Theory of Quantum Imaginary-Time Mpemba Effect

TL;DR

This paper develops a theoretical framework for the quantum imaginary-time Mpemba effect, revealing conditions under which certain states relax faster despite being initially farther from the ground state, impacting quantum state preparation efficiency.

Contribution

It introduces a unified theory and criteria for the quantum Mpemba effect in QITE, including necessary and sufficient conditions and practical sufficient conditions for finite-time crossings.

Findings

Derived a simple necessary and sufficient condition for the quantum Mpemba effect.

Identified criteria for initial states that enable faster relaxation in QITE.

Discovered phenomena like multiple crossings and simultaneous intersections in multi-level systems.

Abstract

Quantum imaginary-time evolution (QITE) is a fundamental framework for preparing ground and thermal states, yet its computational cost scales significantly with the evolution duration $\tau$. Reducing this duration is critical for practical quantum advantage. Here, we establish a unified theoretical framework for the Mpemba effect in QITE -- a counterintuitive phenomenon where a state initially farther from the ground state relaxes to it faster than one initially closer. We derive a remarkably simple necessary and sufficient condition for the occurrence of this effect, showing it is uniquely determined by the population ratios of excited states to the ground state. For practical state preparation, we introduce a rigorous sufficient condition for the finite-time Mpemba effect, ensuring the crossing occurs before reaching a prescribed proximity threshold. Furthermore, we unveil unique dynamical features, including a multiple-crossing phenomenon in multi-level systems and simultaneous intersections for collinear initial states. Our results provide criteria for identifying favorable initial states in QITE and offer deep insights into the speed limit of quantum state preparation.