Simulation of classical thermal states on a quantum computer: A transfer-matrix approach

Abstract

We present a hybrid quantum-classical algorithm to simulate thermal states of classical Hamiltonians on a quantum computer. Our scheme employs a sequence of locally controlled rotations, building up the desired state by adding qubits one at a time. We identify a class of classical models for which our method is efficient and avoids potential exponential overheads encountered by Groverlike or quantum Metropolis schemes. Our algorithm also gives an exponential advantage for 2D Ising models with magnetic field on a square lattice, compared with the previously known Zalka's algorithm.PACS numbers: 03.67.Ac, 05.10.CcSimulation of a finite-temperature physical system with a controllable quantum device is one of the most important goals of quantum simulation [1, 2]. Classical Markov-Chain Monte Carlo (MCMC) algorithms are powerful tools for sampling Gibbs distributions. They are efficient provided that the gap ∆ of the transition matrix is non-vanishing; the running time typically scales as τ ∼ O (1/∆). A quantum generalization [3] of MCMC has recently been explored by the quantum information community due to the connection to quantum walks [4]. Richter [5] developed a method for sampling from the Gibbs distribution for periodic lattices. Somma et al.[6] combined quantum walk and quantum Zeno effect to achieve quantum speedup. Wocjan and Abeyesinghe [7] improved it by using fixed point quantum search. Generally, these quantum algorithms allow the running time to scale as τ ∼ O(1/ √ ∆), a quadratic speedup compared with the classical counterparts. However, for many problems of practical interest, such as optimization problems and spin glasses, the gap ∆ may become exponentially small when the system size increases, making it unpractical to use MCMC algorithms for solving them (see Fig. 1). Therefore, gap-independent methods are more desirable for solving these problems.A class of gap-independent methods is called belief propagation [8], which generalizes the transfer matrix methods in statistical physics. For problems involving a regular geometry, it can be very efficient, e.g. onedimensional spin chains. This property will be exploited in this letter, where a different way for obtaining samples from the thermal state is discussed. This approach is a generalization of the state preparation method by Lidar and Biham [9], and Zalka [10]. We show that in some cases, the structure of the system under investigation allows for large speedups over the general methods. This is because the cost of our method is independent of the temperature and the gap size. Our proposed strategy is to construct a coherent encodings of a thermal state (CETS) |ψ CET S directly, rather than sampling from the thermal probability distribution:where s = {0, 1} N , β = 1/k B T is the inverse temperature, H(s) is the eigen-energy of some classical spin Hamiltonian for the N -spin configuration s = s 1 s 2 · · · s N and Z is the partition function. This CETS can be transformed into the corresponding thermal state ρ = e −βH /Tr e −βH by including a set...

A quantum–quantum Metropolis algorithm

“…CETS I (see also ref. 23) is similar to CETS II, but the basis states fjiig are replaced by the computational basis. (iv) By "quantum speedup", we consider only the quantum speedup with respect to the gap δ of the transition matrix of the Markov chain.…”

Low depth quantum circuits for Ising models

“…in Ref. [21] the authors provide an algorithm which does so but is exponential in the square root of the system size (see also [22]). …”

“…However, corner measurement is a quantum process which assumes the thermal state of the classical Hamiltonian has been prepared. Some earlier work [21,38,39] provides quantum algorithms to simulate thermal states of classical spin models. However as mentioned in Sec.…”

NMR Quantum Information Processing

“…This way of reading off elements from the fourth carbon's spectra was useful due to its well resolved spectra compared to the other three. The starting state was a coherent encoding of the thermal states [97,98]. Different unitary evolutions corresponded to variation in the Hamiltonian parameters.…”