Completeness is Unnecessary for Fast Nonlinear Quantum Search

TL;DR

The paper demonstrates that nonlinear quantum effects can accelerate search algorithms on graphs that are not fully complete, maintaining quantum speedup even with some structural imperfections.

Contribution

It shows that nonlinearities in quantum mechanics enable faster search on sufficiently complete graphs, extending quantum speedup beyond ideal complete graphs.

Findings

Nonlinearities can speed up quantum search on sufficiently complete graphs.

Quantum speedup persists despite noncompleteness of the graph.

Speedup depends on the form of nonlinearity and graph structure.

Abstract

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 $\Theta(\sqrt{N})$ time as on the complete graph, the latter of which is precisely Grover's algorithm. We show that physically realistic nonlinearities of the form $f(|\psi|^2)\psi$ 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.