Mpemba Effects in Quantum Complexity

TL;DR

The paper shows that Mpemba-like relaxation phenomena can appear in quantum resource measures, extending the Mpemba concept beyond thermodynamics to abstract facets of quantum complexity. By framing dynamics in quantum resource theories and using random brickwork and Clifford circuits, it demonstrates QMEs for coherence and imaginarity and Pontus-Mpemba effects across several resources via preheating protocols. These findings reveal that more resourceful quantum states can dissipate their local complexity faster under free dynamics, with implications for understanding quantum relaxation and potential applications in near-term devices. The work also provides scalable computational methods and points to future analytical and open-system explorations of quantum Mpemba physics.

Abstract

The Mpemba effect is the phenomenon whereby systems farther from equilibrium may relax faster. In this work, we show that this counterintuitive behavior appears in the very measures that define quantum complexity. Using the framework of quantum resource theories, we study the dynamics of coherence, imaginarity, non-Gaussianity, and magic state resources in random circuit models. Our results reveal that coherence and imaginarity display a quantum Mpemba effect when the system is initialized in resourceful product states, while non-Gaussianity and magic do not. Strikingly, all four resources exhibit the so-called Pontus-Mpemba effect: an initial "preheating" stage accelerates relaxation compared to direct "cooling" dynamics. Taken together, our findings show that Mpemba physics extends beyond thermodynamics and asymmetry, emerging broadly in the resource theories that capture aspects of quantum complexity.

Figures (5)

• Figure 1: (a) Quantum Resources: We study coherence, imaginarity, non-Gaussianity and magic resources to extend the Mpemba effect to measures of quantum complexity. (b) Setup: a resourceful state $|\Psi(\theta)\rangle$ is prepared, with $\theta$ controlling its resource content. For the quantum Mpemba effect (QME), different initial states evolve under free operations (grey), and resources dissipate locally in subsystem $A$ (light orange) while remaining conserved globally. In the Pontus-Mpemba effect, the same state is first "heated" in $A$ and its complement before free evolution. (c) Quantum Mpemba Effects: A more resourceful state dissipates faster than a less resourceful one. (d) Pontus-Mpemba Effects: Preheating accelerates local dissipation compared to free evolution alone.
• Figure 2: The QME for quantum resources in a 1D chain of $N=20$ qubits: coherence (a), imaginarity (b), and non-Gaussianity (c); and for $N=12$ qutrits for quantum magic resources (d). The resource content of a subsystem of size $N_A=2$ is evaluated with the suitable resource monotones, starting from the initial state $\ket{\Psi(\theta}$. The dynamics are governed by the circuit $U_t$ comprising free unitary operations, and the results are averaged over $4000$ circuit realizations. While these choices of initial states reveal the characteristic crossings of the QME for various tilting angles, $\theta$, for the coherence and imaginarity, see Fig. (a-b), no comparable effect is observed in the case of non-Gaussianity and magic resources, as shown in Fig. (c-d).
• Figure 3: The QPME for quantum resources in a 1D chain of $N=20$ qubits: non-Gaussianity (a); and for $N=12$ qutrits for quantum magic resources (b), for subsystem size $N_A=2$. The initial state is $\ket{\Psi(\theta)}$ and the dynamics is now driven by a two-step protocol: first, with the resourcefulness increasing circuit $\tilde{U}^{A}_{t}\otimes \tilde{U}_{t}^{B}$ of depth $T$ and then, with $U_{t}$, comprising resource-free gates. All the results are averaged over $4000$ circuit realizations. We observe a faster decay of the local resource content for increasing $T$ --- the signature of the QPME.
• Figure 4: The QME for quantum resources in a 1D chain of $N=512$ qubits using the Clifford simulation and the QPME with exact vector simulation for $N=20$ for two resources: coherence (b) and imaginarity (c), and coherence again, however, with a slightly setup from the earlier one (d). In the first case, we monitor the resource content of a subsystem of size $N_A=4$ (a) and $N_A=2$ for the last three cases (b-d), where the results are averaged over $4000$ disorder realizations. In Fig. (a), the system is initialized with qubits in $\ket{+}$ (probability $p$) or $\ket{0}$ (probability $1-p$), and evolves under free operations, demonstrating the robustness of QME for large sizes. In Figs. (b–c), starting from a common initial state $\ket{\Psi(\theta)}$, the system is evolving for $T$ layers with the circuit $\tilde{U}_{t}^{A}\otimes \tilde{U}_{t}^{B}$ including resourceful gates, then under free operations, yielding the characteristic QPME crossings for coherence and imaginarity. Finally, Fig. (d) shows that preheating restricted to subsystem $B$ already suffices to capture the QPME crossing.
• Figure 5: Markov dynamics in case of Coherence-preserving Floquet evolution for $N=6$ and $N_A=2$ for a fixed gate realization in OBCs. Left panel: The time evolution of the resource monotone for the reduced dynamics of the subsystem showcasing the Mpemba effect. (b): The overlap of the slowest-moving mode of the superoperator with the initial state decreases with increasing resources, i.e., $\theta$-values.