Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schr\"odingerization

TL;DR

This paper introduces a quantum algorithm called Nonlinear Schr"odingerization for simulating complex classical oscillator systems, including nonlinear and time-dependent cases, by reducing them to linear Schr"odinger equations in higher dimensions.

Contribution

It extends quantum simulation techniques to nonlinear and non-conservative classical systems using a novel reduction to linear Schr"odinger equations.

Findings

Polynomial complexity in the number of oscillators

Almost linear in evolution time for most systems

Applicable to systems with time-dependent and nonlinear interactions

Abstract

We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems containing $2^n$ coupled oscillators (Eg: $2^n$ masses coupled by springs), including those with time-dependent forces, time-varying stiffness matrices, and weak nonlinear interactions. This generalization of the Harmonic oscillator simulation algorithm is achieved through an approach that we call ``Nonlinear Schr\"{o}dingerization'', which involves reduction of the dynamical system to a nonlinear Schr\"{o}dinger equation and then reduced to a time-independent Schrodinger Equation through perturbative techniques. The linearization of the equation is performed using an approach that allows the dynamics of a nonlinear Schr\"odinger equation to be approximated as a linear Schr\"odinger equation in a higher dimensional space. This allows Hamiltonian Simulation algorithms to be applied to simulate the dynamics of resulting system. When the properties of the classical dynamical systems can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in $n$, and almost linear in evolution time for most dynamical systems. Our work extends the applicability of quantum algorithms to simulate the dynamics of non-conservative and nonlinear classical systems, addressing key limitations in previous approaches.