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arXiv:2005.14059 · Physical Review Letters

Quantum Sampling Algorithms for Near-Term Devices

Dominik S. Wild, Dries Sels, Hannes Pichler, Cristian Zanoci, Mikhail D. Lukin

Published Sep 2, 2021• 31 citations
arXivHTMLPDF

Abstract

Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.

DOI
10.1103/physrevlett.127.100504

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Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.
Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can be solved in principle on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hard-sphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states.
The Markov-chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed up classical mixing processes based on Markov chains. To this end, we present a quantum algorithm, making it possible to prepare a quantum sample---i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic-state generation. We also show that our methods provide a greater speedup over a recently proposed method for obtaining the ground states of (classical) Hamiltonians.
Sampling from complicated probability distributions is a hard computational problem arising in many fields, including statistical physics, optimization, and machine learning. Quantum computers have recently been used to sample from complicated distributions that are hard to sample from classically, but which seldom arise in applications. Here we introduce a quantum algorithm to sample from distributions that pose a bottleneck in several applications, which we implement on a superconducting quantum processor. The algorithm performs Markov chain Monte Carlo (MCMC), a popular iterative sampling technique, to sample from the Boltzmann distribution of classical Ising models. In each step, the quantum processor explores the model in superposition to propose a random move, which is then accepted or rejected by a classical computer and returned to the quantum processor, ensuring convergence to the desired Boltzmann distribution. We find that this quantum algorithm converges in fewer iterations than common classical MCMC alternatives on relevant problem instances, both in simulations and experiments. It therefore opens a new path for quantum computers to solve useful--not merely difficult--problems in the near term.
Sampling tasks are a natural class of problems for quantum computers due to the probabilistic nature of the Born rule. Sampling from useful distributions on noisy quantum hardware remains a challenging problem. A recent paper [Layden, D. et al. Nature 619, 282-287 (2023)] proposed a quantum-enhanced Markov chain Monte Carlo algorithm where moves are generated by a quantum device and accepted or rejected by a classical algorithm. While this procedure is robust to noise and control imperfections, its potential for quantum advantage is unclear. Here we show that there is no speedup over classical sampling on a worst-case unstructured sampling problem. We present an upper bound to the Markov gap that rules out a speedup for any unital quantum proposal.
Gibbs state preparation has emerged as a potential avenue for quantum advantage in the study of physical systems. Although there is known advantage for computing local properties of worst-case Gibbs states, recent results suggest a different picture for average-case local systems: at temperatures where their Gibbs states can be efficiently prepared quantumly, quasi-polynomial time classical algorithms can also compute local expectation values. Yet this leaves open the possibility of an advantage under a stronger notion of simulation: sampling. Gibbs sampling underlies Boltzmann machine-based quantum learning algorithms and quantum supremacy proposals, but its classical average-case complexity for physically relevant models remains unresolved. Here, we give strong evidence against an exponential quantum advantage in Gibbs sampling for an ensemble of local systems known as the quantum ppp-spin model. We describe a classical algorithm for sampling from the Gibbs state to any polynomially small error in total variation distance. Our algorithm is based on a meta-algorithm known as algorithmic stochastic localization (ASL). Our main technical result is a generalized convergence theorem for ASL which reduces approximate sampling from any distribution on the hypercube to estimating the mean of a related "tilted distribution" to sufficiently low additive error. We then extend known quasi-polynomial time algorithms for computing local expectation values of a related disordered model to compute these "tilted means" for the quantum ppp-spin model. Finally, we give a non-rigorous physics argument that this mean-estimation algorithm works down to the spin glass transition temperature. As efficient quantum algorithms also fail at this transition, this suggests they cannot Gibbs sample the quantum ppp-spin model at temperatures below that for which classical algorithms are also (almost) efficient.
Sampling tasks are a natural class of problems for quantum computers due to the probabilistic nature of the Born rule. Sampling from useful distributions on noisy quantum hardware remains a challenging problem. A recent paper [D. Layden et al., Nature (London) 619, 282 (2023).] proposed a quantum-enhanced Markov-chain Monte Carlo algorithm where moves are generated by a quantum device and accepted or rejected by a classical algorithm. While this procedure is robust to noise and control imperfections, its potential for quantum advantage is unclear. Here we show that there is no speedup over classical sampling on a worst-case unstructured sampling problem. We present an upper bound to the Markov gap that rules out a speedup for any unital quantum proposal.