Non-stoquastic Hamiltonians in quantum annealing via geometric phases

• Published in: npj quantum information 2017-09, Vol.3 (1), p.1-6, Article 38
• Main Authors: Vinci, Walter, Lidar, Daniel A.
• Format: Article
• Language: English
• Subjects: 639/766/483/2802; 639/766/483/3926; Adiabatic; Algorithms; Annealing; Classical and Quantum Gravitation; Computer applications; Physics; Physics and Astronomy; Quantum Computing; Quantum Field Theories; Quantum Information Technology; Quantum Physics; Relativity Theory; Spintronics; String Theory
• ISSN: 2056-6387; 2056-6387
• DOI: 10.1038/s41534-017-0037-z

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. Quantum annealing: engineering non-stoquastic interactions via geometric phases Quantum annealing is a promising approach to quantum computation that is particularly suited to solve optimization and sampling problems. Researchers from University of Southern California show that the low-energy effective Hamiltonian describing the quantum anneal of a system implemented with continuous variables includes terms of geometric origin. The inclusion of such geometric effects makes the effective Hamiltonian non-stoquastic. Such Hamiltonians cannot be simulated with Monte Carlo algorithms due to the existence of a sign problem. The implementation of quantum annealing with non-stoquastic Hamiltonians is thus particularly important from a computational complexity perspective. The direct implementation of non-stoquastic interactions is challenging. Their results suggest an alternative path for the implementation of non-stoquasticity via geometric phases that can be exploited for computational purposes.