30 research outputs found
Topological Quantum Programming in TED-K
While the realization of scalable quantum computation will arguably require
topological stabilization and, with it, topological-hardware-aware quantum
programming and topological-quantum circuit verification, the proper
combination of these strategies into dedicated topological quantum programming
languages has not yet received attention. Here we describe a fundamental and
natural scheme that we are developing, for typed functional (hence verifiable)
topological quantum programming which is topological-hardware aware -- in that
it natively reflects the universal fine technical detail of topological q-bits,
namely of symmetry-protected (or enhanced) topologically ordered Laughlin-type
anyon ground states in topological phases of quantum materials.
What makes this work is: (1) our recent result that wavefunctions of
realistic and technologically viable anyon species -- namely of su(2)-anyons
such as the popular Majorana/Ising anyons but also of computationally universal
Fibonacci anyons -- are reflected in the twisted equivariant differential (TED)
K-cohomology of configuration spaces of codimension=2 nodal defects in the host
material's crystallographic orbifold; (2) combined with our earlier observation
that such TED generalized cohomology theories on orbifolds interpret
intuitionistically-dependent linear data types in cohesive homotopy type theory
(HoTT), supporting a powerful modern form of modal quantum logic.
In this short note we give an exposition of the basic ideas, a quick review
of the underlying results and a brief indication of the basic language
constructs for anyon braiding via TED-K in cohesive HoTT. The language system
is under development at the "Center for Quantum and Topological Systems" at the
Research Institute of NYU, Abu Dhabi.Comment: 8 pages, 1 figure; extended abstract for contribution to: PlanQC2022
https://icfp22.sigplan.org/home/planqc-202
Venus Homotopically
The identity concept developed in the Homotopy Type theory (HoTT) supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological significance of this distinction and, finally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the Quantum Mechanics
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
Higher Structures in M-Theory
The key open problem of string theory remains its non-perturbative completion
to M-theory. A decisive hint to its inner workings comes from numerous
appearances of higher structures in the limits of M-theory that are already
understood, such as higher degree flux fields and their dualities, or the
higher algebraic structures governing closed string field theory. These are all
controlled by the higher homotopy theory of derived categories, generalised
cohomology theories, and -algebras. This is the introductory chapter
to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in
M-Theory. We first review higher structures as well as their motivation in
string theory and beyond. Then we list the contributions in this volume,
putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham
Symposium Higher Structures in M-Theory, August 2018, references update