Nonlinear Quantum Search

TL;DR

This paper explores how nonlinear quantum equations like the Gross-Pitaevskii equation affect quantum search algorithms, revealing potential speedups and resource bounds in systems with effective nonlinear dynamics.

Contribution

It demonstrates that nonlinear effective equations can enable faster quantum search algorithms and establishes information-theoretic bounds for these nonlinear quantum systems.

Findings

Nonlinear equations can solve unstructured search in constant time.

Optimizing measurement precision improves overall search scaling to N^{1/4}.

Certain nonlinear systems and graph structures retain quantum speedup benefits.

Abstract

Although quant mech is linear, there are nevertheless quant sys with multiple interacting particles in which the effective evo of a single particle is governed by a nonlinear eq. This includes Bose-Einstein condensates, which are gov by the Gross-Pitaevskii eq (GPE), which is a cubic nonlin Schrodinger eq (NLSE) with a term propto $|\psi|^2\psi$. Evo by this eq solves the unstruct search prob in const time, but at the novel expense of increasing the time-measurement precision. Jointly optimizing these resources results in an overall scaling of $N^{1/4}$, which is a significant, but not unreasonable, improvement over the $N^{1/2}$ scaling of Grover's algo. Since the GPE effectively approx the multi-particle Schrodinger eq, for which Grover's algo is optimal, our result leads to a quant info-theoretic bound on the num of particles needed for this approx to hold, asymp. The GPE is not the only nonlin of the form $f(|\psi|^2)\psi$ that arises in effective eqs for the evo of real quant phys sys, however: The cubic-quintic NLSE describes light propagation in nonlin Kerr media with defocusing corrections, and the log NLSE describes Bose liquids under certain cond. Analysis of comput with such sys yields some surprising results; e.g., when time-measurement precision is included in the resource accounting, searching a "database" when there is a single correct ans may be easier than searching when there are multiple correct ans. These results lead to quant info-theoretic bounds on the phys resources required for these effective nonlin theories to hold, asymp. Further, strongly reg graphs, which have no global sym, are sufficiently complete for quant search on them to asymp behave like unstruct search. Certain suff complete graphs retain the improved runtime and resource scalings for some nonlin, so our scheme for nonlin, analog quant comput retains its benefits even when some struct is introduced.