  The application of quantum annealing to the optimization of
continuous-variable functions is a relatively unexplored area of research. We
test the performance of quantum annealing applied to a one-dimensional
continuous-variable function with a rugged energy landscape. After domain-wall
encoding to map a continuous variable to discrete Ising variables, we first
benchmark the performance of the real hardware, the D-Wave 2000Q, against
several state-of-the-art classical optimization algorithms designed for
continuous-variable problems to find that the D-Wave 2000Q matches the
classical algorithms in a limited domain of computation time. Beyond this
domain, classical global optimization algorithms outperform the quantum device.
Next, we examine several optimization algorithms that are applicable to the
Ising formulation of the problem, such as the TEBD (time-evolving block
decimation) to simulate ideal coherent quantum annealing, simulated annealing,
simulated quantum annealing, and spin-vector Monte Carlo. The data show that
TEBD's coherent quantum annealing achieves far better results than the other
approaches, demonstrating the effectiveness of coherent tunneling. From these
two types of benchmarks, we conclude that the hardware realization of quantum
annealing has the potential to significantly outperform the best classical
algorithms if thermal noise and other imperfections are sufficiently suppressed
and the device operates coherently, as demonstrated in recent short-time
quantum simulations.
