Formulation and evaluation of ocean dynamics problems as optimization problems for quantum annealing machines

Recent advancements in quantum computing suggest the potential to revolutionize computational algorithms across various scientific domains including oceanography and atmospheric science. The field is still relatively young and quantum computation is so different from classical computation that suitable frameworks to represent oceanic and atmospheric dynamics are yet to be explored. Quantum annealing (QA), one of the major paradigms, focuses on combinatorial optimization tasks. Given its potential to excel in NP-hard problems, QA may significantly accelerate the calculation of ocean and atmospheric systems described by the Navier-Stokes equations in the future. In this paper, we apply both QA and simulated annealing (SA), its classical counterpart, to a simplified ocean model known as the Stommel problem. We use the Stommel problem, which is not an NP problem and therefore does not benefit from QA today, just as an example, a first step in exploring QA for more intricate problems governed by the Navier-Stokes equations. We cast the linear partial differential equation governing the Stommel model into an optimization problem by the least-squares method and discretize the cost function in two ways: finite difference and truncated basis expansion. In either case, SA successfully reproduces the expected solution when appropriate parameters are chosen. In contrast, QA using the D-Wave quantum annealing machine fails to obtain good solutions for some cases owing to hardware limitations; in particular, the highly limited connectivity graph of the machine limits the size of the solvable problems, at least under currently available algorithms. Either expanding the machine's connectivity graph or improving the graph-embedding algorithms would probably be necessary for quantum annealing machines to be usable for oceanic and atmospheric dynamics problems.