No Arabic abstract
Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function --- which is essential to make many-body PQMC simulations computationally feasible --- affects the tunneling rate. The chosen testbeds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically-exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.
We show that the formalism of tensor-network states, such as the matrix product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations. Using a stochastic optimization method, we demonstrate the potential of this approach by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality, using periodic boundary conditions and D*D matrices with D up to 48. The computational cost of our scheme formally scales as ND^3, whereas standard MPS approaches and the related density matrix renromalization group method scale as ND^5 and ND^6, respectively, for periodic systems.
Modeling the dynamics of a quantum system connected to the environment is critical for advancing our understanding of complex quantum processes, as most quantum processes in nature are affected by an environment. Modeling a macroscopic environment on a quantum simulator may be achieved by coupling independent ancilla qubits that facilitate energy exchange in an appropriate manner with the system and mimic an environment. This approach requires a large, and possibly exponential number of ancillary degrees of freedom which is impractical. In contrast, we develop a digital quantum algorithm that simulates interaction with an environment using a small number of ancilla qubits. By combining periodic modulation of the ancilla energies, or spectral combing, with periodic reset operations, we are able to mimic interaction with a large environment and generate thermal states of interacting many-body systems. We evaluate the algorithm by simulating preparation of thermal states of the transverse Ising model. Our algorithm can also be viewed as a quantum Markov chain Monte Carlo (QMCMC) process that allows sampling of the Gibbs distribution of a multivariate model. To demonstrate this we evaluate the accuracy of sampling Gibbs distributions of simple probabilistic graphical models using the algorithm.
In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/Delta^2$, where $Delta$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests there is no quantum advantage in using QAs w.r.t. quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model, where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving the door open for potential quantum speedup, even for stoquastic models. In this work, we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/Delta$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain points at an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.
We discuss a projector Monte Carlo method for quantum spin models formulated in the valence bond basis, using the S=1/2 Heisenberg antiferromagnet as an example. Its singlet ground state can be projected out of an arbitrary basis state as the trial state, but a more rapid convergence can be obtained using a good variational state. As an alternative to first carrying out a time consuming variational Monte Carlo calculation, we show that a very good trial state can be generated in an iterative fashion in the course of the simulation itself. We also show how the properties of the valence bond basis enable calculations of quantities that are difficult to obtain with the standard basis of Sz eigenstates. In particular, we discuss quantities involving finite-momentum states in the triplet sector, such as the dispersion relation and the spectral weight of the lowest triplet.
We present and motivate an efficient way to include orbital dependent many--body correlations in trial wave function of real--space Quantum Monte Carlo methods for use in electronic structure calculations. We apply our new orbital--dependent backflow wave function to calculate ground state energies of the first row atoms using variational and diffusion Monte Carlo methods. The systematic overall gain of correlation energy with respect to single determinant Jastrow-Slater wave functions is competitive with the best single determinant trial wave functions currently available. The computational cost per Monte Carlo step is comparable to that of simple backflow calculations.