Completeness is Unnecessary for Fast Nonlinear Quantum Search

Although strongly regular graphs and the hypercube are not complete, they are "sufficiently complete" such that a randomly walking quantum particle asymptotically searches on them in the same $Θ(\sqrt{N})$ time as on the complete graph, the latter of which is precisely Grover's algorithm. We show that physically realistic nonlinearities of the form $f(|ψ|^2)ψ$ can speed up search on sufficiently complete graphs, depending on the nonlinearity and graph. Thus nonlinear (quantum) computation can retain its power even when a degree of noncompleteness is introduced.

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