Measuring Distance between Quantum States by Fuzzy Similarity Operators

This paper introduces a study on fuzzy-based approaches aimed at addressing a crucial task in quantum computation: the evaluation of the similarity between quantum states. A quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement of a quantum algorithm. Because quantum computers are still characterized by high noise in computation, output quantum states generated by quantum algorithms could be very far to be close to the ideal output quantum state computed by a noiseless quantum computer. As a consequence, there is a strong emergence for measures capable of assessing the similarity level of two quantum states, one ideal and the other real, to infer the quality of a quantum device in performing precise calculations and design appropriate quantum error correction schemes. This research proves that fuzzy methods are fully suitable to face this crucial challenge in a pioneering scenario such as that of quantum computing, as proved by their application on well-known quantum algorithms, such as Bernstein-Vazirani and Grover's algorithm.