Variational quantum algorithms for PDE measurement extraction

Classical-quantum hybrid algorithms have garnered attention for their ability to combine classical simulations of minimization with quantum simulation of cost functions, which are broadly characterized by various amalgamations of quantum and classical computing protocols for obtaining readout of solutions. In particular, one recent VQA due to Lubasch et al in 2019 exhibits a theoretical and empirically focused study for obtaining solutions to the Schrodinger and Inviscid Burgers equations, through optimizing for the ground state of cost functions parametrized in each PDE that is being solved. With approximations of nonlinearities that are encoded with expectation values of the cost function, the authors report of high fidelity as their computational quantum-driven methods are capable of reliably predicting classically determined solutions. To determine whether the VQA is suitable for more complicated nonlinearities of other PDEs, in addition to corresponding solutions that are posed with different initial conditions, we discuss investigations of other cost functions that can be clasically minimized. For each nonlinear problem of interest, the quantum state encoded by the cost function is representative of new physical phenomena that can be studied from time-evolving quantum states that are represented with sequences of unitary gates acting on qubits initialized within quantum circuit registers. To avoid encountering barren plateaus and inescapable minima in the optimization landscape that have been throughouly characterized when time-evolving quantum representations of solutions by other areas of the quantum computing community, we form mathematically precise characterizations of the performance of the algorithm for other initial value problems, through simulation of several nonlinear behaviors. To study other nonlinearities, we formulate cost functions dependent upon time-evolved variational states of solutions, and form comparisons with clasically determined solutions.