Quantum Replica Exchange

TL;DR

The paper develops a quantum analogue of replica exchange to speed up Gibbs-state sampling in the presence of local energy barriers. By coupling a slow, barrier-laden Lindbladian to a fast, barrier-free replica via a swap on a localized region, the authors prove a rigorous lower bound on the spectral gap of the joint dynamics, showing exponential improvement in barrier-height dependence under a commuting cut. They illustrate the method with defection-based Ising and Heisenberg models, showing that mixing times shift from exponential to polynomial in barrier strength while preserving near-term implementability through local swaps. The framework blends Gaussian and Metropolis-style Lindbladian filtering with a swap operation and is positioned to generalize to broader barrier types and multi-replica temperature ladders, offering a practical acceleration tool for quantum Gibbs sampling on near-term devices.

Abstract

The presence of energy barriers in the state space of a physical system can lead to exponentially slow convergence for sampling algorithms like Markov chain Monte Carlo (MCMC). In the classical setting, replica exchange (or parallel tempering) is a powerful heuristic to accelerate mixing in these scenarios. In the quantum realm, preparing Gibbs states of Hamiltonians faces a similar challenge, where bottlenecks can dramatically increase the mixing time of quantum dynamical semigroups. In this work, we introduce a quantum analogue of the replica exchange method. We define a Lindbladian on a joint system of two replicas and prove that it can accelerate mixing for a class of Hamiltonians with local energy barriers. We provide a rigorous lower bound on the spectral gap of the combined system's Lindbladian, which leads to an exponential improvement in spectral gap with respect to the barrier height. We showcase the applicability of our method with several examples, including the defected 1D Ising model at arbitrary constant temperature, and defected non-commuting local Hamiltonians at high temperature.

Figures (4)

• Figure 1: Illustration of a local energy barrier with replica exchange. Region $A$ of the state space has a local energy barrier which slows down mixing in $H$. However, by coupling to $I_A$, which has a flat energy landscape, the barrier can be traversed by going to the second copy, going across $A$ there, and then returning to the first copy (path in green). More generally, $H_2$ may be replaced by any fast-mixing Hamiltonian that assists in overcoming barriers in $H_1$, as discussed in \ref{['sec:Implicationsgeneralizations']}.
• Figure 2: When a local barrier is introduced to the initial system, it creates a bottleneck in $A$ which decreases the gap. The application of replica exchange amplifies the gap by introducing a copy of region $A$ ($A'$, in green), which removes the bottleneck and restores the gap of the fast-mixing subsystem $B$, up to a constant factor.
• Figure 3: Defected 1D Ising model (schematic)
• Figure 4: Heisenberg model with a commuting separating cut. The green vertices are in $A$, while the remaining black vertices are in $B$. The blue edges are $\sigma_i^x \sigma^x_j + \sigma^y_i \sigma^y_j + \sigma^z_i \sigma^z_j$, while the red and green edges are $\sigma^z_i \sigma^z_j$ interactions. Yellow edge is $J\sigma_{i^\star}^z\sigma_{j^\star}^z$ with a large coupling $J\gg 1$. Moreover, the red edges form $H_{AB}$.

Theorems & Definitions (61)

• Proposition 1: Slow mixing Hamiltonians, informal
• Theorem 4.1: Main result
• Lemma 1: Lemma 3 of ding2024polynomial, Lemmas 1, 2 of alicki2009thermalization
• proof
• Remark 1
• Lemma 2
• proof
• Lemma 3: Gap bounds imply matching mixing-time bounds
• proof
• Lemma 4: Variant of Lemma 2.1 of gamarnik2024slow