Entanglement of Periodic States, the Quantum Fourier Transform and Shor's Factoring Algorithm

TL;DR

This paper investigates the entanglement properties of periodic states in Shor's algorithm, revealing that the quantum Fourier transform has minimal impact on their entanglement, which is crucial for quantum speedup.

Contribution

It derives a formula to evaluate the Groverian entanglement measure for periodic states and explains the limited effect of the quantum Fourier transform on their entanglement.

Findings

Groverian entanglement of periodic states is only slightly affected by the quantum Fourier transform.

The derived formula enables precise evaluation of entanglement in these states.

Entanglement analysis clarifies its role in quantum algorithm efficiency.

Abstract

The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.