We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing convergence of the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models, including Schrödinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians, and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.
We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing convergence of the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models, including Schrödinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians, and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.
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−cn, where n 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.
Systems in thermal equilibrium at non-zero temperature are described by their Gibbs state. For classical many-body systems, the Metropolis-Hastings algorithm gives a Markov process with a local update rule that samples from the Gibbs distribution. For quantum systems, sampling from the Gibbs state is significantly more challenging. Many algorithms have been proposed, but these are more complex than the simple local update rule of classical Metropolis sampling, requiring non-trivial quantum algorithms such as phase estimation as a subroutine. Here, we show that a dissipative quantum algorithm with a simple, local update rule is able to sample from the quantum Gibbs state. In contrast to the classical case, the quantum Gibbs state is not generated by converging to the fixed point of a Markov process, but by the states generated at the stopping time of a conditionally stopped process. This gives a new answer to the long-sought-after quantum analogue of Metropolis sampling. Compared to previous quantum Gibbs sampling algorithms, the local update rule of the process has a simple implementation, which may make it more amenable to near-term implementation on suitable quantum hardware. This dissipative Gibbs sampler works for arbitrary quantum Hamiltonians, without any assumptions on or knowledge of its properties, and comes with certifiable precision and run-time bounds. We also show that the algorithm benefits from some measure of built-in resilience to faults and errors (``fault resilience''). Finally, we also demonstrate how the stopping statistics of an ensemble of runs of the dissipative Gibbs sampler can be used to estimate the partition function.
The preparation of quantum Gibbs states is a crucial task in quantum computing. In this work, we prove that a recently introduced, efficiently implementable dissipative evolution thermalizes to the Gibbs state in time scaling polynomially or even logarithmically with system size at high enough temperatures for any Hamiltonian that satisfies a Lieb-Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or "thermofield double"states. To the best of our knowledge, these are the first results rigorously establishing the efficient preparation of high-temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of dissipative evolutions for inverse temperatures polynomial in the system's size is computationally equivalent to standard quantum computations. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian, and then connecting its convergence to that of the infinite temperature limit. We further present an alternative proof that is based on showing the exponential decay of the so-called oscillator norm, yielding convergence in logarithmic times. For low temperature, we instead perform a perturbation at zero temperature and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that a family of quasi-local dissipative evolutions efficiently prepares a large class of quantum many-body states of interest, and has the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.
We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of n sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error ε in time O(log4(n)+log(1/ε)). This improves the previous O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log−4(n)).
Over the past decades, the problem of classical Gibbs sampling has led to one of the most powerful algorithmic families of the computer age: the Markov Chain Monte Carlo algorithms. Recent theoretical advances in quantum Gibbs sampling have, understandably, sparked considerable excitement. A recent breakthrough provided the first Lindbladians that provably drive a general quantum system to its Gibbs state, efficiently up to the mixing time. In this thesis, we bridge the gap between theory and implementation, with both analytical and numerical contributions. We work out every detail -- from transition rate choices, through quadrature, Trotter, and generator-splitting errors, to quantum circuits that could run on early fault-tolerant quantum computers. Using our fast Krylov-based Lindblad simulators and analysers, we numerically confirm existing theoretical Gibbs sampling results, cross-check our theoretical contributions, and do extrapolatory simulations to understand how the structure of these Gibbs samplers affects convergence rates. Our open-source framework, QuantumFurnace.jl, is built to work hand in hand with theory, providing the community with a tool to further advance quantum Gibbs sampling.
Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian H and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature β. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width W=λmax(H)−λmin(H) give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.