Coherence and decoherence in generalized and noisy Shor's algorithm

TL;DR

This paper investigates the roles of coherence and decoherence in generalized and noisy versions of Shor's quantum algorithm, deriving bounds and relations for their performance.

Contribution

It introduces bounds on the performance of generalized Shor's algorithm and explores the impact of noise and initial states on its success probability.

Findings

Derived bounds on algorithm performance with arbitrary initial states.

Established relations between success probabilities in different initializations.

Analyzed the effects of noise on coherence and algorithm success.

Abstract

Quantum coherence constitutes a fundamental physical mechanism essential to the study of quantum algorithms. We study the coherence and decoherence in generalized Shor's algorithm where the register $A$ is initialized in arbitrary pure state, or the combined register $AB$ is initialized in any pseudo-pure state, which encompasses the standard Shor's algorithm as a special case. We derive both the lower and upper bounds on the performance of the generalized Shor's algorithm, and establish the relation between the probability of calculating $r$ when the register $AB$ is initialized in any pseudo-pure state and the one when the register $A$ initialized in arbitrary pure state. Moreover, we study the coherence and decoherence in noisy Shor's algorithm and give the lower bound of the probability that we can calculate $r$.