Quantum algorithm for solving high-dimensional linear stochastic differential equations via amplitude encoding of the noise term

TL;DR

This paper introduces quantum algorithms that encode solutions to high-dimensional stochastic differential equations in amplitudes, leveraging quantum circuits for noise simulation to achieve potential speed-ups over classical methods.

Contribution

It develops two quantum methods, Dyson series and Euler-Maruyama, for amplitude encoding of SDE solutions, incorporating quantum noise simulation and efficient expectation estimation.

Findings

Uses quantum circuits for amplitude encoding of SDE solutions.

Achieves polylogarithmic query complexity in the dimension N.

Provides methods for expectation estimation of functions of the solution.

Abstract

This work studies quantum algorithms to solve high-dimensional stochastic differential equations (SDEs) $\mathrm{d} \mathbf{X}_t = A(t) \mathbf{X}_t \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$. Aiming for a speed-up in the dimension $N$ of $\mathbf{X}_t$, we generate quantum states that encode $\mathbf{X}_t$ in the amplitudes, while most of the existing quantum methods for SDEs employ binary encoding. A key challenge is the amplitude encoding of the noise term, and we address this by utilizing the quantum circuit implementation of a pseudorandom number generator (PRNG). We propose two methods: the Dyson series-based method and the Euler-Maruyama (EM)-based method. In the former, we express the noise term via the Dyson series approximation of the time evolution operator, while in the latter, it is approximated using the EM time discretization. Both methods use the quantum linear systems solver to generate the amplitude-encoding state of $\mathbf{X}_t$, making only ${\rm polylog}(N)$ queries to the PRNG circuit and the block-encodings of $A$ and $B$. Additionally, going beyond state preparation, we present methods to estimate expectations of functions of $\mathbf{X}_t$ using the state.