Discrete stochastic processes (DSP) are instrumental for modeling 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 reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
Bibliographical noteFunding Information:
We acknowledge the use of IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team. We thank Philipp Leser, Mile Gu, and Jayne Thompson for fruitful discussions, and James Crutchfield for sharing references. This research is supported by the National Research Foundation of Korea (No. 2019R1I1A1A01050161 and 2021M3H3A1038085), Quantum Computing Development Program (No. 2019M3E4A1080227), and the South African Research Chair Initiative of the Department of Science and Technology and the National Research Foundation (UID: 64812).
© 2021, The Author(s).
All Science Journal Classification (ASJC) codes
- Computer Science (miscellaneous)
- Statistical and Nonlinear Physics
- Computer Networks and Communications
- Computational Theory and Mathematics