No Infinite Tail Beats Optimal Spatial Search

TL;DR

This paper demonstrates that the optimal spatial quantum search algorithm remains effective even when an infinite tail is attached, showing robustness and obliviousness to the presence of such an external system.

Contribution

It extends the known optimality of spatial search algorithms to scenarios with an infinite tail, highlighting their robustness and obliviousness to external infinite systems.

Findings

Spatial search remains optimal with an infinite tail attached.

The algorithm is oblivious to the presence or position of the tail.

Spatial search is robust against a one-dimensional infinite probe.

Abstract

Farhi and Gutmann (Physical Review A, 57(4):2403, 1998) proved that a continuous-time analogue of Grover search (also called spatial search) is optimal on the complete graphs. We extend this result by showing that spatial search remains optimal in a complete graph even in the presence of an infinitely long path (or tail). If we view the latter as an external quantum system that has a limited but nontrivial interaction with our finite quantum system, this suggests that spatial search is robust against a coherent infinite one-dimensional probe. Moreover, we show that the search algorithm is oblivious in that it does not need to know whether the tail is present or not, and if so, where it is attached to.