Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition

TL;DR

This work studies the problem of efficiently computing the quantum Gibbs free energy and sampling from the Gibbs state of local quantum spin systems. It introduces the Markov Entropy Decomposition (MED), a classical convex relaxation, and proves that exponential decay of the Hamiltonian of mean force (the effective interaction) guarantees efficient convergence of MED in 1D at any temperature and in high-temperature commuting settings in higher dimensions; this decay implies decay of the conditional mutual information and enables a rounding procedure based on rotated Petz recovery maps to reconstruct a global Gibbs state from MED marginals. As a byproduct, the authors obtain (quasi-)polynomial-time classical algorithms for approximating the free energy and constant-diameter marginals, and they show how to implement an efficient quantum Gibbs sampler via a channel-concatenation rounding scheme. The results rigorously connect correlation-decay physics and information-theoretic properties to algorithmic efficiency for thermal-state problems, providing a framework for provable Gibbs-state preparation in relevant quantum many-body models.

Abstract

We revisit the Markov Entropy Decomposition, a classical convex relaxation algorithm introduced by Poulin and Hastings to approximate the free energy in quantum spin lattices. We identify a sufficient condition for its convergence, namely the decay of the effective interaction. The effective interaction, also known as Hamiltonians of mean force, is a widely established correlation measure, and we show our decay condition in 1D at any temperature as well as in the high-temperature regime under a certain commutativity condition on the Hamiltonian building on existing results. This yields polynomial and quasi-polynomial time approximation algorithms in these settings, respectively. Furthermore, the decay of the effective interaction implies the decay of the conditional mutual information for the Gibbs state of the system. We then use this fact to devise a rounding scheme that maps the solution of the convex relaxation to a global state and show that the scheme can be efficiently implemented on a quantum computer, thus proving efficiency of quantum Gibbs sampling under our assumption of decay of the effective interaction.

Figures (5)

• Figure 1: Overview of the connections proven in this work. We prove a decay condition on the effective interaction and use it for the design of classical and quantum algorithms for computing the free energy $F$. Here, $\rho$ and $\rho_A$ denote the Gibbs state and its marginal, respectively, and $I(A:C|B)=S(AB)+S(BC)-S(ABC)-S(B)$ is the von Neumann conditional mutual information.
• Figure 2: Illustration of the Markov shield $S_k$ as defined in \ref{['eq:Skshield']} with $\ell_S=2$.
• Figure 3: System $ABC$ in a chain.
• Figure 4: $N$-partite system $A_1 \ldots A_N$ in a chain.
• Figure 5: An interval $\Lambda$ split into three subintervals $\Lambda=ABC$ such that $B$ shields $A$ from $C$, where $B$ is further split into five subintervals $B_1,\ldots, B_5$ with $B_1B_2$ and $B_4B_5$ corresponding to the left and right effective interaction terms respectively.

Theorems & Definitions (40)

• Theorem 1: Informal version of Theorem \ref{['thm:errorEff']}
• Theorem 2: Informal version of Theorem \ref{['thm:rounding']}
• Remark 1
• Theorem 3
• proof
• Remark 2
• Definition 1
• Definition 2
• Definition 3
• Example 1