Necessary condition for the quantum adiabatic approximation

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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“…[47]). -To adiabatically prepare a final eigenstate using a Hamiltonian evolution HðsÞ requires time that scales at least as OðL=ΔÞ.…”

Section: A Numerical Argumentmentioning

confidence: 99%

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

2016

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“…It asserts that, if the rate of change of the Hamiltonian scales with the energy gap ∆ between their two lowest-energy states, |ψ can be prepared with controlled accuracy [6,7]. Such an approximation may also be necessary [8]. However, it could result in undesired overheads if ∆ is small but transitions between the lowestenergy states are forbidden due to selection rules, or if transitions between lowest-energy states can be exploited to prepare |ψ .…”

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Quantum Speedup by Quantum Annealing

2012

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We study the glued-trees problem of Childs, et. al. [1] in the adiabatic model of quantum computing and provide an annealing schedule to solve an oracular problem exponentially faster than classically possible. The Hamiltonians involved in the quantum annealing do not suffer from the socalled sign problem. Unlike the typical scenario, our schedule is efficient even though the minimum energy gap of the Hamiltonians is exponentially small in the problem size. We discuss generalizations based on initial-state randomization to avoid some slowdowns in adiabatic quantum computing due to small gaps.PACS numbers: 03.67. Ac, 03.67.Lx, 42.50.Lc Quantum annealing is a powerful heuristic to solve problems in optimization [2,3]. In quantum computing, the method consists of preparing a low-energy or ground state |ψ of a quantum system such that, after a simple measurement, the optimal solution is obtained with large probability. |ψ is prepared by following a particular annealing schedule, with a parametrized Hamiltonian path subject to initial and final conditions. A ground state of the initial Hamiltonian is then transformed to |ψ by varying the parameter adiabatically. In contrast to more general quantum adiabatic state transformations, the Hamiltonians along the path in quantum annealing are termed stoquastic and do not suffer from the so-called numerical sign problem [4]: for a specified basis, the offdiagonal Hamiltonian-matrix entries are nonpositive [5]. This property is useful for classical simulations [3].A sufficient condition for convergence of the quantum method is given by the quantum adiabatic approximation. It asserts that, if the rate of change of the Hamiltonian scales with the energy gap ∆ between their two lowest-energy states, |ψ can be prepared with controlled accuracy [6,7]. Such an approximation may also be necessary [8]. However, it could result in undesired overheads if ∆ is small but transitions between the lowestenergy states are forbidden due to selection rules, or if transitions between lowest-energy states can be exploited to prepare |ψ . The latter case corresponds to the annealing schedule in this Letter. It turns out that the relevant energy gap for the adiabatic approximation in these cases is not ∆ and can be much bigger.Because of the properties of the Hamiltonians, the annealing can also be simulated using probabilistic classical methods such as quantum Monte-Carlo (QMC) [9]. The goal in QMC is to sample according to the distribution of the ground state, i.e. with probabilities coming from amplitudes squared. While we lack of necessary conditions that guarantee convergence, the power of QMC is widely recognized [3,9,10]. In fact, if the Hamiltonians satisfy an additional frustration-free property, efficient QMC simulations for quantum annealing exist [11,12]. This places a doubt on whether a quantum-computer simulation of general quantum annealing processes can ever be done using substantially less resources than QMC or any other classical simulation.Towards answering this question, we...

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“…If the change is sufficiently slow and there is no environment, the adiabatic theorem of quantum mechanics predicts that the system will remain in its ground state, and an optimal solution is obtained 12,13 . Realistically, one should include the effects of coupling to a thermal environment, that is, consider open system quantum adiabatic evolution [14][15][16][17][18][19] .…”

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Experimental signature of programmable quantum annealing

et al. 2013

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Quantum annealing is a general strategy for solving difficult optimization problems with the aid of quantum adiabatic evolution. Both analytical and numerical evidence suggests that under idealized, closed system conditions, quantum annealing can outperform classical thermalization-based algorithms such as simulated annealing. Current engineered quantum annealing devices have a decoherence timescale which is orders of magnitude shorter than the adiabatic evolution time. Do they effectively perform classical thermalization when coupled to a decohering thermal environment? Here we present an experimental signature which is consistent with quantum annealing, and at the same time inconsistent with classical thermalization. Our experiment uses groups of eight superconducting flux qubits with programmable spin-spin couplings, embedded on a commercially available chip with 4100 functional qubits. This suggests that programmable quantum devices, scalable with current superconducting technology, implement quantum annealing with a surprising robustness against noise and imperfections.

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Necessary condition for the quantum adiabatic approximation

Cited by 44 publications

References 30 publications

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems

Quantum Speedup by Quantum Annealing

Experimental signature of programmable quantum annealing

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