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arXiv:2604.07214

Quantum Gibbs sampling through the detectability lemma

Di Fang, Jianfeng Lu, Yu Tong, Chu Zhao

quant-phmath-phmath.NASubmitted Apr 8, 2026• Announced Apr 9, 2026• 0 citations
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Abstract

Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of $M$ terms, this approach reduces the cost by a factor of $O(M)$. We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.

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Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of MMM terms, this approach reduces the cost by a factor of O(M)O(M)O(M). We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.
Szegedy's quantum walk quadratically accelerates reversible classical Markov chains, but extending this mechanism to efficiently implementable quantum Gibbs samplers has remained challenging beyond special classes. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access and warm-start assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. Moreover, we construct noncommuting Hamiltonians with efficiently preparable warm starts whose refinement under quantum Gibbs samplers requires exponential time. For these models, our algorithm achieves an end-to-end quantum speedup over the worst-case mixing scale.
Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature.Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH H on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian H ( c ) . We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows.– If H is a 2-local qudit CLH, then H ( c ) is a 2-local qudit classical Hamiltonian.– If H is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then H ( c ) is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in O ( n 2 p o l y ( l o g n ) ) time.– If H is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then H ( c ) is a constant-local classical Hamiltonian.
Preparing thermal and ground states is an essential quantum algorithmic task for quantum simulation. In this work, we construct the first efficiently implementable and exactly detailed-balanced Lindbladian for Gibbs states of arbitrary noncommutative Hamiltonians. Our construction can also be regarded as a continuous-time quantum analog of the Metropolis-Hastings algorithm. To prepare the quantum Gibbs state, our algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature βββ, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius ∼β\simβ∼β) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladians yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (i.e., the Thermal Field Double state). These favorable features suggest that our construction serves as a quantum algorithmic counterpart to classical Markov chain Monte Carlo sampling.
Many physical phenomena, including thermalization in open quantum systems and quantum Gibbs sampling, are modeled by Lindbladians approximating a system weakly coupled to a bath. Understanding the convergence speed of these Lindbladians to their steady states is crucial for bounding algorithmic runtimes and thermalization timescales. We study two such families of processes: one characterizing a repeated-interaction Gibbs sampling algorithm, and another modeling open many-body quantum thermalization. We prove that both converge in polynomial time for several non-commuting systems, including high-temperature local lattices, weakly interacting fermions, and 1D spin chains. These results demonstrate that simple dissipative quantum algorithms can prepare complex Gibbs states and that Lindblad dynamics accurately capture thermal relaxation. Our proofs rely on a novel technical result that extrapolates spectral gap lower bounds from quasi-local Lindbladians to the non-local generators governing these dynamics.
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.
Solving for quantum ground states is important for understanding the properties of quantum many-body systems, and quantum computers are potentially well-suited for solving for quantum ground states. Recent work [1] has presented a nearly optimal scheme that prepares ground states on a quantum computer for completely generic Hamiltonians, whose query complexity scales as δ−1 , i.e. inversely with their normalized gap. Here we consider instead the ground state preparation problem restricted to a special subset of Hamiltonians, which includes those which we term "nearly-frustration-free": the class of Hamiltonians for which the ground state energy of their block-encoded and hence normalized Hamiltonian α−1H is within δy of -1, where δ is the spectral gap of α−1H and 0≤y≤1 . For this subclass, we describe an algorithm whose dependence on the gap is asymptotically better, scaling as δy/2−1 , and show that this new dependence is optimal up to factors of log⁡δ . In addition, we give examples of physically motivated Hamiltonians which live in this subclass. Finally, we describe an extension of this method which allows the preparation of excited states both for generic Hamiltonians as well as, at a similar speedup as the ground state case, for those which are nearly frustration-free.