  A black-box optimization algorithm such as Bayesian optimization finds
extremum of an unknown function by alternating inference of the underlying
function and optimization of an acquisition function. In a high-dimensional
space, such algorithms perform poorly due to the difficulty of acquisition
function optimization. Herein, we apply quantum annealing (QA) to overcome the
difficulty in the continuous black-box optimization. As QA specializes in
optimization of binary problems, a continuous vector has to be encoded to
binary, and the solution of QA has to be translated back. Our method has the
following three parts: 1) Random subspace coding based on axis-parallel
hyperrectangles from continuous vector to binary vector. 2) A quadratic
unconstrained binary optimization (QUBO) defined by acquisition function based
on nonnegative-weighted linear regression model which is solved by QA. 3) A
penalization scheme to ensure that the QA solution can be translated back. It
is shown in benchmark tests that its performance using D-Wave Advantage$^{\rm
TM}$ quantum annealer is competitive with a state-of-the-art method based on
the Gaussian process in high-dimensional problems. Our method may open up a new
possibility of quantum annealing and other QUBO solvers including quantum
approximate optimization algorithm (QAOA) using a gated-quantum computers, and
expand its range of application to continuous-valued problems.
