Quantum-enhanced analysis of discrete stochastic processes

TL;DR

This paper introduces a quantum algorithm for analyzing discrete stochastic processes that offers linear growth in circuit complexity, eliminates importance sampling, and provides quadratic speed-up, with applications demonstrated in finance and random walks.

Contribution

The paper presents a novel quantum algorithm for calculating the characteristic function of DSPs, reducing complexity and improving variance estimation over classical methods.

Findings

Quantum algorithm computes characteristic functions with linear circuit growth.

Eliminates the need for importance sampling in stochastic analysis.

Demonstrated applications in finance and correlated random walks.

Abstract

Discrete stochastic processes (DSP) are instrumental for modelling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm takes all stochastic trajectories into account and hence eliminates the need of importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. Both of these strategies improve variance beyond classical capabilities. The quantum method can be combined with Fourier approximation to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented to exemplify the usefulness of our results. Proof-of-principle experiments are performed using the IBM quantum cloud platform.