Understanding quantum tunneling using diffusion Monte Carlo simulations

Abstract

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/∆ 2 , where ∆ 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/∆, 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.Difficult optimization problems are ubiquitous in science and in engineering. Relevant examples are protein folding, the traveling salesman problem, and portfolio optimization. Such problems can often be formulated as the search of the lowest-energy spin configuration in an Ising glass [1], a task that has been proven to be NP-hard in the case of non-planar graphs [2]. While exact classical algorithms are believed to require computational times that exponentially grow with the problem size (unless P = NP), various heuristic methods can often provide quite accurate (but possibly not exact) solutions in a feasible time. Perhaps, the most versatile of such heuristic methods is simulated classical annealing (SCA) [3], which exploits thermal fluctuations in a Markov chain Monte Carlo simulation to escape local minima and, hopefully, find the lowest energy state at the end of the annealing process when the temperature has been reduced to zero.Also adiabatic quantum computers, such as the quantum annealers (QAs) built using superconducting flux qubits [4-6] -or, potentially, with Rydberg atoms trapped in arrays of optical tweezers [7] -can be used to solve complex combinatorial optimization problems. They implement a quantum annealing process [8][9][10], in which quantum mechanical tunneling through tall barriers is used to escape local minima, and quantum fluctuations are gradually removed by reducing to zero the transverse field of a quantum Ising model. While in problems with energy landscapes characterized by tall but thin barriers quantum tunneling definitely makes QAs more efficient than classical optimization methods such as SCA [11,12], cert...

Learnability scaling of quantum states: Restricted Boltzmann machines

“…For the maximum bond dimension that we em- ploy (2000), the expected Z 2 symmetry is not realized below certain values of the transverse field strength h when the number of qubits is large. Furthermore, a similar phenomenon has been observed previously in studies of the relative energy in diffusion Monte Carlo [15] and a recent variational imaginary time ansatz [19]. It would be an interesting topic of future study.…”

“…where σ x,y,z are Pauli operators, defined over N sites (or qubits), and ij denotes nearest-neighbor pairs on a onedimensional lattice with open boundary conditions. This model is thoroughly studied in the condensed matter and quantum information literature, and serves as a standard benchmark for many numerical methods, such as quantum Monte Carlo [14,15], Tensor Networks (TNs) [16], or more recent quantum optimization algorithms [17][18][19]. We generate training data from a density matrix renormalization group (DMRG) simulation [20] for various values of h/J using the ITensor library [21].…”

Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

“…This scaling relation has previously been identified in Ref. [26] in PQMC simulations of Ising-type models, again performed without a GWF. Next, we run DMC simulations with a GWF.…”

Rapid counter-diabatic sweeps in lattice gauge adiabatic quantum computing

“…We note that the improved ground state fidelity does not stem from an increase of the minimal energy gap between ground and first excited state and cannot be understood in incoherent quantum annealing or with path integral Monte Carlo methods [66][67][68]. The additional counter-diabatic term rather compensates for the Berry curvature which causes transitions between eigenstates [30,52,69].…”