  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.
