  To solve nonlinear partial differential equations (PDEs) is one of the most
common but important tasks in not only basic sciences but also many practical
industries. We here propose a quantum variational (QuVa) PDE solver with the
aid of machine learning (ML) schemes to synergise two emerging technologies in
mathematically hard problems. The core quantum processing in this solver is to
calculate efficiently the expectation value of specially designed quantum
operators. For a large quantum system, we only obtain data from measurements of
few control qubits to avoid the exponential cost in the measurements of the
whole quantum system and optimise a pathway to find possible solution sets of
the desired PDEs using ML techniques. As an example, a few different types of
the second-order DEs are examined with randomly chosen samples and a regression
method is implemented to chase the best candidates of solution functions with
another trial samples. We demonstrated that a three-qubit system successfully
follows the pattern of analytical solutions of three different DEs with high
fidelity since the variational solutions are given by a necessary condition to
obtain the exact solution of the DEs. Thus, we believe that final solution
candidate sets are efficiently extracted from the QuVa PDE solver with the
support of ML techniques and this algorithm could be beneficial to search for
the solutions of complex mathematical problems as well as to find good ansatzs
for eigenstates in large quantum systems (e.g., for quantum chemistry).

Quantum Physics