Compact quantum algorithms for time-dependent differential equations

Many claims of computational advantages have been made for quantum computing over classical, but they have not been demonstrated for practical problems. Here, we present algorithms for solving time-dependent PDEs, with particular reference to fluid equations. We build on an idea based on linear combination of unitaries to simulate non-unitary, non-Hermitian quantum systems, and generate hybrid quantum-classical algorithms that efficiently perform iterative matrix-vector multiplication and matrix inversion operations. These algorithms are end-to-end, with relatively low-depth quantum circuits that demonstrate quantum advantage, with the best-case asymptotic complexities, which we show are near-optimal. We demonstrate the performance of the algorithms by conducting: (a) fully gate level, state-vector simulations using an in-house, high performance, quantum simulator called QFlowS; (b) experiments on a real quantum device; and (c) noisy simulations using Qiskit Aer. We also provide device specifications such as error-rates (noise) and state sampling (measurement) to accurately perform convergent flow simulations on noisy devices. The results offer evidence that the proposed algorithm is amenable for use on near-term quantum devices.