Compact quantum algorithms for time-dependent differential equations

Abstract: 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.