Mean Field Approximation for Identical Bosons on the Complete Graph

TL;DR

This paper investigates the validity of the mean field approximation for identical bosons on a complete graph, revealing conditions under which the approximation accurately describes the system's quantum states.

Contribution

It characterizes the states of bosons on a complete graph using Young diagrams and analyzes the conditions for the mean field approximation to hold.

Findings

Isolated particles are necessary for accurate mean field approximation.

Matrix fidelity agreement is limited to at most 50% without isolated particles.

States are parameterized by Young diagrams, enabling detailed analysis.

Abstract

Non-linear dynamics in the quantum random walk setting have been shown to enable conditional speedup of Grover's algorithm. We examine the mean field approximation required for the use of the Gross-Pitaevskii equation on identical bosons evolving on the complete graph. We show that the states of such systems are parameterized by the basis of Young diagrams and determine their one- and two-party marginals. We find that isolated particles are required for good agreement with the mean field approximation, proving that without isolated particles the matrix fidelity agreement is bounded from above by $1/2$.