Prospects for the application of mean-field nonlinear quantum solvers to computational fluid dynamics problems
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Due to the fundamentally linear nature of quantum computers, nonlinear data processing, in particular the integration of differential equations, has proven challenging. Over the past years, various algorithmic concepts have been developed that can largely be assigned to one of the three categories: hybrid quantum-classical solvers, linear embedding solvers and mean-field solvers. Hybrid solvers have already been demonstrated on simple models of flows (e.g. Burger’s equation, MFE model), yet the full extend of their quantum advantage remains open. Linear embedding solvers (e.g. Carleman) have exact mathematical scaling bounds but can not yet be implemented on current quantum hardware. Mean-field solvers are based on symmetric interactions between a number of identical quantum register copies. For certain nonlinear systems, the initial form of the solver based on quantum matrix inversion offers a quantum advantage. An amended time-marching version of the solver has been demonstrated on low dimensional models. Yet, a full analysis of the resource requirements for the application of mean-field solvers to fluid systems has not been carried out. In this talk, we shall present such initial resource estimates. We will consider explicit forms of the mean-field solver required to integrate fluid systems and find that even-order nonlinearities pose a significant challenge. A number of work-arounds are presented and the suitability of different types of quantum hardware will be discussed. Generally, platforms with inherently exchange-symmetric quantum states and large connectivity (e.g. photon or trapped-ion quantum computers) are found to be more suitable than other platforms. We conclude that mean-field solvers can in principle be applied to computational fluid dynamics problems but are likely to require quantum hardware resources not available in the near-term future.
