Quantum Optimisation for Continuous Multivariable Functions by a Structured Search

Solving optimisation problems is a promising near-term application of quantum computers. Quantum variational algorithms leverage quantum superposition and entanglement to optimise over exponentially large solution spaces using an alternating sequence of classically tunable unitaries. However, prior work has primarily addressed discrete optimisation problems. In addition, these algorithms have been designed generally under the assumption of an unstructured solution space, which constrains their speedup to the theoretical limits for the unstructured Grover's quantum search algorithm. In this paper, we show that quantum variational algorithms can efficiently optimise continuous multivariable functions by exploiting general structural properties of a discretised continuous solution space with a convergence that exceeds the limits of an unstructured quantum search. We introduce the Quantum Multivariable Optimisation Algorithm (QMOA) and demonstrate its advantage over pre-existing methods, particularly when optimising high-dimensional and oscillatory functions.