969 research outputs found
A functional quantum programming language
We introduce the language QML, a functional language for quantum computations
on finite types. Its design is guided by its categorical semantics: QML
programs are interpreted by morphisms in the category FQC of finite quantum
computations, which provides a constructive semantics of irreversible quantum
computations realisable as quantum gates. QML integrates reversible and
irreversible quantum computations in one language, using first order strict
linear logic to make weakenings explicit. Strict programs are free from
decoherence and hence preserve superpositions and entanglement - which is
essential for quantum parallelism.Comment: 15 pages. Final version, to appear in Logic in Computer Science 200
Data types with symmetries and polynomial functors over groupoids
Polynomial functors are useful in the theory of data types, where they are
often called containers. They are also useful in algebra, combinatorics,
topology, and higher category theory, and in this broader perspective the
polynomial aspect is often prominent and justifies the terminology. For
example, Tambara's theorem states that the category of finite polynomial
functors is the Lawvere theory for commutative semirings. In this talk I will
explain how an upgrade of the theory from sets to groupoids is useful to deal
with data types with symmetries, and provides a common generalisation of and a
clean unifying framework for quotient containers (cf. Abbott et al.), species
and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan.
The multi-variate setting also includes relations and spans, multispans, and
stuff operators. An attractive feature of this theory is that with the correct
homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits,
etc. - the groupoid case looks exactly like the set case. After some standard
examples, I will illustrate the notion of data-types-with-symmetries with
examples from quantum field theory, where the symmetries of complicated tree
structures of graphs play a crucial role, and can be handled elegantly using
polynomial functors over groupoids. (These examples, although beyond species,
are purely combinatorial and can be appreciated without background in quantum
field theory.) Locally cartesian closed 2-categories provide semantics for
2-truncated intensional type theory. For a fullfledged type theory, locally
cartesian closed \infty-categories seem to be needed. The theory of these is
being developed by D.Gepner and the author as a setting for homotopical
species, and several of the results exposed in this talk are just truncations
of \infty-results obtained in joint work with Gepner. Details will appear
elsewhere.Comment: This is the final version of my conference paper presented at the
28th Conference on the Mathematical Foundations of Programming Semantics
(Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer
Science. 16p
The semantics of jitter in anticipating time itself within nano-technology
The development of nano-technology calls for a careful examination of anticipatory systems at this small scale. For the characteristics of time at the boundary between classical and quantum domains are quite critical for the advancement of the new technology. It has long been well recognised that time is not absolute even in classical subjects like navigation and dynamics where idealised concepts like mean solar time, International Atomic Time and Newtonâs dynamical time have had to be used. Time is the data of the Universe and belongs in the semantics of its extensional form. At the boundary between classical and quantum behaviour the uncertainty of time data becomes a significant effect and this is why it is of great importance in nanotechnology, in areas such as the interoperability of different time domains in hardware, where noise in the form of jitter causes a system to behave in an unpredictable fashion, a severe and expensive problem for anticipating how time is to be handled. A fundamental difficulty is that jitter is represented using numbers, giving rise to undecidability and incompleteness according to Gödelâs theorems. To escape the clutches of Gödel undecidability it is necessary to advance to cartesian closed categories beyond the category of sets to represent the relationship between different times as adjoint endofunctors in monad and comonad constructions
An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces
Explicit expressions are given for the actions and radial matrix elements of
basic radial observables on multi-dimensional spaces in a continuous sequence
of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also
given for SO(N)-reduced matrix elements of basic orbital observables. These
developments make it possible to determine the matrix elements of polynomial
and a other Hamiltonians analytically, to within SO(N) Clebsch-Gordan
coefficients, and to select an optimal basis for a particular problem such that
the expansion of eigenfunctions is most rapidly convergent.Comment: 19 pages, 8 figure
- âŠ