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High-Temperature Fermionic Gibbs States are Mixtures of Gaussian States

Akshar Ramkumar, Yiyi Cai, Yu Tong, Jiaqing Jiang

TL;DR

The paper proves that high-temperature Gibbs states of bounded-degree fermionic Hamiltonians decompose as convex mixtures of fermionic Gaussian states, enabling efficient classical sampling. It introduces an adaptive telescoping and pinning framework that yields an unnormalized Gaussian operator with the Gibbs state as its expectation, and then converts this structural result into an efficient sampling algorithm that outputs normalized Gaussian-state components with the correct mixture weights. A key technical advance is the adaptive Majorana-pairing (partial matching) construction, which defines Gaussian states tied to varying matchings and guarantees positivity of normalization constants under a constant temperature threshold. The work also establishes a contrasting non-Gaussian behavior for the dense SYK model, showing the limits of the structural phenomenon. Together, these results sharpen the boundary between quantum advantage and classical simulability for fermionic Gibbs states and provide a concrete algorithmic pathway for classical Gibbs sampling at high temperature.

Abstract

Efficient simulation of a quantum system generally relies on structural properties of the quantum state. Motivated by the recent results by Bakshi et al. on the sudden death of entanglement in high-temperature Gibbs states of quantum spin systems, we study the high-temperature Gibbs states of bounded-degree local fermionic Hamiltonians, which include the special case of geometrically local fermionic systems. We prove that at a sufficiently high temperature that is independent of the system size, the Gibbs state is a probabilistic mixture of fermionic Gaussian states. This forms the basis of an efficient classical algorithm to prepare the Gibbs state by sampling from a distribution of fermionic Gaussian states. As a contrasting example, we show that high-temperature Gibbs states of the Sachdev-Ye-Kitaev (SYK) model are not convex mixtures of Gaussian states.

High-Temperature Fermionic Gibbs States are Mixtures of Gaussian States

TL;DR

The paper proves that high-temperature Gibbs states of bounded-degree fermionic Hamiltonians decompose as convex mixtures of fermionic Gaussian states, enabling efficient classical sampling. It introduces an adaptive telescoping and pinning framework that yields an unnormalized Gaussian operator with the Gibbs state as its expectation, and then converts this structural result into an efficient sampling algorithm that outputs normalized Gaussian-state components with the correct mixture weights. A key technical advance is the adaptive Majorana-pairing (partial matching) construction, which defines Gaussian states tied to varying matchings and guarantees positivity of normalization constants under a constant temperature threshold. The work also establishes a contrasting non-Gaussian behavior for the dense SYK model, showing the limits of the structural phenomenon. Together, these results sharpen the boundary between quantum advantage and classical simulability for fermionic Gibbs states and provide a concrete algorithmic pathway for classical Gibbs sampling at high temperature.

Abstract

Efficient simulation of a quantum system generally relies on structural properties of the quantum state. Motivated by the recent results by Bakshi et al. on the sudden death of entanglement in high-temperature Gibbs states of quantum spin systems, we study the high-temperature Gibbs states of bounded-degree local fermionic Hamiltonians, which include the special case of geometrically local fermionic systems. We prove that at a sufficiently high temperature that is independent of the system size, the Gibbs state is a probabilistic mixture of fermionic Gaussian states. This forms the basis of an efficient classical algorithm to prepare the Gibbs state by sampling from a distribution of fermionic Gaussian states. As a contrasting example, we show that high-temperature Gibbs states of the Sachdev-Ye-Kitaev (SYK) model are not convex mixtures of Gaussian states.
Paper Structure (19 sections, 36 theorems, 141 equations, 3 algorithms)

This paper contains 19 sections, 36 theorems, 141 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Technical highlights.
  4. Contribution and open problems
  5. Overview and organization
  6. Problem Setup
  7. Telescoping and Sampling
  8. Structural Algorithm
  9. Overview
  10. Algorithmic Description
  11. Analysis
  12. Notation
  13. Proof of Correctness
  14. Sampling Algorithm
  15. Overview
  16. ...and 4 more sections

Key Result

Theorem 1.1

Consider a system of $2n$ Majorana operators on a constant-dimension lattice, and a fermionic Hamiltonian $H=\sum_a H_a$ where each $H_a$ involves only a constant number of Majorana operators on nearby lattice sites. Then, there exists a constant critical temperature such that, above this temperatur where $\{q^{(i)}\}_i$ is a classical distribution, $\rho^{(i)}$ belongs to a special class of Gauss

Theorems & Definitions (36)

  • Theorem 1.1: Informal version of Theorem \ref{['thm:structural']}
  • Theorem 1.2: Informal version of Theorem \ref{['thm:efficient_gibbs_sampling']}
  • Theorem 1.3: Informal version of Theorem \ref{['thm:SYK_gaussian']}
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Theorem 4.2
  • Lemma 4.3
  • Corollary 4.4
  • Corollary 4.5
  • ...and 26 more