We show how to obtain perfect samples from a quantum Gibbs state on a quantum computer. To do so, we adapt one of the ``Coupling from the Past''-algorithms proposed by Propp and Wilson. The algorithm has a probabilistic run-time and produces perfect samples without any previous knowledge of the mixing time of a quantum Markov chain. To implement it, we assume we are able to perform the phase estimation algorithm for the underlying Hamiltonian and implement a quantum Markov chain such that the transition probabilities between eigenstates only depend on their energy. We provide some examples of quantum Markov chains that satisfy these conditions and analyze the expected run-time of the algorithm, which depends strongly on the degeneracy of the underlying Hamiltonian. For Hamiltonians with highly degenerate spectrum, it is efficient, as it is polylogarithmic in the dimension and linear in the mixing time. For non-degenerate spectra, its runtime is essentially the same as its classical counterpart, which is linear in the mixing time and quadratic in the dimension, up to a logarithmic factor in the dimension. We analyze the circuit depth necessary to implement it, which is proportional to the sum of the depth necessary to implement one step of the quantum Markov chain and one phase estimation. This algorithm is stable under noise in the implementation of different steps. We also briefly discuss how to adapt different ``Coupling from the Past''-algorithms to the quantum setting.
We show how to obtain perfect samples from a quantum Gibbs state on a quantum computer. To do so, we adapt one of the Coupling from the Past''-algorithms proposed by Propp and Wilson. The algorithm has a probabilistic run-time and produces perfect samples without any previous knowledge of the mixing time of a quantum Markov chain. To implement it, we assume we are able to perform the phase estimation algorithm for the underlying Hamiltonian and implement a quantum Markov chain such that the transition probabilities between eigenstates only depend on their energy. We provide some examples of quantum Markov chains that satisfy these conditions and analyze the expected run-time of the algorithm, which depends strongly on the degeneracy of the underlying Hamiltonian. For Hamiltonians with highly degenerate spectrum, it is efficient, as it is polylogarithmic in the dimension and linear in the mixing time. For non-degenerate spectra, its runtime is essentially the same as its classical counterpart, which is linear in the mixing time and quadratic in the dimension, up to a logarithmic factor in the dimension. We analyze the circuit depth necessary to implement it, which is proportional to the sum of the depth necessary to implement one step of the quantum Markov chain and one phase estimation. This algorithm is stable under noise in the implementation of different steps. We also briefly discuss how to adapt different Coupling from the Past''-algorithms to the quantum setting.
Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm [Chen et al., 2023] provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n × n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.
Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm [Chen et al., 2023] provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n × n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.
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.
We present an algorithm for doing Gibbs sampling on a quantum computer. The algorithm combines phase estimation for a Szegedy operator, and Grover's algorithm. For any ε>0, the algorithm will sample a probability distribution in O(δ1) steps with precision O(ε). Here δ is the distance between the two largest eigenvalue magnitudes of the transition matrix of the Gibbs Markov chain used in the algorithm. It takes O(δ1) steps to achieve the same precision if one does Gibbs sampling on a classical computer.
We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in Nβ/Z and polynomial in log(1/ε), where N is the Hilbert space dimension, β is the inverse temperature, Z is the partition function, and ε is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on 1/ε and quadratically improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix P, it runs in time almost linear in 1/(εΔ3/2), where ε is the absolute precision in the estimation and Δ is a parameter determined by P, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on 1/ε and 1/Δ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.
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.