Related papers: On quantum computation, anyons, and categories
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a…
Formalized $1$-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," rely on…
We compare two possible ways of defining a category of 1-combs, the first intensionally as coend optics and the second extensionally as a quotient by the operational behaviour of 1-combs on lower-order maps. We show that there is a full and…
Quantum theory is a well-defined local theory with a clear interpretation. No "measurement problem" or any other foundational matters are waiting to be settled.
This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented…
We establish and advocate for a novel branch of category theory, centered around strong dinatural transformations (herein known as "paranatural transformations"). Paranatural transformations generalize natural transformations to…
Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant,…
Using von Neumann algebras, we extend the theory of quantum computation on a graph to a theory of computation on an arbitrary topological space.
Anyons in one spatial dimension can be defined by correctly identifying the configuration space of indistinguishable particles and imposing Robin boundary conditions. This allows an interpolation between the bosonic and fermionic limits. In…
Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction…
The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among…
We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with…
Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those…
We make use of a higher version of the Yoneda embedding to construct, from a given quasicategory, a tribe, as a subcategory of a well-behaved simplicial model category, that presents the same $(\infty,1)$-category as the former…
A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories…
Quantum theory provides an extremely accurate description of fundamental processes in physics. It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which it has been tested experimentally. Here we…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed.…
We show how it is possible to realize quantum computations on a system in which most of the parameters are practically unknown. We illustrate our results with a novel implementation of a quantum computer by means of bosonic atoms in an…