An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. Higher-order quantum operations\unicodex2014quantum operations that transform quantum operations\unicodex2014are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
Quantum computing represents a transformative paradigm that exploits the principles of quantum mechanics—superposition, entanglement, and quantum interference—to perform computations beyond the capabilities of classical computers. This review presents a comprehensive overview of the fundamental concepts underlying quantum computation, including qubits, quantum gates, quantum algorithms and measurement theory. It further examines major physical implementations of quantum computers, such as superconducting circuits, trapped ions, photonic systems, spin-based qubits, and topological approaches, highlighting their operational principles, advantages, and technological limitations. Key challenges hindering large-scale quantum computation—decoherence, error rates, qubit scalability, and fault tolerance—are critically discussed. Additionally, recent progress in quantum error correction, hybrid quantum–classical algorithms, and hardware optimization is summarized. The review concludes by outlining future research directions and potential applications of quantum computing in cryptography, materials science, optimization, and drug discovery. By integrating theoretical foundations with experimental advancements, this article aims to provide a clear and accessible reference for students, researchers, and scientists entering the rapidly evolving field of quantum computing.
Quantum convolutional neural networks (QCNNs) are a highly appealing architecture that combines quantum computing and deep learning. Inspired by classical convolutional neural network (CNN) hierarchical feature extraction, QCNNs use quantum operations such as entanglement, superposition, and measurement to capture complex correlations in high-dimensional datasets. Since their introduction, QCNNs have evolved into several architectural variants, including fully quantum, variational, hybrid, and graph-based models, and have been investigated in applications ranging from quantum many-body physics to classical machine learning tasks such as image classification, speech recognition, and time-series forecasting. The development of software ecosystems such as Qiskit Machine Learning, Pennylane, and TensorFlow Quantum (TFQ) has facilitated rapid development and experimentation, speeding up research. Despite these developments, existing research remains divided, with many focused on individual implementations and lacking a unifying taxonomy or full review. This article addresses that gap by providing a systematic and holistic survey on QCNN, offering comparative insights across key architectures, applications, and toolboxes, and outlining open challenges. We also identify potential study areas, such as scalable architectures, domain-knowledge integration, fault tolerance, and security. This survey seeks to serve as a basic reference for furthering QCNN research in the current near-term quantum era and beyond, providing both architectural insights and application-driven viewpoints.
Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.
Quantum computing represents a paradigm shift in computational capabilities, promising to solve problems that are intractable for classical computers. This paper explores the architecture, implementation, and operational aspects of quantum computing, focusing on its applications in various fields. I conducted a comprehensive literature review to identify existing research gaps and propose methodologies for effective quantum computing implementation. My findings highlight the potential of quantum computing in optimization, cryptography, and machine learning, among other areas. This research paper provides a comprehensive examination of the architecture, implementation, and operational strategies of quantum computing, with a focus on its diverse applications in optimization, cryptography, and machine learning. Through an extensive literature review, I identify critical research gaps and propose innovative methodologies for effective quantum computing implementation. I conclude with strategic recommendations for future research directions, emphasizing the importance of enhancing quantum hardware, developing hybrid algorithms, and fostering interdisciplinary collaborations. This foundational study not only deepens the understanding of quantum computing’s current state but also paves the way for future advancements in this rapidly evolving field. The insights gained from the literature review, methodology, and results sections contribute to a deeper understanding of the current state of quantum computing and its potential applications.
This paper presents a comprehensive systematic and bibliometric review of the rapidly expanding literature on quantum computing. The study addresses six research questions concerning prominent themes, temporal evolution, types of key findings, frequently reported research gaps, higher-order gap typologies, and future research directions. Utilizing a researcher-curated database, the final corpus comprises 153 studies compiled from non-Scopus, Scopus and Web of Science. The results demonstrate that quantum hardware and architecture, algorithms, and theoretical foundations dominate the current literature. Furthermore, publication output has risen significantly between 2018 and 2025. Among the reported findings, algorithmic innovations, performance claims, and hardware advancements are the most common. Despite this progress, the field faces substantial barriers; technical, scalability, fault-tolerance, and methodological-standardization gaps are the most frequently reported. Based on these identified gaps, eight actionable future research directions emerge to guide the scientific community. Ultimately, this review concludes that quantum computing is actively transitioning from a purely physics-driven domain into a multidisciplinary, engineering- and application-oriented science. However, its continued advancement is presently constrained by a hardware-methodology-application triad, wherein reporting practices often outpace empirical verification and standardized benchmarking.