25,624 research outputs found
Two-Bit Gates are Universal for Quantum Computation
A proof is given, which relies on the commutator algebra of the unitary Lie
groups, that quantum gates operating on just two bits at a time are sufficient
to construct a general quantum circuit. The best previous result had shown the
universality of three-bit gates, by analogy to the universality of the Toffoli
three-bit gate of classical reversible computing. Two-bit quantum gates may be
implemented by magnetic resonance operations applied to a pair of electronic or
nuclear spins. A ``gearbox quantum computer'' proposed here, based on the
principles of atomic force microscopy, would permit the operation of such
two-bit gates in a physical system with very long phase breaking (i.e., quantum
phase coherence) times. Simpler versions of the gearbox computer could be used
to do experiments on Einstein-Podolsky-Rosen states and related entangled
quantum states.Comment: 21 pages, REVTeX 3.0, two .ps figures available from author upon
reques
On the formal theory of pseudomonads and pseudodistributive laws
We contribute to the formal theory of pseudomonads, i.e. the analogue for
pseudomonads of the formal theory of monads. In particular, we solve a problem
posed by Steve Lack by proving that, for every Gray-category K, there is a
Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad
transformations and pseudomonad modifications in K. We then establish a
triequivalence between Psm(K) and the Gray-category of pseudomonads introduced
by Marmolejo. Finally, these results are applied to give a clear account of the
coherence conditions for pseudodistributive laws. 40 pages. Comments welcome.Comment: This submission replaces arXiv:0907:1359v1, titled "On the coherence
conditions for pseudo-distributive laws". 40 page
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
The Syntax of Coherence
This article tackles categorical coherence within a two-dimensional
generalization of Lawvere's functorial semantics. 2-theories, a syntactical way
of describing categories with structure, are presented. From the perspective
here afforded, many coherence results become simple statements about the
quasi-Yoneda lemma and 2-theory-morphisms. Given two 2-theories and a
2-theory-morphism between them, we explore the induced relationship between the
corresponding 2-categories of algebras. The strength of the induced
quasi-adjoints are classified by the strength of the 2-theory-morphism. These
quasi-adjoints reflect the extent to which one structure can be replaced by
another. A two-dimensional analogue of the Kronecker product is defined and
constructed. This operation allows one to generate new coherence laws from old
ones.Comment: 44 pages, LaTeX; XY-Pic (with 2-cells). Corrected typos and small
change
Types and forgetfulness in categorical linguistics and quantum mechanics
The role of types in categorical models of meaning is investigated. A general
scheme for how typed models of meaning may be used to compare sentences,
regardless of their grammatical structure is described, and a toy example is
used as an illustration. Taking as a starting point the question of whether the
evaluation of such a type system 'loses information', we consider the
parametrized typing associated with connectives from this viewpoint.
The answer to this question implies that, within full categorical models of
meaning, the objects associated with types must exhibit a simple but subtle
categorical property known as self-similarity. We investigate the category
theory behind this, with explicit reference to typed systems, and their
monoidal closed structure. We then demonstrate close connections between such
self-similar structures and dagger Frobenius algebras. In particular, we
demonstrate that the categorical structures implied by the polymorphically
typed connectives give rise to a (lax unitless) form of the special forms of
Frobenius algebras known as classical structures, used heavily in abstract
categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure
Turing Automata and Graph Machines
Indexed monoidal algebras are introduced as an equivalent structure for
self-dual compact closed categories, and a coherence theorem is proved for the
category of such algebras. Turing automata and Turing graph machines are
defined by generalizing the classical Turing machine concept, so that the
collection of such machines becomes an indexed monoidal algebra. On the analogy
of the von Neumann data-flow computer architecture, Turing graph machines are
proposed as potentially reversible low-level universal computational devices,
and a truly reversible molecular size hardware model is presented as an
example
Classical Structures Based on Unitaries
Starting from the observation that distinct notions of copying have arisen in
different categorical fields (logic and computation, contrasted with quantum
mechanics) this paper addresses the question of when, or whether, they may
coincide. Provided all definitions are strict in the categorical sense, we show
that this can never be the case. However, allowing for the defining axioms to
be taken up to canonical isomorphism, a close connection between the classical
structures of categorical quantum mechanics, and the categorical property of
self-similarity familiar from logical and computational models becomes
apparent.
The required canonical isomorphisms are non-trivial, and mix both typed
(multi-object) and untyped (single-object) tensors and structural isomorphisms;
we give coherence results that justify this approach.
We then give a class of examples where distinct self-similar structures at an
object determine distinct matrix representations of arrows, in the same way as
classical structures determine matrix representations in Hilbert space. We also
give analogues of familiar notions from linear algebra in this setting such as
changes of basis, and diagonalisation.Comment: 24 pages,7 diagram
Pictures of complete positivity in arbitrary dimension
Two fundamental contributions to categorical quantum mechanics are presented.
First, we generalize the CP-construction, that turns any dagger compact
category into one with completely positive maps, to arbitrary dimension.
Second, we axiomatize when a given category is the result of this construction.Comment: Final versio
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