Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

TL;DR

This paper proposes a method to embed nonlinear dynamical systems into linear systems for potential quantum computing simulation, enabling more efficient approximation of complex systems with fewer qubits.

Contribution

It introduces a novel embedding technique for nonlinear systems into linear systems and analyzes its potential for efficient quantum simulation.

Findings

Quantum simulation of nonlinear systems is feasible with logarithmic qubits.

Embedding methods can approximate nonlinear dynamics with finite linear systems.

Efficiency depends on the strength of nonlinearity and truncation accuracy.

Abstract

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously-studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.