Quantum Metropolis sampling

Abstract

The original motivation to build a quantum computer came from Feynman, who imagined a machine capable of simulating generic quantum mechanical systems--a task that is believed to be intractable for classical computers. Such a machine could have far-reaching applications in the simulation of many-body quantum physics in condensed-matter, chemical and high-energy systems. 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 problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibbs states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that has basically acquired a monopoly on the simulation of interacting particles. Here we demonstrate how to implement a quantum version of the Metropolis algorithm. This algorithm permits sampling directly from the eigenstates of the Hamiltonian, and thus evades the sign problem present in classical simulations. A small-scale implementation of this algorithm should be achievable with today's technology.

A quantum–quantum Metropolis algorithm

“…As we shall see, the pair structure jφ i ijφ i i of the eigenvectors provides a unique feature for our construction of the quantum algorithm. For example, through the phase estimation algorithm (PEA), the value of eigenvalue E i can be obtained either from jφ i i or jφ i i; this is the key property introduced in this paper, which relaxes the constraints of the previous quantum Metropolis algorithm (34) in applying it to Szegedy's method.…”

“…Motivation, Results, and Outline of the Quantum-Quantum Metropolis Algorithm (Q 2 MA) To summarize, before this work, the existing quantum algorithms were capable of either (i) preparing thermal states of classical systems with quantum speedup (31,32), or (ii) preparing thermal states of quantum systems without quantum speedup (33,34); this suggests that the potential of quantum computers may not have been fully exploited. This problem motivates us to further explore the power of quantum computation by providing a quantum algorithm that allows us to prepare thermal states of quantum systems with quantum speedup.…”

“…However, one limitation of their result is that the key step involves too many energy nonlocal transitions, making it inefficient for solving practical problems. Recently, Temme et al (34) addressed this problem by introducing a measurement-based rejection method, which makes the algorithm more realistic for simulating many-body problems. This result shows that quantum computers can solve many-body problems beyond the limits of classical computers.…”

“…However, preparation of the thermal states of quantum Hamiltonians (20,21,33,34), where the eigenvectors are in nontrivial superposition of the computational basis vectors, is a much more challenging problem for classical computation. Classically, one would need to solve for the full eigenvalue problem (i.e., look for all eigenvalues and eigenvectors) for the quantum Hamiltonian first, which often requires more computational resources than directly computing the corresponding partition functions.…”

Contractivity properties of a quantum diffusion semigroup

“…[25] and references therein), estimating the rate at which the purity of a state decreases under the action of a quantum dynamical semigroup, estimating run times of quantum algorithms based on quantum Markov processes (e.g. [41]), etc. Functional inequalities are related to general formulations of certain convergence properties of the underlying semigroup.…”