel initial state |ψ1(ϕ)⟩N . We compare the results for ϕ = 0.3π, 0.4π, and 0.5π. The parameters are fixed to N = 18, NA = 4, θ = π/3, m/g = 4, and ag = 1/4. For the Suzuki–Trotter decomposition, we use the time step δt = 0.02/g. entanglement asymmetry persists in a physically admissible fermionic setting. In a fermionic theory, physical states cannot contain coherence between sectors with even and odd fermion parity [80, 81]. We therefore consider the following fermion-parity-preserving…
Figure 12. Charge-block coherence weights CqA,q′ A = ∥ΠqA ρAΠq ′ A ∥ 2 HS at t = 0 for the spin-product ferromagnetic initial state |ψ1(ϕ)⟩F (left) and the fermion-parity-preserving ferromag￾netic initial state |ψ2(ϕ)⟩F (right). The two panels use the same color scale, with white corresponding to zero coherence. We use NA = 4, and ϕ = 0.5π. Thus, these states preserve fermion parity while still breaking the continuous U(1) symmetry. To make the resulting constraint on the symmetry-breaking coherence explicit, we de￾compose the reduced density matrix into subsystem-charge blocks, ρA;qA,q′ A =…
Figure 13. Time evolution of the second R´enyi entanglement asymmetry ∆S (2) A (t) for the fermion￾parity-preserving initial states in (3.27). The left and right panels show the ferromag￾netic and N´eel families, respectively, for ϕ = 0.3π, 0.4π, and 0.5π. The parameters are fixed to N = 18, NA = 4, θ = π/3, m/g = 4, and ag = 1/4. The Suzuki–Trotter time step is δt = 0.02/g. characterize the breaking of the subsystem U(1) symmetry. The spin-product state exhibits coherence between both equal- and opposite-parity charge sectors. By contrast, the fermion￾parity-preserving state displays the…
Figure 14. Commutator bound for the second-order product formula. (Left) The commutator bound αcom defined in (4.10) as a function of the system size N on a log-log scale. The symbols show the numerical values, while the lines show power-law fits of the form αcom = ANα. Different colors correspond to different choices of m/g with fixed ag = 0.25 and θ = π/3. The fitted values of α and A are given in
Figure 15. Empirical Trotter-step requirement defined in (4.20). We set tg = 1, ag = 0.25 and ϵalg = 0.01 as representative values. (Left) Memp as a function of the system size N on a log-log scale. The symbols denote the numerical results, while the lines show power-law fits of the form Memp = BNβ/2 . Different colors correspond to different choices of m/g at fixed θ = π/3. The fitted values of β and B are given in
Figure 16. Estimated single-copy CNOT count required to implement the Trotterized time evo￾lution with target error ϵalg = 0.01. The green, blue, and orange lines show the estimate obtained from the worst-case bound (4.13), the commutator bound (4.19), and the empirical Trotter number (4.20), respectively. The left and right panels show the results for m = mc and m/g = 4, respectively. Both cases exhibit qualitatively similar behavior. For the other parameters, we use tg = 1, θ = π/3, and ag = 1/4 as a representative parameter set. The full second-R´enyi protocol requires two such…
Figure 17. Estimated single-copy T count required to implement the Trotterized time evolution with target Trotter error ϵTrot = 10−2 and rotation-synthesis error ϵsyn = 10−2 . The left vertical axis shows the single-copy T count, while the right vertical axis shows the corresponding runtime of the full two-copy protocol estimated from (4.33), assuming an aggregate logical T-gate throughput of RT = 1 MHz. The green, blue, and orange lines show the estimates obtained from the worst-case bound (4.13), the commutator bound (4.19), and the empirical Trotter number (4.20), respectively. The left…
Figure 18. Early-FTQC resource estimates for the two-copy Schwinger-model simulation using TE-PAI on STAR ver. 3. (a) Estimated number of physical qubits, and (b) total runtime including the TE-PAI sampling and PEC overheads. We use Q = 1, ϵalg = 0.01, CSMM = 3, τcyc = 1 µs, pph = 10−3 , and αRUS = 0.1. The two TE-PAI circuits are independently sampled and assumed to be executed in parallel. Only parameter sets (tg, N) satisfying P (2) total ≲ 1 are shown, with the physical parameters fixed to ag = 1/4, m/g = 4, and θ = π/3. of substantial physical-qubit overhead. For example, at N = 50 and…
Figure 19. Schematic zero-temperature phase structure of the continuum Schwinger model in the θ–m/g plane. The theory is periodic under θ → θ + 2π. The solid line at θ = π and m > mc denotes the first-order transition line, which terminates at the second-order critical point (θ, m/g) = (π, mc/g), indicated by the red point. B Parameter Dependence of the Symmetry-Restoration Dynamics In the main text, we used N = 18, NA = 4, θ = π/3, m/g = 4, and ag = 1/4 as a representative set of parameters for the lattice Schwinger model. In this appendix, we examine the dependence of the…
Figure 20. Dependence of the symmetry-restoration dynamics on the fermion mass and the θ angle for the spin-product N´eel initial state (3.23). The vertical and horizontal axes denote the second R´enyi entanglement asymmetry ∆S (2) A (t) and the dimensionless time tg, respectively. The upper panels show the mass dependence at fixed θ = π/3 (left) and θ = π (right), while the lower panels show the θ dependence at fixed m/g = 4 (left) and m = mc (right). The remaining parameters are fixed to N = 18, NA = 4, ϕ = 0.5π, ag = 1/4, and δt = 0.02/g. temperature state is ⟨H⟩β=0 = TrH dim H = 0. (C.1)…
Figure 21. Time evolution of the second R´enyi entanglement asymmetry ∆S (2) A (t) for total system sizes N = 12, 14, 16, and 18. We use the fermion-parity-preserving N´eel initial states in (3.27). The left and right panels correspond to ϕ = 0.5π and ϕ = 0.4π, respectively. The remaining parameters are fixed to NA = 4, θ = π/3, m/g = 4, ag = 1/4, and δt = 0.02/g. temperature value. Under the thermal interpretation above, this corresponds to a decrease in |βeff|, with ϕ = 0.5π and ϕ = 0.3π corresponding to the smallest and largest |βeff|, respec￾tively, among the three representative values.…
Figure 22. Energy expectation values of the fermion-parity-preserving initial states in (3.27) as functions of ϕ. The left and right panels correspond to the ferromagnetic and N´eel families, respectively. For the ferromagnetic family, the energy expectation value is independent of the fermion mass m, so the curves for different masses coincide. The remaining parameters are fixed to N = 18, θ = π/3, and ag = 1/4. The second-order product formula is U(δt) = Y K a=1 e −i δt 2 Ha Y 1 b=K e −i δt 2 Hb . (D.2) As discussed in Sec. 4.1, its one-step error satisfies

Figures reproduced from arXiv:2611.05987 by the authors.

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