  A variational quantum algorithm for numerically solving partial differential
equations (PDEs) on a quantum computer was proposed by Lubasch et al. In this
paper, we generalize the method introduced by Lubasch et al. to cover a broader
class of nonlinear PDEs as well as multidimensional PDEs, and study the
performance of the variational quantum algorithm on several example equations.
Specifically, we show via numerical simulations that the algorithm can solve
instances of the Single-Asset Black-Scholes equation with a nontrivial
nonlinear volatility model, the Double-Asset Black-Scholes equation, the
Buckmaster equation, and the deterministic Kardar-Parisi-Zhang equation. Our
simulations used up to $n=12$ ansatz qubits, computing PDE solutions with $2^n$
grid points. We also performed proof-of-concept experiments with a trapped-ion
quantum processor from IonQ, showing accurate computation of two representative
expectation values needed for the calculation of a single timestep of the
nonlinear Black--Scholes equation. Through our classical simulations and
experiments on quantum hardware, we have identified -- and we discuss --
several open challenges for using quantum variational methods to solve PDEs in
a regime with a large number ($\gg 2^{20}$) of grid points, but also a
practical number of gates per circuit and circuit shots.
