Quantum Speedup by Quantum Annealing

Abstract

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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