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

AbstractVariational 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.Keywords: Variational quantum algorithmsVariational quantum Monte CarloMulti-asset Black-Scholes PDE AcknowledgmentsThe Authors thank the anonymous AE and the referees for their suggestions, which helped to improve our paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 VQAs for mult-asset financial derivative pricing have been subsequently explored in Kubo et al. (Citation2022).2 See appendix 1 for a derivation. Similiar ODEs have been introduced in the neural Galerkin method (Bruna et al. Citation2022).3 This can be considered as a special case of the complex-valued case (Hibat-Allah et al. Citation2020, Sharir et al. Citation2020).Additional informationFundingAuthors gratefully acknowledge support from NSF under grants DMS-2038030 and DMS-2006305. This research was supported in part through computational resources and services provided by the Advanced Research Computing (ARC) at the University of Michigan.