A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment

We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$, obtain a target vector $\textbf{\emph{x}}(t)$ as a function of time $t$ according to the constraint $d\textbf{\emph{x}}(t)/dt=\mathcal{M}\textbf{\emph{x}}(t)+\textbf{\emph{b}}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a $4\times4$ linear differential equation using a 4-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.