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arXiv:2109.03007 · Physical Review A

Quantum sampling algorithms, phase transitions, and computational complexity

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

Published Sep 2, 2021• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.104.032602

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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.
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.
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.
A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: At each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. If negative evolution times can be implemented with constant overhead, then the average absolute evolution time required by our method is O(L^2/Delta) for constant error probability, where L is the length of the path of eigenstates and Delta is the minimum spectral gap of the Hamiltonian. Making explicit the dependence on the path length is useful for cases where L does not depend on Delta. The complexity of our method has a logarithmic improvement over previous algorithms of this type. The same cost applies to the discrete-time case, where a family of unitary operators is given and each unitary and its inverse can be used. Restriction to positive evolution times incurs an error that decreases exponentially with the cost. Applications of this method to unstructured search and quantum sampling are considered. In particular, we discuss the quantum simulated annealing algorithm for solving combinatorial optimization problems. This algorithm provides a quadratic speed-up in the gap of the stochastic matrix over its classical counterpart implemented via Markov chain Monte Carlo.
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.
A quantum system coupled to a bath at some fixed, finite temperature converges to its Gibbs state. This thermalization process defines a natural, physically-motivated model of quantum computation. However, whether quantum computational advantage can be achieved within this realistic physical setup has remained open, due to the challenge of finding systems that thermalize quickly, but are classically intractable. Here we consider sampling from the measurement outcome distribution of quantum Gibbs states at constant temperatures, and prove that this task demonstrates quantum computational advantage. We design a family of commuting local Hamiltonians (parent Hamiltonians of shallow quantum circuits) and prove that they rapidly converge to their Gibbs states under the standard physical model of thermalization (as a continuous-time quantum Markov chain). On the other hand, we show that no polynomial time classical algorithm can sample from the measurement outcome distribution by reducing to the classical hardness of sampling from noiseless shallow quantum circuits. The key step in the reduction is constructing a fault-tolerance scheme for shallow IQP circuits against input noise.
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.