Abstract
Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization. These results apply to any adiabatic algorithm which interpolates to a final Hamiltonian that is diagonal in the computational basis.
Highlights
Quantum adiabatic optimization (QAO) [16, 24, 28, 46, 61] is a quantum algorithm that uses adiabatic evolution in Hamiltonian ground states to solve classical combinatorial optimization problems
It has been observed that this stoquastic QAO can in many cases be efficiently classically simulated by quantum Monte Carlo (QMC) methods [40], meaning that QMC simulations are able to effectively follow the instantaneous ground state of the interpolating Hamiltonian with the same computational cost scaling as QAO. (QMC simulations can approximately sample equilibrium states in the computational basis and estimate the expectation values of local observables.) There are several rigorous results on polynomialtime QMC simulations for various classes of equilibrium states of stoquastic Hamiltonians [11,12,13, 19, 20, 32], there are counterexamples where QMC methods fail to converge efficiently [6, 33, 44]
2 Background and Overview In QAO, we assume that the global optima of discrete classical optimization problems are encoded in the ground states of a problem Hamiltonian Hp that is diagonal in the computational basis [24, 26, 34, 37, 71, 72]
Summary
Quantum adiabatic optimization (QAO) [16, 24, 28, 46, 61] is a quantum algorithm that uses adiabatic evolution in Hamiltonian ground states to solve classical combinatorial optimization problems. The standard proposal for QAO uses an adiabatic path defined by a one parameter family of Hamiltonians {H(s)}s∈[0,1] that are all stoquastic, meaning that H(s) has real and non-positive off-diagonal matrix elements in the computational basis for each s ∈ [0, 1] [11]. It has been observed that this stoquastic QAO can in many cases be efficiently classically simulated by quantum Monte Carlo (QMC) methods [40], meaning that QMC simulations are able to effectively follow the instantaneous ground state of the interpolating Hamiltonian with the same computational cost scaling as QAO. Adiabatic computation based on non-stoquastic Hamiltonians can be universal for quantum computing [1, 9] and is not efficiently simulable by QMC due to the sign problem [31, 52, 54, 67]. Spectral graph theory, and other analytical techniques we develop an explanation of this phenomenon based on the low-energy spectrum of stoquastic and non-stoquastic Hamiltonians
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