Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps

TL;DR

This paper introduces a quantum algorithm for solving multidimensional transport equations that significantly reduces the number of required time-steps, enabling more efficient quantum simulations of physical phenomena.

Contribution

It presents a novel Trotter-based quantum scheme with exponential reduction in time-steps and provides theoretical analysis, efficient circuits, and experimental validation.

Findings

Exponential reduction in time-steps compared to previous methods

Successful implementation on real quantum hardware for 1D convection

Effective simulation of nonlinear ODEs via Liouville equation

Abstract

The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits compared to previously established operator-norm analysis, thereby significantly lowering the projected computational resources. We also present efficient quantum circuits and numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.