Simulating continuous quantum systems by mean field fluctuations

TL;DR

This paper demonstrates how finite qubit systems' mean field fluctuations can accurately simulate continuous quantum systems, with convergence results surpassing previous Gaussian or polarized state methods, especially for quadratic Hamiltonian dynamics.

Contribution

It introduces a convergence framework showing fluctuation operators of finite qubits approximate continuous quantum variables, extending simulation capabilities beyond Gaussian or polarized state restrictions.

Findings

Fluctuation operators converge to continuous variables in expectation values.

Approximate time evolution of continuous systems is achievable with finite qubit fluctuations.

Results hold for small times under quadratic Hamiltonian dynamics.

Abstract

In this paper we are discussing the question how a continuous quantum system can be simulated by mean field fluctuations of a finite number of qubits. On the kinematical side this leads to a convergence result which states that appropriately chosen fluctuation operators converge in a certain weak sense (i.e. we are comparing expectation values) to canonical position and momentum Q, P of one-degree of freedom, continuous quantum system. This result is substantially stronger than existing methods which rely either on central limit theorem arguments (and are therefore restricted to the Gaussian world) or are valid only if the states of the ensembles are close to the "fully polarized" state. Dynamically this relationship keeps perfectly intact (at least for small times) as long as the continuous system evolves according to a quadratic Hamiltonian. In other words we can approximate the corresponding (Heisenberg picture) time evolution of the canonical operators Q, P up to arbitrary accuracy by the appropriately chosen time evolution of fluctuation operators of the finite systems.