Quantum Walk Search through Potential Barriers

TL;DR

This paper analyzes how potential energy barriers affect quantum walk algorithms, showing that the success of quantum search depends on the tunneling fidelity, which must scale appropriately with database size to maintain quantum speedup.

Contribution

It proves that potential barriers must be sufficiently small for quantum walk search algorithms to retain their quadratic speedup, applicable to both discrete and continuous-time models.

Findings

Failure amplitude must scale as O(1/√N) for quantum speedup

Higher failure probability degrades search to classical performance

Results apply to both discrete- and continuous-time quantum walks

Abstract

An ideal quantum walk transitions from one vertex to another with perfect fidelity, but in physical systems, the particle may be hindered by potential energy barriers. Then the particle has some amplitude of tunneling through the barriers, and some amplitude of staying put. We investigate the algorithmic consequence of such barriers for the quantum walk formulation of Grover's algorithm. We prove that the failure amplitude must scale as $O(1/\sqrt{N})$ for search to retain its quantum $O(\sqrt{N})$ runtime; otherwise, it searches in classical $O(N)$ time. Thus searching larger "databases" requires increasingly reliable hop operations or error correction. This condition holds for both discrete- and continuous-time quantum walks.