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.
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.
Quantum computation has attracted numerous attentions and develops rapidly in the recent decades. To against the decoherence and the control errors upon the qubits, quantum error corrections are adopted. Such approaches require lots of redundant qubits, accurate measurement and timely feedback. Here we investigate a new framework of quantum computation that is associated with fuzzy processing. It will benefit significantly from three aspects: the fuzzy recognition of qubit states reduce the required gate fidelity; the fuzzy encoding encodes the information of the qubits into a distribution of probability, suppressing the fluctuations in the output of long quantum circuits; the fuzzy feedback offers a more efficient way to control the qubits when precision information of quantum states are absent. Furthermore, the fuzzy processing can be integrated into quantum error correction, eliminating the need for immediate correction operations. The proposed scheme will be fairly suitable for the solution of decision problems, which has significant applications in the optimization problems and control problems.
Among clustering algorithms, fuzzy clustering stands out for its ability to offer a nuanced representation of the data by assigning degrees of membership to clusters, providing a more flexible and adaptive approach than the rigid partitioning of hard clustering algorithms. This has proved highly advantageous, particularly for image segmentation problems. Numerous approaches have been proposed to improve the Fuzzy C-means (FCM) algorithm using quantum computing, some are quantum-inspired and others can be run on quantum simulators. In this paper, a study was conducted on Quantum Fuzzy Means (QFCM) approaches. Then, a novel QFCM algorithm is introduced to address the challenges associated with these current algorithms, particularly in handling large datasets and incorporating genuine fuzzy system principles. Using concepts from quantum computing, our approach aims to improve distance calculations between data points by using a quantum distance measure. This method enables significant acceleration of the clustering process especially when dealing with extensive datasets. Moreover, our proposed algorithm integrates a structured fuzzy system framework into the membership matrix calculation, enhancing the precision and interpretability of the clustering results. Furthermore, unlike other FCM algorithms, which often lack explicit representation of fuzzy logic principles, our approach incorporates a well-defined fuzzy system to capture the inherent uncertainty and ambiguity in real-world data.
Recently, a quantum algorithm called Quantum Fuzzy Inference Engine (QFIE) has been introduced with the main goal of providing exponential speedup in the execution of a Mamdani fuzzy inference engine. This quantum algorithm achieves this result by modeling a fuzzy rule base with a quantum oracle, a black box that is widely used to estimate functions using quantum mechanical principles. Although QFIE offers the possibility of performing efficient fuzzy computation on quantum computers, its real-world applicability is limited by the high levels of noise still present in current quantum computers. Consequently, it is necessary to introduce technological arrangements to make QFIE fully functional in practical cases until noise-free quantum computers are released. This paper addresses this issue by designing a distributed version of QFIE, based on the D-NISQ reference model, to distribute the computation of subsets of fuzzy rules across multiple quantum processors and minimize the negative impact of quantum noise. Experimental results prove that this distributed version of QFIE is able to significantly improve the accuracy of fuzzy computation on quantum devices, making QFIE applicable in real-world scenarios.
This review is designed to introduce mathematicians and computational scientists to quantum computing (QC) through the lens of uncertainty quantification (UQ) by presenting a mathematically rigorous and accessible narrative for understanding how noise and intrinsic randomness shape quantum computational outcomes in the language of mathematics. By grounding quantum computation in statistical inference, we highlight how mathematical tools such as probabilistic modeling, stochastic analysis, Bayesian inference, and sensitivity analysis, can directly address error propagation and reliability challenges in today's quantum devices. We also connect these methods to key scientific priorities in the field, including scalable uncertainty-aware algorithms and characterization of correlated errors. The purpose is to narrow the conceptual divide between applied mathematics, scientific computing and quantum information sciences, demonstrating how mathematically rooted UQ methodologies can guide validation, error mitigation, and principled algorithm design for emerging quantum technologies, in order to address challenges and opportunities present in modern-day quantum high performance and fault-tolerant quantum computing paradigms.
Recently, Quantum Computing is entered in the so-called Noisy Intermediate-Scale Quantum (NISQ) era, where devices characterized by a few number of qubits are potentially able to overcome classical computers in performing specific tasks. However, noise in quantum operators still limits the size of quantum circuits that can be run in a reliable way. Consequently, there is a strong need for error mitigation approaches aimed at increasing reliability in quantum computation and making this paradigm really useful and productive in real world applications. In this paper, a fuzzy method, such as Fuzzy C-Means (FCM) clustering, has been used, for the very first time, to support the identification of matrices for error mitigation in quantum measurement. As shown in experiments, mitigation matrices identified with the support of FCM are able to strongly reduce error in computation when compared to mitigation matrices conventionally identified, like those used by IBM in its quantum library named Qiskit.
From Pharmacology to Cryptography and from Geology to Astronomy are some of the scientific fields in which Quantum Computing potentially will take off and fly high. Big Quantum Computing vendors invest a large amount of money in improving the hardware and they claim that soon enough a quantum program will be hundreds of thousands of times faster than a typical one we know nowadays. But still the reliability of such systems is the main obstacle. In this work, the reliability of real quantum devices is tested and techniques of noise and error correction are presented while measurement error mitigation is explored. In addition, a well-known string matching algorithm (Bernstein-Vazirani) was applied to the real quantum computing device in order to measure its accuracy and reliability. Simulated environments were also used in order to evaluate the results. The results obtained, even if these were not 100% accurate, are very promising which proves that even these days a quantum computer working side by side with a typical one is reliable and especially when error mitigation techniques are applied.