Introduction of Quantum Annealing and Ideas for Improving its Performance

Shu Tanaka

Tadashi Kadowaki

We propose a method to interpolate dynamics of von Neumann and classical master equations with an arbitrary mixing parameter to investigate the thermal effects in quantum dynamics. The two dynamics are mixed by intervening to continuously modify their solutions, thus coupling them indirectly instead of directly introducing a coupling term. This maintains the quantum system in a pure state even after the introduction of thermal effects and obtains not only a density matrix but also a state vector representation. Further, we demonstrate that the dynamics of a two-level system can be rewritten as a set of standard differential equations, resulting in quantum dynamics that includes thermal relaxation. These equations are equivalent to the optical Bloch equations at the weak coupling and asymptotic limits, implying that the dynamics cause thermal effects naturally. Numerical simulations of ferromagnetic and frustrated systems support this idea. Finally, we use this method to study thermal effects in quantum annealing, revealing nontrivial performance improvements for a spin glass model over a certain range of annealing time. This result may enable us to optimize the annealing time of real annealing machines.

Tadashi Kadowaki, Hidetoshi Nishimori

A shorter processing time is desirable for quantum computation to minimize the effects of noise. We propose a simple procedure to variationally determine a set of parameters in the transverse-field Ising model for quantum annealing (QA) appended with a field along the [Formula: see text]-axis. The method consists of greedy optimization of the signs of coefficients of the [Formula: see text]-field term based on the outputs of short annealing processes. We test the idea in the ferromagnetic system with all-to-all couplings and spin-glass problems, and find that the method outperforms the traditional form of QA and simulated annealing in terms of the success probability and the time to solution, in particular, in the case of shorter annealing times, achieving the goal of improved performance while avoiding noise. The non-stoquastic [Formula: see text] term can be eliminated by a rotation in the spin space, resulting in a non-trivial diabatic control of the coefficients in the stoquastic transverse-field Ising model, which may be feasible for experimental realization. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

秀稔 西森, Hidetoshi Nishimori

Shu Tanaka

Yoshiki Matsuda, Hidetoshi Nishimori, Helmut G. Katzgraber

We study the performance of quantum annealing for systems with ground-state degeneracy by directly solving the Schrödinger equation for small systems and quantum Monte Carlo simulations for larger systems. The results indicate that naive quantum annealing using a transverse field may not be well suited to identify all degenerate ground-state configurations, although the value of the ground-state energy is often efficiently estimated. An introduction of quantum transitions to all states with equal weights is shown to greatly improve the situation, but with a sacrifice in annealing time. We also clarify the relation between the spin configurations in degenerate ground states and the probabilities that those states are obtained by quantum annealing. The strengths and weaknesses of quantum annealing for problems with degenerate ground states are discussed in comparison with classical simulated annealing.