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arXiv:2604.08408

Algorithmic Aspects of the Fermi--Hubbard Model

Ainesh Bakshi, Ankur Moitra, Xinyu Tan

quant-phcs.DSmath-phSubmitted Apr 9, 2026• Announced Apr 10, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Apr 9, 2026v2 · Oct 6, 2026
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Abstract

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 $n$-site nearest-neighbor lattice of fixed dimension, we construct a quantum Gibbs sampler that mixes in $\mathcal{O}(\log(n/ε))$ 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.

Comments
Substantially supersedes v1, with additional rapid mixing results to the fermionic systems and in particular the Fermi--Hubbard model

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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.
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.
Recently, there have been several advancements in quantum algorithms for Gibbs sampling. These algorithms simulate the dynamics generated by an artificial Lindbladian, which is meticulously constructed to obey a detailed-balance condition with the Gibbs state of interest, ensuring it is a stationary point of the evolution, while simultaneously having efficiently implementable time steps. The overall complexity then depends primarily on the mixing time of the Lindbladian, which can vary drastically, but which has been previously bounded in the regime of high enough temperatures [Rouzé et al. arXiv:2403.12691 and arXiv:2411.04885]. In this work, we calculate the spectral gap of the Lindbladian for free fermions using third quantisation, and also prove a logarithmic bound on its mixing time by analysing corresponding covariance matrices. Then we prove a constant gap of the perturbed Lindbladian corresponding to interacting fermions up to some maximal coupling strength. This is achieved by using theorems about stability of the gap for lattice fermions. Our methods apply at any constant temperature and independently of the system size. The gap then provides an upper bound on the mixing time, and hence on the overall complexity of the quantum algorithm, proving that the purified Gibbs state of weakly interacting (quasi-)local fermionic systems of any dimension can be prepared in O~(n3polylog⁡(1/ε))\widetilde{\mathcal{O}} (n^3 \operatorname{polylog}(1/ε))O(n3polylog(1/ε)) time on O(n)\mathcal{O}(n)O(n) qubits, where nnn denotes the size of the system and εεε the desired accuracy. As an application, we explain how to calculate partition functions for the considered systems. We provide exact numerical simulations for small system sizes supporting the theory and also identify different suitable jump operators and filter functions for the sought-after regime of intermediate coupling in the Fermi-Hubbard model.
While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states, provided that the spectral gap scales favourably with the number of modes. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. For the Gibbs sampler corresponding to the full Bose-Hubbard model, we obtain a spectral-gap lower bound Ce−cnCe^{-cn}Ce−cn, where nnn is the number of modes. We apply our results to provide a Gibbs-state preparation algorithm on qubit hardware, with runtime polynomial in the number of modes and the inverse spectral gap, and thereby obtain a quantum algorithm to compute thermal properties of the model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.
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.
Fermionic atoms in optical lattices provide a native implementation of Fermi-Hubbard (FH) models that can be used as analog quantum simulators of many-body fermionic systems. Recent experimental advances include the time-dependent local control of chemical potentials and tunnelings, and thus enable us to operate this platform digitally as a programmable quantum simulator. Here, we explore these opportunities and develop ground-state preparation algorithms for different fermionic models, based on the ability to implement both single-particle and many-body high-fidelity fermionic gates, as provided by the native FH Hamiltonian. In particular, we first design variational precompiled quantum circuits to prepare the ground state of the natively implemented FH model, with significant speedups relative to competing adiabatic protocols. Besides, the versatility of this variational approach enables us to target extended FH models, i.e., including terms that are not natively realized on the platform. As an illustration, we include next-nearest-neighbor tunnelings at finite dopings, relevant in the context of d -wave superconductivity. Furthermore, we discuss how to approximate the imaginary-time evolution using variational fermionic circuits, both as an alternative state-preparation strategy and as a subroutine for the quantum Lanczos algorithm to further improve the energy estimation. We benchmark our protocols for ladder geometries, though they can be readily applied to two-dimensional experimental setups to address regimes beyond the capabilities of current classical methods. These results pave the way for more efficient and comprehensive explorations of relevant many-body phases with existing programmable fermionic quantum simulators.
Preparing the thermal density matrix ρ β ∝ e − β H corresponding to a given Hamiltonian H is a task of central interest across quantum many-body physics, and is particularly salient when attempting to study it with quantum computers. Although solved in principle by recent constructions of efficiently simulable Lindblad master equations—that provably have ρ β as a steady state [C.-F. Chen , ]—the implementation of these “exact Gibbs samplers” requires large-scale quantum computing resources and is hence challenging in practice on current or even near-term quantum devices. Here, we propose a scheme for approximately simulating an exact Gibbs sampler up to a rigorously bounded error that only requires the (repeated) implementation of three readily available ingredients: (a) analog simulation of H ; (b) strictly local but time-dependent couplings to ancilla qubits; and (c) reset of the ancillas. We give rigorous guarantees on the difference between the fixed point reached by our protocol and the exact thermal state, which only depend on parameters of the protocol and its . The procedure is efficiently implementable on near-term devices if H is local and the mixing time scales mildly with both system size and protocol parameters. While guaranteeing the latter for Hamiltonians of interest remains an important problem for future work, here we lay the groundwork for developing fully efficient thermal state preparation protocols on quantum simulators.