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One of the distinct features of quantum mechanics is that the probabilityamplitude can have both positive and negative signs, which has no classicalcounterpart as the classical probability must be positive. Consequently, onepossible way to achieve quantum speedup is to explicitly harness this feature.Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes(with respect to the computational basis), a non-stoquastic Hamiltonian can beeventually stoquastic or properly non-stoquastic when its ground state has bothpositive and negative amplitudes. In this paper, we describe that, for somehard instances which are characterized by the presence of an anti-crossing (AC)in a transverse-field quantum annealing (QA) algorithm, how to design anappropriate XX-driver graph (without knowing the prior problem structure) withan appropriate XX-coupler strength such that the resulting non-stoquastic QAalgorithm is proper-non-stoquastic with two bridged anti-crossings (adouble-AC) where the spectral gap between the first and second level is largeenough such that the system can be operated diabatically in polynomial time.The speedup is exponential in the original AC-distance, which can besub-exponential or exponential in the system size, over the stoquastic QAalgorithm, and possibly the same order of speedup over the state-of-the-artclassical algorithms in optimization. This work is developed based on the novelcharacterizations of a modified and generalized parametrization definition ofan anti-crossing in the context of quantum optimization annealing introduced in[4].

Essentiality of the Non-stoquastic Hamiltonians and Driver Graph Design in Quantum Optimization Annealing