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Non-linear dynamics in the quantum random walk setting have been shown toenable conditional speedup of Grover's algorithm. We examine the mean fieldapproximation required for the use of the Gross-Pitaevskii equation onidentical bosons evolving on the complete graph. We show that the states ofsuch systems are parameterized by the basis of Young diagrams and determinetheir one- and two-party marginals. We find that isolated particles arerequired for good agreement with the mean field approximation, proving thatwithout isolated particles the matrix fidelity agreement is bounded from aboveby $1/2$.

Mean Field Approximation for Identical Bosons on the Complete Graph