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arXiv:1011.1468 · Proceedings of the National Academy of Sciences

A quantum–quantum Metropolis algorithm

Man‐Hong Yung, Alán Aspuru‐Guzik

Published Jan 3, 2012• 149 citations
arXivHTMLPDF

Abstract

The classical Metropolis sampling method is a cornerstone of many statistical modeling applications that range from physics, chemistry, and biology to economics. This method is particularly suitable for sampling the thermal distributions of classical systems. The challenge of extending this method to the simulation of arbitrary quantum systems is that, in general, eigenstates of quantum Hamiltonians cannot be obtained efficiently with a classical computer. However, this challenge can be overcome by quantum computers. Here, we present a quantum algorithm which fully generalizes the classical Metropolis algorithm to the quantum domain. The meaning of quantum generalization is twofold: The proposed algorithm is not only applicable to both classical and quantum systems, but also offers a quantum speedup relative to the classical counterpart. Furthermore, unlike the classical method of quantum Monte Carlo, this quantum algorithm does not suffer from the negative-sign problem associated with fermionic systems. Applications of this algorithm include the study of low-temperature properties of quantum systems, such as the Hubbard model, and preparing the thermal states of sizable molecules to simulate, for example, chemical reactions at an arbitrary temperature.

DOI
10.1073/pnas.1111758109

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The classical Metropolis sampling method is a cornerstone of many statistical modeling applications that range from physics, chemistry, and biology to economics. This method is particularly suitable for sampling the thermal distributions of classical systems. The challenge of extending this method to the simulation of arbitrary quantum systems is that, in general, eigenstates of quantum Hamiltonians cannot be obtained efficiently with a classical computer. However, this challenge can be overcome by quantum computers. Here, we present a quantum algorithm which fully generalizes the classical Metropolis algorithm to the quantum domain. The meaning of quantum generalization is twofold: The proposed algorithm is not only applicable to both classical and quantum systems, but also offers a quantum speedup relative to the classical counterpart. Furthermore, unlike the classical method of quantum Monte Carlo, this quantum algorithm does not suffer from the negative-sign problem associated with fermionic systems. Applications of this algorithm include the study of low-temperature properties of quantum systems, such as the Hubbard model, and preparing the thermal states of sizable molecules to simulate, for example, chemical reactions at an arbitrary temperature.
The original motivation to build a quantum computer came from Feynman who envisaged a machine capable of simulating generic quantum mechanical systems, a task that is intractable for classical computers. Such a machine would have tremendous applications in all physical sciences, including condensed matter physics, chemistry, and high energy physics. Part of Feynman's challenge was met by Lloyd who showed how to approximately decompose the time-evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the more fundamental problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibb's states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that basically acquired a monopoly for the simulation of interacting particles. In this talk, I will demonstrate that the corresponding quantum problem can be solved by a quantum Metropolis algorithm. This validates the quantum computer as a universal simulator, and proves that the so-called sign ! problem occurring in quantum Monte Carlo methods can be resolved with a quantum computer.
Sampling from thermal equilibrium distributions of quantum systems is a fundamental problem in multiple areas of science, such as condensed matter physics, quantum chemistry, and high-energy physics. Classical methods can degrade quickly, due to the fermionic sign problem or classical approximation errors, motivating the search for more efficient alternatives on quantum hardware. The presented work develops the Quantum Metropolis algorithm proposed by Temme et al. (2011) further to arbitrary energy precision while also reducing demands on quantum resources by requiring only a single, non-persistent auxiliary register. A general formalism for quantum Markov Chain Monte Carlo (qMCMC) maps is introduced and used to prove correctness via primitivity and quantum detailed balance. The simplified rejection mechanism results in additional conditions on the step operators, in particular requiring them to be permutations on the eigenbasis. Numerical simulations validate convergence behavior and robustness under ideal and approximate step operators. However, since constructing step operators directly based on the eigenbasis is not feasible in general, future work could explore variational learning of approximate step operators, structured Hamiltonians, or hybrid classical-quantum walk sampling.
Abstract Quantum computers promise to tackle quantum simulation problems that are classically intractable 1 . Although a lot of quantum algorithms 2–4 have been developed for simulating quantum dynamics, a general-purpose method for simulating low-temperature quantum phenomena remains unknown. In classical settings, the analogous task of sampling from thermal distributions has been largely addressed by Markov Chain Monte Carlo (MCMC) methods 5,6 . Here we propose an efficient quantum algorithm for thermal simulation that—akin to MCMC methods—exhibits detailed balance, respects locality and serves as a toy model for thermalization in open quantum systems. The enduring impact of MCMC methods suggests that our new construction may play an equally important part in quantum computing and applications in the physical sciences and beyond.
The quest for improved sampling methods to solve statistical mechanics problems of physical and chemical interest has proceeded with renewed efforts since the invention of the Metropolis algorithm, in 1953. In particular, the understanding of thermally activated rare-event processes between long-lived metastable states, such as protein folding, is still elusive. In this case, one needs both the finite-temperature canonical distribution function and the reaction current between the reactant and product states to completely characterize the dynamic. Here we show how to tackle this problem using a quantum computer. We use the connection between a classical stochastic dynamics and the Schr\"odinger equation, also known as stochastic quantization, to variationally prepare quantum states, allowing us to unbiasedly sample from a Boltzmann distribution. Similarly, reaction rate constants can be computed as ground-state energies of suitably transformed operators, following the supersymmetric extension of the formalism. Finally, we propose a hybrid quantum-classical sampling scheme to escape local minima and explore the configuration space in both real-space and spin Hamiltonians. We indicate how to realize the quantum algorithms constructively, without assuming the existence of oracles or quantum walk operators. The quantum advantage concerning the above applications is discussed.
It has been shown that the Metropolis algorithm can be implemented on quantum computers in a way that avoids the sign problem. However, flat histogram techniques are often preferred as they don't suffer from the same limitations that afflict Metropolis for problems of real-world interest and provide a host of other benefits. In particular, the Wang-Landau method is known for its efficiency and accuracy. In this work we design, implement, and validate a quantum algorithm for Wang-Landau sampling, greatly expanding the range of quantum many body problems solvable by Monte Carlo simulation.
Quantum computing and quantum Monte Carlo (QMC) are respectively the state-of-the-art quantum and classical computing methods for understanding many-body quantum systems. Here, we propose a hybrid quantum-classical algorithm that integrates these two methods, inheriting their distinct features in efficient representation and manipulation of quantum states and overcoming their limitations. We first introduce non-stoquasticity indicators (NSIs) and their upper bounds, which measure the sign problem, the most notable limitation of QMC. We show that our algorithm could greatly mitigate the sign problem, which decreases NSIs with the assistance of quantum computing. Meanwhile, the use of quantum Monte Carlo also increases the expressivity of shallow quantum circuits, allowing more accurate computation that is conventionally achievable only with much deeper circuits. We numerically test and verify the method for the N2_22​ molecule (12 qubits) and the Hubbard model (16 qubits). Our work paves the way to solving practical problems with intermediate-scale and early-fault tolerant quantum computers, with potential applications in chemistry, condensed matter physics, materials, high energy physics, etc.