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We study the role of Hamiltonian complexity in the performance of quantumannealers. We consider two general classes of annealing Hamiltonians:stoquastic ones, which can be simulated efficiently using the quantum MonteCarlo algorithm, and nonstoquastic ones, which cannot be treated efficiently.We implement the latter by adding antiferromagnetically coupled two-spin driverterms to the traditionally studied transverse-field Ising model, and comparetheir performance to that of similar stoquastic Hamiltonians withferromagnetically coupled additional terms. We focus on a model of long-rangeIsing spin glass as our problem Hamiltonian and carry out the comparisonbetween the annealers by numerically calculating their success probabilities insolving random instances of the problem Hamiltonian in systems of up to 17spins. We find that, for a small percentage of mostly harder instances,nonstoquastic Hamiltonians greatly outperform their stoquastic counterparts andtheir superiority persists as the system size grows. We conjecture that theobserved improved performance is closely related to the frustrated nature ofnonstoquastic Hamiltonians.

Nonstoquastic Hamiltonians and Quantum Annealing of an Ising Spin Glass