Non-stoquastic Hamiltonians in quantum annealing via geometric phases

Walter Vinci,Daniel A Lidar

doi:10.1038/s41534-017-0037-z

Abstract

We argue that a complete description of quantum annealing implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquasticity in the effective quantum Ising Hamiltonians that are typically used to describe quantum annealing with flux qubits. We explicitly demonstrate the effect of this geometric non-stoquasticity when quantum annealing is performed with a system of one and two coupled flux qubits. The realization of non-stoquastic Hamiltonians has important implications from a computational complexity perspective, since it is believed that in many cases quantum annealing with stoquastic Hamiltonians can be efficiently simulated via classical algorithms such as Quantum Monte Carlo. It is well known that the direct implementation of non-stoquastic Hamiltonians with flux qubits is particularly challenging. Our results suggest an alternative path for the implementation of non-stoquasticity via geometric phases that can be exploited for computational purposes.

Highlights

It is well known that the solution of computational problems can be encoded into the ground state of a time-dependent quantum Hamiltonian
A Hamiltonian H is stoquastic with respect to a given basis if H has only real nonpositive off-diagonal matrix elements in that basis. This means that its ground state can be expressed as a classical probability distribution,[14, 15] and that the standard quantum-to-classical mapping[16] used in quantum Monte Carlo algorithms does not result in a “sign problem”
In many cases ground-state stoquastic adiabatic quantum computation (AQC) can be efficiently simulated by classical algorithms, but certain exceptions are known.[23, 24]

Summary

INTRODUCTION

The geometric effect can be considered as a type of non-stoquastic catalyst,[5] with the potential to lead to quantum speedups.[25,26,27,28,29,30,31] From a more general experimental perspective, the presence of the geometric terms studied in this paper has clear signatures that must be taken into account in the validation and and tf denotes the final time. It follows directly from Eq (5b) that G(t)dt = G(s)ds (see SM, section B), which shows that.

RESULTS

DISCUSSION

METHODS