A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms

TL;DR

This paper introduces a quantum algorithm employing probabilistic imaginary-time evolution to efficiently solve advection-diffusion equations, demonstrating exponential speedup over classical methods and providing numerical validation and extensions.

Contribution

It presents a novel approximate PITE operator, explicit quantum circuit implementation with logarithmic gate complexity, and compares its performance to quantum linear system algorithms.

Findings

Achieves exponential speedup in matrix size over classical methods.

Provides numerical simulations validating the algorithm.

Extends to coupled advection-diffusion systems for practical use.

Abstract

Simulating differential equations on classical computers becomes an intractable problem if the grid size is extremely large. Quantum computers are believed to achieve a possibly exponential speedup in the matrix operation. In this paper, we propose a quantum algorithm for solving the advection-diffusion-reaction equation by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. First, the effectiveness of the proposed approximate PITE operator is justified by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit to realize the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. Compared to the existing algorithms for the quantum linear system problem, our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound. Moreover, numerical simulations using gate-based quantum emulator for 1D/2D examples are also provided to verify our algorithm. Finally, we extend our algorithm to the coupled system of advection-diffusion equations to show the prospects for practical applications.