Quantum Walk Search through Potential Barriers

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.

13 pages, 7 figures

References in corpus (7)

• Exponential algorithmic speedup by quantum walk
• Spatial search by quantum walk
• Universal computation by multi-particle quantum walk
• Quantum algorithms for hidden nonlinear structures
• Laplacian versus Adjacency Matrix in Quantum Walk Search
• Quantum Search with Multiple Walk Steps per Oracle Query
• Quantum Walk on the Line through Potential Barriers

Cited by in corpus (4)

• On the optimality of spatial search by continuous-time quantum walk
• Equivalence of Szegedy's and Coined Quantum Walks
• Continuous-time quantum walks on planar lattices and the role of the magnetic field
• Nonlinear flip-flop quantum walks through potential barriers