Dividing quantum circuits for time evolution of stochastic processes by orthogonal series density estimation

Quantum Monte Carlo integration (QMCI) is a quantum algorithm to estimate expectations of random variables, with applications in various industrial fields such as financial derivative pricing. When QMCI is applied to expectations concerning a stochastic process X(t), e.g., an underlying asset price in derivative pricing, the quantum circuit U_X(t) to generate the quantum state encoding the probability density of X(t) can have a large depth. With time discretized into N points, using state preparation oracles for the transition probabilities of X(t), the state preparation for X(t) results in a depth of O(N), which may be problematic for large N. Moreover, if we estimate expectations concerning X(t) at N time points, the total query complexity scales on N as O\(N²\), which is worse than the O(N) complexity in the classical Monte Carlo method. In this paper, to improve this, we propose a method to divide U_X(t) based on orthogonal series density estimation. This approach involves approximating the densities of X(t) at N time points with orthogonal series, where the coefficients are estimated as expectations of the orthogonal functions by QMCI. By using these approximated densities, we can estimate expectations concerning X(t) by QMCI without requiring deep circuits. Our error and complexity analysis shows that to obtain the approximated densities at N time points, our method achieves the circuit depth and total query complexity scaling as O\(sqrt{N}\) and O\(N^3/2\), respectively.