Related papers: Braids, Motions and Topological Quantum Computing
We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to…
Topological quantum computation started as a niche area of research aimed at employing particles with exotic statistics, called anyons, for performing quantum computation. Soon it evolved to include a wide variety of disciplines. Advances…
In this paper we present a hybrid scheme for topological quantum computation in a system of cold atoms trapped in an atomic lattice. A topological qubit subspace is defined using Majorana fermions which emerge in a network of atomic Kitaev…
We consider a two-dimensional spin system that exhibits abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically the model…
Despite the evident necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well…
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function.…
We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well…
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…
Braiding has attracted significant attention in physics because of its important role in describing the fundamental exchange of particles. Infusing the braiding with topological protection will make it robust against imperfections and…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…
While there is a general consensus about the structure of one qubit operations in topological quantum computer, two qubits are as usual a more difficult and complex story of different attempts with varying approaches, problems and…
This paper is an introduction to relationships between quantum topology and quantum computing. We take a foundational approach, showing how knots are related not just to braiding and quantum operators, but to quantum set theoretical…
We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by…
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of…
The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that Clifford algebras based on involutive braids can be naturally endowed with a braided quantum group…
Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding…
In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the…
We study the problem of universality in the anyon model described by the $SU(2)$ Witten-Chern-Simons theory at level $k$. A classic theorem of Freedman-Larsen-Wang states that for $k \geq 3, \ k \neq 4$, braiding of the anyons of…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For…