Implementing the Koopman-von Neumann approach on continuous-variable photonic quantum computers

TL;DR

This paper explores implementing the Koopman-von Neumann formalism on continuous-variable photonic quantum computers to simulate classical nonlinear dynamics efficiently using quantum algorithms.

Contribution

It demonstrates the feasibility of applying KvN formalism on photonic quantum architectures for simulating classical systems.

Findings

Successful implementation of KvN on photonic quantum hardware.

Simulation of harmonic oscillator dynamics using quantum algorithms.

Potential for efficient classical-to-quantum mappings in nonlinear dynamics.

Abstract

The Koopman-von Neumann (KvN) formalism recasts classical mechanics in a Hilbert space framework using complex wavefunctions and linear operators, akin to quantum mechanics. Instead of evolving probability densities in phase space (as in Liouville's equation), KvN uses a Schr\"odinger-like equation for a classical wavefunction, with commuting position and momentum operators. Mapped to quantum computing, KvN offers a promising route to simulate classical dynamical systems using quantum algorithms by leveraging unitary evolution and quantum linear algebra tools, potentially enabling efficient classical-to-quantum mappings without invoking full quantum uncertainty. In this work, we specifically explore the implementation of the KvN approach on continuous-variable photonic quantum computing architectures, with the goals of leveraging quantum simulation for both sampling and computing intractable nonlinear dynamics. We will demonstrate its implementation and feasibility with two problems: the harmonic oscillator and a 1D partial differential equation governing nonlinear dynamics.