Methods and systems are disclosed for solving a differential problem using a
hybrid computer
system. The differential problem defines a
differential equation, and a target function defines an approximate solution to the
differential equation. The target function is defined as a sum of basis functions with parameterized coefficients. The method comprises receiving or determining a
quantum circuit defining operations of a parameterized
quantum model of the
differential equation, and determining the solution to the differential problem based on the parameterized
quantum model and, optionally, a
loss function associated with the differential problem. The
quantum circuit defines operations for encoding primitive elements (e.g., terms such as f(x), f(x) / dx, g(x), etc.) of the differential equation in a plurality of quantum states of quantum elements of the quantum processor. Each
quantum state of the plurality of quantum states corresponds to a set of (parametrized or unparametrized) coefficients associated with the basis functions, of which at least one
quantum state of the plurality of quantum states corresponds to a set of parametrized coefficients associated with the basis functions. The determination of the solution includes varying the variational parameters to determine a set of optimal parameters. The varying of the variational parameters includes: translating the operations of the
quantum circuit into control signals for control and readout of the quantum elements of the quantum register; applying the operations of the
quantum circuit to the quantum elements based on the control signals; determining classical measurement data by measuring one or more comparisons, e.g. overlaps, between pairs of quantum states from the plurality of quantum states; computing a loss value based on the
loss function and the classical measurement data; and adjusting the variational parameters based on the loss value.