Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems

TL;DR

This paper introduces a quantum algorithm leveraging Hamiltonian simulation techniques to efficiently model both linear and nonlinear differential equations, including advection-diffusion and Koopman-von Neumann systems, with detailed implementation strategies.

Contribution

It presents a novel quantum algorithm that combines QSP and QSVT methods for simulating nonlinear dynamics and includes explicit block-encoding implementations for these models.

Findings

Simulated quantum circuits demonstrate feasibility on fault-tolerant computers.

Algorithm effectively models dissipation and avoids oscillations in advection equations.

Broad applicability to various linear and nonlinear differential equations.

Abstract

We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman-von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.