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

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.