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arXiv:2609.35159

Weakly interacting fermionic Gibbs states are Gaussian mixtures and classically simulable

Zhengyi Han, Yuanchen Zhao, Zi-Wen Liu

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

Identifying regimes in which interacting fermionic systems admit efficient classical simulation is a crucial task in quantum many-body physics. In this work, we rigorously establish the Gaussianity and classical simulability of sufficiently weakly interacting fermions. For the Gibbs state of a local interacting fermionic Hamiltonian $H=H_0+V$ at any finite temperature and in any spatial dimension, where $H_0$ is quadratic and $V$ is the interaction term, we show that there exists a positive interaction strength below which the Gibbs state is a convex mixture of fermionic Gaussian states. Moreover, based on this structure, we construct a polynomial-time classical sampler for the Gibbs state via Metropolis updates on a tree whose leaves are Gaussian states.

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Identifying regimes in which interacting fermionic systems admit efficient classical simulation is a crucial task in quantum many-body physics. In this work, we rigorously establish the Gaussianity and classical simulability of sufficiently weakly interacting fermions. For the Gibbs state of a local interacting fermionic Hamiltonian H=H0+VH=H_0+VH=H0​+V at any finite temperature and in any spatial dimension, where H0H_0H0​ is quadratic and VVV is the interaction term, we show that there exists a positive interaction strength below which the Gibbs state is a convex mixture of fermionic Gaussian states. Moreover, based on this structure, we construct a polynomial-time classical sampler for the Gibbs state via Metropolis updates on a tree whose leaves are Gaussian states.
We study the structure of Gibbs states in weakly perturbed interacting fermionic systems. First, for a sparse Hamiltonian H=H0+VH=H_0+VH=H0​+V with a quadratic term H0H_0H0​ and a non-quadratic perturbation VVV of scale εεε, we show that the Gibbs state ρβρ_βρβ​ decomposes into a convex combination of Gaussian states whenever the inverse temperature satisfies β≤O(log⁡(1/ε))β\le O(\log(1/ε))β≤O(log(1/ε)). Moreover, we prove that this bound is asymptotically tight by establishing that β≤Θ(log⁡(1/ε))β\le Θ(\log(1/ε))β≤Θ(log(1/ε)) is necessary for certain sparse Hamiltonians. This general framework applies directly to the weak-coupling (small-∣U∣\vert{}U\vert{}∣U∣) regime of the Fermi--Hubbard model with hopping ttt and on-site interaction UUU on any graph of maximum degree DDD. Complementarily, in the strong-coupling (small-∣t∣\vert{}t\vert{}∣t∣) regime, we show that the Gibbs state remains convex-Gaussian up to β≤O(∣U∣−1log⁡(∣U∣/(D∣t∣)))β\le O\big(\vert{}U\vert{}^{-1}\log(\vert{}U\vert{}/(D\vert{}t\vert{}))\big)β≤O(∣U∣−1log(∣U∣/(D∣t∣))), revealing a mechanism for convex-Gaussianity distinct from the weak-coupling setting.
Preparing Gibbs states on a quantum computer offers a way to computationally probe how microscopic interactions give rise to macroscopic phenomena at finite temperature. However, an uncomfortable tension has set in: our strongest general guarantees for Gibbs state preparation hold in regimes where the resulting states can also be simulated classically. We propose investigating physical models with competing energy scales. The Fermi--Hubbard model is a natural target: the competition between electron hopping, on-site repulsion and chemical potential encapsulates many mysteries of condensed matter physics, including Mott insulators, striping and superconductivity. Our main result identifies a simple guiding principle: the temperature scale for efficient preparation is set by the couplings between sites, while the on-site energy scale is unrestricted. For the Fermi--Hubbard model on an nnn-site nearest-neighbor lattice of fixed dimension, we construct a quantum Gibbs sampler that mixes in O(log⁡(n/ε))\mathcal{O}(\log(n/ε))O(log(n/ε)) time whenever the temperature is at least a constant factor larger than the hopping strength, but is allowed to be independent of the repulsion and chemical potential. We obtain analogous guarantees for local qubit systems with bounded intersite interactions and arbitrary external fields. As a consequence, our algorithms can efficiently prepare Fermi--Hubbard Gibbs states that are provably outside the convex hull of fermionic Gaussian states and qubit Gibbs states for locally interacting Hamiltonians outside the set of product states.
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.
We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable AAA at time ttt. By analyzing the convergence of the perturbative expansion in the interaction strength λλλ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime λ∣t∣2D+1=O(1)λ|t|^{2D+1}=\mathcal{O}(1)λ∣t∣2D+1=O(1), when the Hamiltonian is geometrically local on a DDD-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to λ∣t∣=O(1)λ|t|=\mathcal{O}(1)λ∣t∣=O(1) with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to λ∣t∣=O(1)λ|t|=\mathcal{O}(1)λ∣t∣=O(1), modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.
Quantum computers hold the potential to revolutionise the simulation of quantum many-body systems, with profound implications for fundamental physics and applications like molecular and material design. However, demonstrating quantum advantage in simulating quantum systems of practical relevance remains a significant challenge. In this work, we introduce a quantum algorithm for preparing Gibbs states of interacting fermions on a lattice with provable polynomial resource requirements. Our approach builds on recent progress in theoretical computer science that extends classical Markov chain Monte Carlo methods to the quantum domain. We derive a bound on the mixing time for quantum Gibbs state preparation by showing that the generator of the quantum Markovian evolution is gapped at any temperature up to a maximal interaction strength. This enables the efficient preparation of low-temperature states of weakly interacting fermions and the calculation of their free energy. We present exact numerical simulations for small system sizes that support our results and identify well-suited algorithmic choices for simulating the Fermi-Hubbard model beyond our rigorous guarantees.
Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Further, we adapt the techniques from separable qudits to the fermionic setting and prove rapid mixing of the strongly-interacting regime of the Fermi-Hubbard model, for which we also explicitly evaluate the guaranteed parameter regimes. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.