The geometric phase stands as a foundational concept in quantum physics, revealing deep connections between geometric structures and quantum dynamical evolution. Unlike dynamical phases, geometric phases exhibit intrinsic resilience to certain types of perturbation, making them particularly valuable for quantum information processing, where maintaining coherent quantum operations is essential. This article provides a review of geometric phases in the context of universal quantum gate construction, i.e., the geometric quantum computation (GQC), with special attention to recent progress in nonadiabatic implementations that enhance gate fidelity and/or operational robustness. We first review a unified theoretical framework that can encompass all existing nonadiabatic GQC approaches, then systematically examine the design principles of nonadiabatic geometric gates with a particular focus on how optimal control techniques can be leveraged to improve the accuracy and noise resistance. In addition, we conducted detailed numerical comparisons of various nonadiabatic GQC protocols, offering a quantitative assessment of their respective performance characteristics and practical limitations. Through this focused investigation, our aim is to provide researchers with both fundamental insights and practical guidance for advancing geometric approaches in quantum computing.
In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of 10 − 10 10^{-10} and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics, and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.
Geometric and topological methods play an increasingly important role in quantum information science and quantum computation. Beyond the conventional Hilbert space formalism, a variety of mathematical frameworks, including group representations, mapping class groups, modular tensor categories, and character varieties, have been proposed to describe quantum states and quantum gates in a structurally robust manner. This review surveys the development of topological and geometric approaches to quantum information, with particular emphasis on representations of fundamental groups into SL(2,C), their associated character varieties, and the algebraic surfaces arising from trace coordinates, such as Fricke and Cayley cubic surfaces. These structures provide a geometric encoding of quantum degrees of freedom and offer alternative perspectives on topological quantum computing beyond anyon-based models. We also examine connections with integrable systems and isomonodromic deformations, where Painlevé equations and monodromy data supply a dynamical viewpoint on quantum state evolution. A critical comparison is provided with other geometric and topological approaches to quantum information, including geometric quantum mechanics, information geometry, tensor network geometry, and category-theoretic formulations. By synthesising results from topology, algebraic geometry, and mathematical physics, this review aims to clarify the conceptual landscape of topological quantum information geometry and to identify open problems and emerging directions in the field.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimental proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed. The notion of geometric phases is extended to cover mixed states undergoing unitary evolutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase concepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps. The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approximation. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation. The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field. We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated.
Quantum Information Geometry (QIG) proposes a novel framework integrating quantum information theory with geometric analysis, positioning quantum states as fundamental building blocks for understanding quantum entanglement and quantum computation. This paper explores a new approach to leveraging the geometric properties of quantum states to unlock deeper insights into the complexities of quantum mechanics. We introduce a conceptual model centered around the concept of "geometric entanglement," where entanglement isn't just a correlation but a geometric phenomenon inherent in the quantum state. The core mechanism involves developing quantum geometry algorithms to analyze and manipulate these geometric properties, ultimately aiming to provide a more intuitive and mathematically rigorous understanding of quantum information. The paper then highlights the significance of this approach, emphasizing its potential to revolutionize quantum algorithm design and the exploration of quantum information's underlying structure.