A quantum spectral method for simulating stochastic processes, with applications to Monte Carlo
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for Monte Carlo estimation combine a quantum simulation of a stochastic process with amplitude estimation to improve mean estimation. In this work we study quantum algorithms for simulating stochastic processes which are compatible with Monte Carlo methods. We introduce a new ``analog'' quantum representation of stochastic processes, in which the value of the process at time t is stored in the amplitude of the quantum state, enabling an exponentially efficient encoding of process trajectories. We show that this representation allows for highly efficient quantum algorithms for simulating certain stochastic processes, using spectral properties of these processes combined with the quantum Fourier transform. In particular, we show that we can simulate $T$ timesteps of fractional Brownian motion using a quantum circuit with gate complexity $\text{polylog}(T)$, which coherently prepares the superposition over Brownian paths. We then show this can be combined with quantum mean estimation to create end to end algorithms for estimating certain time averages over processes in time $O(\text{polylog}(T)\epsilon^{-c})$ where $3/2<c<2$ for certain variants of fractional Brownian motion, whereas classical Monte Carlo runs in time $O(T\epsilon^{-2})$ and quantum mean estimation in time $O(T\epsilon^{-1})$. Along the way we give an efficient algorithm to coherently load a quantum state with Gaussian amplitudes of differing variances, which may be of independent interest.