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 $O(N^2)$, 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.