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Stochastic processes play a fundamental role in physics, mathematics,engineering and finance. One potential application of quantum computation is tobetter approximate properties of stochastic processes. For example, quantumalgorithms for Monte Carlo estimation combine a quantum simulation of astochastic process with amplitude estimation to improve mean estimation. Inthis work we study quantum algorithms for simulating stochastic processes whichare compatible with Monte Carlo methods. We introduce a new ``analog'' quantumrepresentation of stochastic processes, in which the value of the process attime t is stored in the amplitude of the quantum state, enabling anexponentially efficient encoding of process trajectories.  We show that this representation allows for highly efficient quantumalgorithms for simulating certain stochastic processes, using spectralproperties of these processes combined with the quantum Fourier transform.  In particular, we show that we can simulate $T$ timesteps of fractionalBrownian motion using a quantum circuit with gate complexity$\text{polylog}(T)$, which coherently prepares the superposition over Brownianpaths.  We then show this can be combined with quantum mean estimation to create endto 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 offractional 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 quantumstate with Gaussian amplitudes of differing variances, which may be ofindependent interest.

A quantum spectral method for simulating stochastic processes, with applications to Monte Carlo