Abstract
Stochastic differential equations (SDEs), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state while the previous studies did the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speed-up for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.
Highlights
Stochastic differential equations (SDEs), which describe the time evolution of random variables, are among the most important mathematical tools for modeling uncertain systems in diverse fields, such as finance [1], physics [2], and biology [3]
This paper proposed a quantum-classical hybrid algorithm that simulates SDEs based on variational quantum simulation (VQS)
A continuous stochastic process was discretized in a trinomial tree model and was reformulated as a linear differential equation
Summary
Stochastic differential equations (SDEs), which describe the time evolution of random variables, are among the most important mathematical tools for modeling uncertain systems in diverse fields, such as finance [1], physics [2], and biology [3]. To solve a SDE with quantum algorithms, we apply a tree model approximation [25] and obtain a linear differential equation describing the probability distribution of SDE solutions. This differential equation is solved by a variational quantum simulation (VQS) [26,27,28,29,30]. Appendix C evaluates the error of expectation values from piecewise polynomial approximation
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