Quantum search algorithms on the hypercube

TL;DR

This paper analyzes quantum search algorithms on the hypercube, demonstrating that a class of such algorithms can find marked vertices in roughly square root of the total vertices time, using symmetry and avoided crossings.

Contribution

It introduces a class of quantum search algorithms on the hypercube that operate efficiently via symmetry reduction and avoided crossings, improving understanding of quantum spatial search.

Findings

Search time scales as √N, where N=2^n.

A class of algorithms achieves optimal search times.

Estimates on quantum states at avoided crossings are provided.

Abstract

We investigate a set of discrete-time quantum search algorithms on the n-dimensional hypercube following a proposal by Shenvi, Kempe and Whaley. We show that there exists a whole class of quantum search algorithms in the symmetry reduced space which perform a search of a marked vertex in time of order $\sqrt{N}$ where $N = 2^n$, the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level sub-space of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the 2-level sub-spaces at the avoided crossings and derive improved estimates on the search times.