Permutation matrix representation quantum Monte Carlo

Abstract

We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter errorfree. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.

Quantum Algorithm for Simulating Hamiltonian Dynamics with an Off-diagonal Series Expansion

“…We proposed a quantum algorithm for simulating the dynamics of general time-independent Hamiltonians. Our approach consisted of expanding the time evolution operator using an off-diagonal series; a parameter-free Trotter error-free method that was recently developed in the context of quantum Monte Carlo simulations [3][4][5]. This expansion enabled us to simulate the time evolution of states under general Hamiltonians using alternating segments of diagonal and off-diagonal evolutions, with the latter implemented using the LCU technique [1].…”

“…We next derive an expansion of the time evolution operator based on the off-diagonal series expansion recently introduced in Refs. [3][4][5] in the context of quantum Monte Carlo simulations. While we focus in what follows on time-independent Hamiltonians for simplicity, we note that an extension of the following derivation to include time-dependent Hamiltonians also exists [6].…”

“…4 for a detailed comparison). We accomplish this by utilizing a series expansion of the quantum time-evolution operator in its off-diagonal elements wherein the operator is expanded around its diagonal component [3][4][5]. This expansion allows one to effectively integrate out the diagonal component of the evolution, thereby reducing the overall gate and qubit complexities of the algorithm as compared to existing methods.…”

Calculating elements of matrix functions using divided differences

“…, 0] T , d j (z ) is a complex-valued coefficient, and |z = |z is a basis state of unit norm. The above representation, which we refer to as a 'permutation matrix representation' is general and can be applied to any given matrix [7]. Now consider the action of f (M ) on a basis state |z , assuming that f (•) obeys a Maclaurin series expansion with a region of convergence containing the eigenvalues of M :…”

“…We will demonstrate that our algorithm is applicable in cases where the above methods fail because the vectors M j |z get more and more dense as j increases and can no longer be stored, even if |z has only a single nonzero entry. Our approach builds on the recently introduced off-diagonal series expansion [5][6][7] which provides a systematic memory efficient way of obtaining individual matrix elements by summing a series in which every summand may be interpreted as a walk on a graph.…”