Quantum Mpemba Effect

• Quantum Mpemba Effect is a non-equilibrium phenomenon where a two-step evolution protocol accelerates relaxation by first breaking and then restoring symmetry.
• The engineered transient symmetry breaking redistributes charge sectors, enabling faster decay of observables and more efficient convergence to thermal or ground states.
• This effect offers practical benefits in quantum simulations and state preparation by reducing computational resources and mitigating thermalization bottlenecks.

The quantum Pontus-Mpemba effect (QPME) is a non-equilibrium phenomenon wherein a quantum system relaxes more rapidly—either in real or imaginary time—using a two-step evolution protocol than it would through direct evolution under a symmetric Hamiltonian. This effect is distinct from standard quantum Mpemba effects in that the acceleration arises from an engineered sequence of time evolution: a transient symmetry-breaking stage followed by a switch to symmetric dynamics. The QPME has been theoretically demonstrated in both real-time and imaginary-time quantum many-body evolution, providing insight into the role of transient symmetry breaking in accelerating relaxation and thermalization dynamics (Yu et al., 2 Sep 2025).

1. Definition and Conceptual Framework

The QPME is defined as follows: a quantum system, initialized in a single fixed U(1)-asymmetric state, is evolved according to two different protocols:

• Direct symmetric evolution: evolution under a U(1)-symmetric Hamiltonian $H_\mathrm{sym}$, yielding $$|\psi(t)\rangle = e^{-iH_\mathrm{sym}t}|\psi_i(\theta)\rangle$$ (real time) or $$|\psi(\tau)\rangle = \frac{e^{-H_\mathrm{sym}\tau}|\psi_i(\theta)\rangle}{\|e^{-H_\mathrm{sym}\tau}|\psi_i(\theta)\rangle\|}$$ (imaginary time).
• Two-step protocol: initial time evolution under a symmetry-breaking Hamiltonian $H_\mathrm{asym}$ for a time $t_\mathrm{switch}$ (or $\tau_\mathrm{switch}$), then at $t_\mathrm{switch}$ abruptly switching to the symmetric Hamiltonian $H_\mathrm{sym}$ for the remainder.

The remarkable feature is that, for suitably chosen parameters (notably the initial state's symmetry properties and the strength/duration of symmetry breaking), the two-step protocol results in faster relaxation to the thermal (or ground) state than the symmetric protocol alone.

This sharply contrasts with conventional quantum Mpemba protocols, which focus on comparisons between different initial states under a fixed Hamiltonian. Here, the initial state is kept fixed; all acceleration is induced by the engineered time-dependent Hamiltonian sequence.

2. Dynamical Mechanism and Mathematical Structure

The enhanced relaxation arises from a transient breaking of global symmetry, which broadens the state's distribution across conserved charge sectors before restoration of symmetry. This “charge spreading” effect is central:

• For a tilted ferromagnetic initial state (e.g., $$|\psi_i(\theta)\rangle = e^{-i(\theta/2)\sum_j \sigma_j^y}|000\ldots0\rangle$$), pure symmetric evolution (with $H_\mathrm{sym}$, for example the XXZ model with $U(1)$ symmetry) restricts the reachable subspace due to initial charge localization (referred to as a “Hilbert subspace imprint”, HSI). As a result, thermalization is slowed.
• A pre-evolution under $H_\mathrm{asym}$ (which explicitly breaks $U(1)$, e.g., with anisotropic parameters $\gamma\neq1$ in the XXZ spin chain) redistributes probability weight over many charge sectors, artificially “prethermalizing” the system and erasing the HSI. Subsequent evolution under $H_\mathrm{sym}$ proceeds from a more “ergodic” state, promoting faster relaxation.

In real time, this mechanism leads to a faster decay of relevant observables (such as entanglement asymmetry and energy expectation value). In imaginary time, the same two-step protocol accelerates projection onto the ground state, as measured by faster convergence of $\langle H \rangle$ to the ground energy.

Formally, for imaginary time,

$$|\psi(\tau)\rangle = \frac{e^{-H_\mathrm{sym}(\tau-\tau_\mathrm{switch})} e^{-H_\mathrm{asym}\tau_\mathrm{switch}}|\psi_i(\theta)\rangle}{\left\| e^{-H_\mathrm{sym}(\tau-\tau_\mathrm{switch})} e^{-H_\mathrm{asym}\tau_\mathrm{switch}}|\psi_i(\theta)\rangle \right\| }$$

will, for suitable parameters, yield faster decay of the excitation density and energy variance than the symmetric-only protocol.

