The quadratic speedup in Grover's search algorithm from the entanglement perspective

TL;DR

This paper investigates how entanglement dynamics underpin Grover's quadratic speedup, showing that entanglement change directly correlates with the algorithm's efficiency and establishing its optimality from an entanglement perspective.

Contribution

It provides a novel entanglement-based framework to understand Grover's speedup and offers a necessary and sufficient condition linking entanglement change to algorithmic efficiency.

Findings

Entanglement change governs the evolution towards the target state.

A necessary and sufficient condition for quadratic speedup based on bipartite entanglement.

Reestablishes Grover's optimality from an entanglement perspective.

Abstract

We analyze the role played by entanglement in the dynamical evolution of Grover's search algorithm in the space of qubits. We show that the algorithm can be equivalently described as an iterative change of the entanglement between the qubits, which governs the evolution of the initial state towards the target state, and where the entanglement can be quantified in terms a single entanglement monotone. We also provide a necessary and sufficient condition for the quadratic speedup, which illustrates how the change in the bipartite entanglement of the state after each iteration determines the corresponding increase in the probability of finding the target state. This allows us to reestablish from the entanglement perspective that Grover's search algorithm is the only optimal pure state search algorithm.