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Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully…
Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge…
We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic…
The application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry (quantum gravity). The geometry also plays an…
I review recent works showing that information geometry is a useful framework to characterize quantum coherence and entanglement. Quantum systems exhibit peculiar properties which cannot be justified by classical physics, e.g. quantum…
Due to its unique computing principles, quantum computing technology will profoundly change the spectacle of color art. Focusing on experimental exploration of color qubit representation, color channel processing, and color image generation…
In this paper, we use a topological quantum field theory (TQFT) to define families of new homology theories of a $2$-dimensional CW complex of a smooth closed surface. The dimensions of these homology groups can be used to count the number…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being…
In the lectures we will be concerned with some aspects of physical implementations of quantum gate operations which are necessary for quantum information processing. We will discuss two possible realizations. One of them is based on qubits…
Here, we reveal our recent progress on a geometrical approach of quantum physics and topological crystals linking with Dirac magnetic monopoles and gauge fields through classical electrodynamics. The Bloch sphere of a quantum spin-1/2…
Recent hardware demonstrations and advances in circuit compilation have made quantum computing with higher-dimensional systems (qudits) on near-term devices an attractive possibility. Some problems have more natural or optimal encodings…
This paper introduces an efficient quantum computing method for reducing special graphs in the context of the graph coloring problem. The special graphs considered include both symmetric and non-symmetric graphs where the axis passes…
Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical…
The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter $\hbar$ to produce a new (classical)…
Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-$D$…
Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by…
We summarize our work on spherically symmetric midi-superspaces in loop quantum gravity. Our approach is based on using inhomogeneous slicings that may penetrate the horizon in case there is one and on a redefinition of the constraints so…