Quantum Circuits for the Black-Scholes equations via Schr\"{o}dingerisation

TL;DR

This paper develops quantum circuits for solving the Black-Scholes equations in finance using Schr"odingerisation, enabling high-dimensional quantum simulations with potential computational advantages.

Contribution

It introduces a novel quantum algorithm that converts the Black-Scholes PDE into a unitary form via Schr"odingerisation, facilitating quantum simulation of financial models.

Findings

Quantum circuits successfully simulate Black-Scholes equations.

Complexity analysis shows potential quantum advantage.

Numerical experiments validate the approach.

Abstract

In this paper, we construct quantum circuits for the Black-Scholes equations, a cornerstone of financial modeling, based on a quantum algorithm that overcome the cure of high dimensionality. Our approach leverages the Schr\"odingerisation technique, which converts linear partial and ordinary differential equations with non-unitary dynamics into a system evolved by unitary dynamics. This is achieved through a warped phase transformation that lifts the problem into a higher-dimensional space, enabling the simulation of the Black-Scholes equation on a quantum computer. We will conduct a thorough complexity analysis to highlight the quantum advantages of our approach compared to existing algorithms. The effectiveness of our quantum circuit is substantiated through extensive numerical experiments.