Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations

Abstract: 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.