3. Conditions for the Effect and Parameter Dependence

QPME is highly dependent on both initial state properties and protocol details:

• Initial state: The effect is most pronounced for tilted ferromagnetic states with small tilt angle $\theta$ (e.g., $\theta = 0.05\pi,\, 0.2\pi$). For these, the HSI is most restrictive and symmetry breaking most effective. If large tilt is used (e.g., $\theta = 0.4\pi$), the state is already broadly delocalized in charge sectors, diminishing any further gain from pre-evolution.
• Type of initial state: Antiferromagnetic (Néel) states, which intrinsically reside in the broadest, “most thermal” charge sector, do not display QPME under this protocol.
• Symmetry-breaking strength and duration: The parameter $\gamma$ (degree of asymmetry) and the optimal switch time are crucial. Insufficient or excessive asymmetry (or duration) prevents the acceleration or may induce a non-monotonic speedup.

The optimal protocol is found by maximizing the “charge broadening” prior to switching to symmetric dynamics, which can be quantified via measures such as entanglement asymmetry and the variance of conserved charge.

4. Numerical Verification and Scaling

Comprehensive simulations demonstrate QPME in both real and imaginary time with various system sizes (e.g., chains with $L=12$ and $L=48$ sites). Observables, including entanglement asymmetry and energy density, exhibit clear dynamical crossing points: following the two-step protocol, observables decay more rapidly compared to direct symmetric evolution, and the total relaxation/computation time to reach target accuracy is reduced.

Scaling analysis confirms that the acceleration afforded by QPME persists in the thermodynamic limit: the relative difference in energy density, for example, remains constant as system size grows, even as absolute energy scales with $L$.

5. Applications in Quantum Simulation and State Preparation

The QPME enables significant improvements for quantum simulation and computational techniques:

• State preparation: For both real and imaginary time, the two-step protocol enables more efficient rapid approach to the thermal (or ground) state, with direct implications for protocols such as time-evolving block decimation (TEBD) and projection quantum Monte Carlo (PQMC).
• Resource reduction: The faster relaxation translates to a reduction in required computational resources and helps to mitigate bottlenecks such as the sign problem in PQMC.
• Accelerating thermalization: In platforms where control over Hamiltonian parameters is possible, e.g., optical lattice emulators or superconducting quantum processors, engineered transient symmetry breaking can be employed as a “shortcut” to equilibrium.

This suggests the QPME may be harnessed in hardware-efficient algorithms, quantum annealing, and other contexts where ground state (or thermal state) computation is central.

6. Implications and Broader Significance

The QPME demonstrates that a controlled sequence of Hamiltonians—specifically exploiting a transient episode of symmetry breaking—can fundamentally alter and accelerate quantum relaxation pathways. Operationally, this provides a practical strategy to surmount Hilbert subspace imprint constraints that hinder ergodicity in many-body systems.

Theoretically, the phenomenon highlights the intricate interplay between symmetry, Hilbert space structure, and non-equilibrium dynamics. The existence of QPME illuminates the deep connections between symmetry restoration rates, charge-sector populations, and thermalization bottlenecks controlled by the initial configuration.

The effect also raises further questions about the universality of the protocol across symmetry classes, the extension to non-Abelian cases, and possible roles of prethermalization and entanglement growth in the persistence and optimization of QPME.

Table: Protocol Comparison

Protocol	Symmetry-breaking stage	Symmetry restoration	Acceleration observed
Direct Quench	None	Standard	No
Two-step (QPME)	Yes (transient)	Standard	Yes

7. Future Directions

Potential avenues for extending the QPME include:

• Optimization of the symmetry-breaking Hamiltonian and switching time for a given target state and system.
• Generalization to higher-rank symmetries, non-integrable (chaotic) models, and systems with additional constraints (e.g., conserved quantities, disorder).
• Exploration of QPME in experimental platforms with tunable Hamiltonians, such as Rydberg arrays, trapped ions, or superconducting qubit processors.
• Application of the protocol in quantum algorithms for state preparation, quantum optimization, and Hamiltonian learning tasks.

The QPME thus offers a new methodology for controlled acceleration of relaxation and optimization in complex quantum many-body systems, demonstrating the profound utility of symmetry manipulation in non-equilibrium quantum dynamics (Yu et al., 2 Sep 2025).