Quantum Monte Carlo simulations of tunneling in quantum adiabatic optimization

Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semiclassical potential arising from the spin-coherent path-integral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoided-level crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spin-vector dynamics, is at least as efficient as diabatic QA. Finally, in a different example with a convex cost function, the diabatic transitions result in a speedup relative to both adiabatic QA with tunneling and classical spin-vector dynamics.

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“…This was studied in Ref. [19], where evidence for the following conjecture for the scaling of the quantum minimum gap g min was presented:…”

Section: Perturbed Hamming Weight Optimization Problems and The Ementioning

confidence: 99%

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

2016

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“…Tunneling events occur also during QMC simulations, similarly to what happens during the quantum annealers' dynamics [18,19]. In various problems characterized by a double-well energy landscape, the tunneling rate of finite-temperature PIMC simulations was found to scale with the system size, or with the height of the energy barrier, as the square of the first energy gap [20][21][22]. This is the same scaling predicted by the theory of incoherent quantum tunneling [23], and it is also the scaling of the inverse of the annealing time required by a coherent quantum annealer to avoid diabatic transitions [24].…”

Section: Introductionmentioning

confidence: 91%

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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“…The quantum algorithm can approach a constant runtime independent of the number of qubits [34,35], while the simulated annealing runtime grows exponentially. It has been shown that the dynamics of spike-like problems can be effectively captured (at least in terms of the separation between exponential and polynomial scaling) by path integral quantum Monte Carlo, a classical simulation algorithm inspired by quantum physics [36][37][38]. While these results make spike problems less interesting from a computational perspective, they still contain interesting many body physics, and may still provide useful tests of how faithfully the underlying quantum dynamics is reproduced in an experimental system.…”

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confidence: 99%

Practical designs for permutation-symmetric problem Hamiltonians on hypercubes

et al. 2019

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We present a method to experimentally realize large-scale permutation-symmetric Hamiltonians for continuous-time quantum protocols such as quantum walk and adiabatic quantum computation. In particular, the method can be used to perform an encoded continuous-time quantum search on a hypercube graph with 2 n vertices encoded into 2n qubits. We provide details for a realistically achievable implementation in Rydberg atomic systems. Although the method is perturbative, the realization is always achieved at second order in perturbation theory, regardless of the size of the mapped system. This highly efficient mapping provides a natural set of problems which are tractable both numerically and analytically, thereby providing a powerful tool for benchmarking quantum hardware and experimentally investigating the physics of continuous-time quantum protocols.Quantum computing based on continuous time evolution rather than discrete gate operations offers a promising route for practical near-term quantum computing. This approach has a wide variety of natural applications including in finance [1-3], aerospace [4], machine learning [5][6][7], theoretical computer science [8], mathematics [9,10], decoding of communications [11] and computational biology [12]. Moreover, experimental quantum annealing has proven highly successful recently [13][14][15][16][17].While continuous-time quantum computing shows great promise, there are few known methods to experimentally implement test problems that can be used to prove the performance of hardware. For quantum computing based on discrete gates, solving unstructured search by Grover's algorithm [18] provides a quadratic speedup over any classical algorithm-the best possible speedup as proven by Bennett et al. [19]. There are continuous-time variants of quantum search algorithms which can obtain the same optimal speedup for both adiabatic quantum computation [20,21] and continuoustime quantum walk [22]. It has recently been shown that these are the two extremes of a continuum of protocols that all achieve the optimal quantum speedup [23].Continuous-time search algorithms are not easy to experimentally implement when encoded into qubits. In contrast, Grover's original algorithm can be efficiently decomposed into quantum gates [24]. A naive decomposition of the continuous-time search problem yields exponentially many terms coupling all possible subsets of qubits. To date, the largest qubit-encoded continuous-time quantum walks have been performed on two qubits [25,26]; neither implemented a search algorithm. Larger encoded discrete-time quantum walks and quantum searches have been experimentally realized [27,28], and alternative encodings have been explored in [29,30].Because of the difficulty of implementing qubitencoded continuous-time quantum search algorithms, this has been considered a toy problem: useful as a theoretical tool, but not practical experimentally. The search Hamiltonian can always be represented in a permutation symmetric basis, by transforming the marked state to either t...

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Quantum Monte Carlo simulations of tunneling in quantum adiabatic optimization

Cited by 31 publications

References 15 publications

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

Practical designs for permutation-symmetric problem Hamiltonians on hypercubes

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