Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

Mazzola, Guglielmo; Smelyanskiy, Vadim; Troyer, Matthias

doi:10.1103/physrevb.96.134305

Cited by 34 publications

(34 citation statements)

References 71 publications

(78 reference statements)

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“…We find that the DMC tunneling time grows proportionally to the inverse of the gap 1/∆ when the system size increases. This behavior is analogous to what was previously found [21,23] The analysis of possible systematic biases of the DMC algorithm, in particular the one due to the finite randomwalker population (see section IV), shows that the maximum relative error in the prediction of the ground-state energy increases with the system size. The convergence to the exact infinite random-walker number limit becomes slower as the system size increases, and the number of random walkers required to maintain a fixed relative error increases asymptotically exponentially with the system size.…”

supporting

confidence: 83%

“…[22]). This 1/∆ 2 scaling was found to hold in ferromagnetic quan-tum Ising models [21], which are characterized by an effective double-well energy landscape (the two symmetric minima are the ground states with opposite magnetizations), and also in one-dimensional and two-dimensional continuous-space double-well models relevant for quantum chemistry applications [23]. Remarkably, this is the same scaling of the time of incoherent quantum tunneling in symmetric double-well models [24].…”

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confidence: 79%

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Understanding quantum tunneling using diffusion Monte Carlo simulations

et al. 2018

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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...

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confidence: 79%

Understanding quantum tunneling using diffusion Monte Carlo simulations

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“…It can be shown [36] that the equilibrium probability distribution of this equation is the desired joint probability distribution p(y) in Eq. (13). The stochastic process corresponding to this equation can be implemented with the following steps:…”

Section: Importance Sampling Guided By Unrestricted Neural Netwomentioning

confidence: 99%

Projective quantum Monte Carlo simulations guided by unrestricted neural network states

et al. 2018

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We investigate the use of variational wave-functions that mimic stochastic recurrent neural networks, specifically, unrestricted Boltzmann machines, as guiding functions in projective quantum Monte Carlo (PQMC) simulations of quantum spin models. As a preliminary step, we investigate the accuracy of such unrestricted neural network states as variational Ansätze for the ground state of the ferromagnetic quantum Ising chain. We find that by optimizing just three variational parameters, independently on the system size, accurate ground-state energies are obtained, comparable to those previously obtained using restricted Boltzmann machines with few variational parameters per spin. Chiefly, we show that if one uses optimized unrestricted neural network states as guiding functions for importance sampling the efficiency of the PQMC algorithms is greatly enhanced, drastically reducing the most relevant systematic bias, namely that due to the finite random-walker population. The scaling of the computational cost with the system size changes from the exponential scaling characteristic of PQMC simulations performed without importance sampling, to a polynomial scaling, even at the ferromagnetic quantum critical point. The important role of the protocol chosen to sample hidden-spins configurations, in particular at the critical point, is analyzed. We discuss the implications of these findings for what concerns the problem of simulating adiabatic quantum optimization using stochastic algorithms on classical computers.

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“…Refs. [21,22,25] explained these results using a semiclassical theory of instantons in PIMC simulations. In the case of PQMC algorithms, the tunneling rate was found to scale linearly with the gap, providing a quadratic speedup compared to the expected behavior of a quantum annealer [26] [27].…”

Section: Introductionmentioning

confidence: 78%

“…We measure ξ with a protocol analogous to the one adopted in Refs. [21,22] for PIMC simulations. All walkers are initially set at the bottom of the left well x = x L .…”

Section: B Tunneling Time In Diffusion Monte Carlo Simulationsmentioning

confidence: 99%

Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

et al. 2019

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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.

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Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

Cited by 34 publications

References 71 publications

Understanding quantum tunneling using diffusion Monte Carlo simulations

Understanding quantum tunneling using diffusion Monte Carlo simulations

Projective quantum Monte Carlo simulations guided by unrestricted neural network states

Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

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