Optimal quantum algorithm for Gibbs state preparation

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced disispative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then employ our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

💡 Research Summary

The paper addresses the problem of efficiently preparing thermal (Gibbs) states of quantum many‑body systems on a quantum computer, focusing on the high‑temperature regime. Building on a recently introduced dissipative Lindbladian evolution L(β) that mimics open‑system thermalization and is known to be efficiently implementable, the authors prove that for sufficiently high temperatures this dynamics mixes rapidly to the Gibbs state. “Rapid mixing” means that for any initial state ρ the trace‑distance to the target Gibbs state σ_β decays as ‖e^{t L(β)}(ρ)−σ_β‖₁ ≤ poly(n) e^{−γt}, where γ is a constant independent of the system size n. Consequently, the mixing time scales only logarithmically with n.

The technical core introduces an “oscillator norm” ‖X‖{osc}=∑{a∈Λ}‖δ_a(X)‖_∞, where δ_a removes the local reduced part on site a. This norm is highly sensitive to local perturbations, making it suitable for a perturbative analysis of the Lindbladian as a β‑dependent deformation of the β=0 limit, which is simply a fully depolarizing channel with known decay rate λ=1/√2 e^{1/4}. By bounding the perturbation strength κ(β) in the oscillator norm, the authors show that for β below a critical value β* the inequality κ<λ holds, leading to exponential decay of the oscillator norm and, via a standard trace‑norm conversion, to rapid mixing.

For (k,l)‑local Hamiltonians on a D‑dimensional lattice the critical temperature is explicitly given by β* = 1/(615 D² J), where J = h k ℓ combines the interaction strength h, the locality size k, and the maximum number ℓ of terms acting on a site. The proof relies on Lieb‑Robinson bounds to control spatial propagation of errors and on explicit estimates of two error functions: Δ(r₀), which quantifies the error from truncating the Hamiltonian to a ball of radius r₀, and η(β), which measures the temperature‑dependent deviation from the depolarizing limit. Both decay rapidly (Δ) or grow linearly (η) with β, allowing the choice of a modest r₀ (e.g., r₀=4) that guarantees κ<λ whenever βJ ≤ 1/(615 D).

The analysis is extended to long‑range interactions that satisfy a power‑law decay with exponent ν. If ν > 4D+2, a Lieb‑Robinson‑type bound still holds, and a similar perturbative argument yields a critical temperature β* = C/g, where C depends only on D, the interaction prefactor g, and ν. Thus, even for non‑local models the same logarithmic mixing time is achieved at sufficiently high temperature.

Algorithmically, the paper leverages the implementation scheme from prior work