Could we exploit quantum Boltzmann machines for physics-constrained nonlinear regression?

The lattice Boltzmann method (LBM) for weakly-compressible flows simulates flow fields by performing two distinct operations at every time step: collision and streaming. Without forcibly assuming its linearity, the collision step amounts to evaluating a multivariate polynomial nonlinear function. An efficient quantum implementation of the nonlinear collision continues to elude researchers. In this work, we train a variational quantum Boltzmann machine (QBM) to this end. This is a generative quantum machine learning (QML) method that learns to reproduce the underlying probability distribution of a dataset. Physical properties of the nonlinear collision step are exploited in using a quantum Boltzmann machine to constrain the process. We obtain a good training accuracy of O($10^{-3}$) and a Kullback-Leibler divergence of O($10^{-2}$). The present variational approach is amenable to near-term (NISQ) applications, utilising solely 8 qubits to perform the collision step for a D2Q9-type LBM. D2Q9 refers to a two-dimensional lattice with 9 discrete velocities.