Abstract The SWAP test is a fundamental tool commonly used in the theory of quantum information and computation while its understanding and applications have been investigated in the literature. This survey focuses on the SWAP test in the studies of quantum complexity theory. We review the basics of the SWAP test, some tests related to it, and their properties, and report their applications in quantum complexity theory.
Abstract The SWAP test is a fundamental tool commonly used in the theory of quantum information and computation while its understanding and applications have been investigated in the literature. This survey focuses on the SWAP test in the studies of quantum complexity theory. We review the basics of the SWAP test, some tests related to it, and their properties, and report their applications in quantum complexity theory.
This is a review of quantum methods for machine learning problems that consists of two parts. The first part, “quantum tools”, presents the fundamentals of qubits, quantum registers, and quantum states, introduces important quantum tools based on known quantum search algorithms and SWAP-test, and discusses the basic quantum procedures used for quantum search methods. The second part, “quantum classification algorithms”, introduces several classification problems that can be accelerated by using quantum subroutines and discusses the quantum methods used for classification.
We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
Abstract Quantum memory devices are essential components of integrated quantum-photonic circuits with applications in quantum computing and quantum information processing. The comprehension and quantification of critical properties such as fidelity, coherence time, storage time, and efficiency of quantum memory are insufficient in the extant literature. In this review, we examine the development and characteristics of various types of quantum memories. Although diverse approaches to storing quantum information are described in detail, we emphasize solid-state quantum memory. Furthermore, we discuss the fabrication and characterization of defects of quantum materials, including two-dimensional materials, for quantum memory applications. We survey theoretical tools such as density functional theory for studying defects to evaluate quantum properties.
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.
Abstract Quantum entanglement is essential to the development of quantum computation, communications, and technology. The controlled SWAP test, widely used for state comparison, can be adapted to an efficient and useful test for entanglement of a pure state. Here we show that the test can evidence the presence of entanglement (and further, genuine n-qubit entanglement), can distinguish entanglement classes, and that the concurrence of a two-qubit state is related to the test’s output probabilities. We also propose a multipartite measure of entanglement that acts similarly for n-qubit states. The average number of copies of the test state required to detect entanglement decreases for larger systems, to four on average for many (n ≳ 8) qubits for maximally entangled states. For non-maximally entangled states, the number of copies required to detect entanglement increases with decreasing entanglement. Furthermore, the results are robust to second order when typical small errors are introduced to the state under investigation.
The second quantum revolution has been built on a foundation of fundamental research at the intersection of physics and information science, giving rise to the discipline we now call quantum information science (QIS).The quest for new knowledge and understanding drove the development of new experimental tools and rigorous theory, which defined the road map for second-wave quantum technologies, including quantum computers, quantum-enhanced sensors, and communication systems.As technology has matured, the race to develop and commercialize near-term applications has accelerated.In the current regime of Noisy Intermediate Scale Quantum (NISQ) devices [J.Preskill, Quantum 2, 79 ( 2018)], the continued necessity of basic research is manifest.Under what conditions can we truly harness quantum complexity and what are its implications for potential useful applications?These questions remain largely unanswered and as the QIS industry ramps up, a continuous feedback between basic science and technology is essential.In this Perspective, I review how curiosity-driven research has led to radical new technologies and why the quest for basic understanding is essential for further progress.