Interference versus success probability in quantum algorithms with imperfections

TL;DR

This paper investigates how errors and decoherence affect the performance and interference in quantum algorithms like Shor's and Grover's, revealing that interference is necessary but not sufficient for quantum advantage.

Contribution

It introduces a measure of interference to analyze the impact of various errors on quantum algorithms, highlighting the complex relationship between interference and algorithm efficiency.

Findings

Unitary errors destroying interference reduce algorithm efficiency.

Additional interference from errors can decrease performance.

Decoherence can convert constructive interference into destructive interference.

Abstract

We study the influence of errors and decoherence on both the performance of Shor's factoring algorithm and Grover's search algorithm, and on the amount of interference in these algorithms using a recently proposed interference measure. We consider systematic unitary errors, random unitary errors, and decoherence processes. We show that unitary errors which destroy the interference destroy the efficiency of the algorithm, too. However, unitary errors may also create useless additional interference. In such a case the total amount of interference can increase, while the efficiency of the quantum computation decreases. For decoherence due to phase flip errors, interference is destroyed for small error probabilities, and converted into destructive interference for error probabilities approaching one, leading to success probabilities which can even drop below the classical value. Our results show that in general interference is necessary in order for a quantum algorithm to outperform classical computation, but large amounts of interference are not sufficient and can even lead to destructive interference with worse than classical success rates.