Constant-Time Quantum Search with a Many-Body Quantum System

TL;DR

This paper demonstrates that a many-body quantum system, such as a Bose-Einstein condensate with nonlinear interactions, can perform database searches in constant time by tuning system parameters, surpassing traditional quantum search limits.

Contribution

It introduces a method to achieve constant-time quantum search using a many-body system with nonlinear interactions, tuning parameters to widen success probability peaks and eliminate high-precision timing.

Findings

Constant-time search achieved with many-body quantum system.

Tuning nonlinear coefficients widens success probability peaks.

Derived lower bound on atom number for effective nonlinearity.

Abstract

The optimal runtime of a quantum computer searching a database is typically cited as the square root of the number of items in the database, which is famously achieved by Grover's algorithm. With parallel oracles, however, it is possible to search faster than this. We consider a many-body quantum system that naturally effects parallel queries, and we show that its parameters can be tuned to search a database in constant time, assuming a sufficient number of interacting particles. In particular, we consider Bose-Einstein condensates with pairwise and three-body interactions in the mean-field limit, which effectively evolve by a nonlinear Schr\"odinger equation with cubic and quintic nonlinearities. We solve the unstructured search problem formulated as a continuous-time quantum walk searching the complete graph in constant time. Depending on the number of marked vertices, however, the success probability can peak sharply, necessitating high precision time measurement to observe the system at this peak. Overcoming this, we prove that the relative coefficients of the cubic and quintic terms can be tuned to eliminate the need for high time-measurement precision by widening the peak in success probability or having it plateau. Finally, we derive a lower bound on the number of atoms needed for the many-body system to evolve by the effective nonlinearity.