Quantum-inspired variational algorithms for partial differential equations: Application to financial derivative pricing

Abstract

Variational quantum Monte Carlo (VMC) combined with neural-network quantum states offers a novel angle of attack on the curse-of-dimensionality encountered in a particular class of partial differential equations (PDEs); namely, the real-and imaginary time-dependent Schrödinger equation. In this paper, we present a simple generalization of VMC applicable to arbitrary time-dependent PDEs, showcasing the technique in the multi-asset Black-Scholes PDE for pricing European options contingent on many correlated underlying assets.

Toward Neural Network Simulation of Variational Quantum Algorithms

“…The exploitation of flexible neural networks as many-body trial wavefunctions has made it possible to leverage the enormous success of machine learning (ML) in solving a variety of quantum many-body eigenvalue problems [3]. The close parallels between ideas developed in the very different fields of VQAs, VQMC, and ML has yielded many opportunities for technology transfer between the three, including the discovery of a VQMC-inspired natural gradient optimization algorithm for VQAs [14] and an expanding list of VQA-inspired algorithms for combinatorial optimization [6,19,8] and partial differential equations [22].…”