Hypercube Quantum Search: Exact Computation of the Probability of Success in Polynomial Time

TL;DR

This paper analyzes quantum search algorithms on hypercube structures, providing an exact, polynomial-time method to compute success probabilities by examining eigenspaces of walk operators.

Contribution

It introduces a detailed analysis of quantum search on hypercubes, enabling exact success probability computation in polynomial time through eigenspace analysis.

Findings

Success probability evolution can be predicted exactly in polynomial time.

The search operates within a small, linearly growing subspace of the Hilbert space.

Quantum walks on hypercubes can be effectively analyzed using eigenspace restrictions.

Abstract

In the emerging domain of quantum algorithms, the Grover's quantum search is certainly one of the most significant. It is relatively simple, performs a useful task and more importantly, does it in an optimal way. However, due to the success of quantum walks in the field, it is logical to study quantum search variants over several kind of walks. In this paper, we propose an in-depth study of the quantum search over a hypercube layout. First, through the analysis of elementary walk operators restricted to suitable eigenspaces, we show that the acting component of the search algorithm takes place in a small subspace of the Hilbert workspace that grows linearly with the problem size. Subsequently, we exploit this property to predict the exact evolution of the probability of success of the quantum search in polynomial time.