43 research outputs found
Quantum Turing automata
A denotational semantics of quantum Turing machines having a quantum control
is defined in the dagger compact closed category of finite dimensional Hilbert
spaces. Using the Moore-Penrose generalized inverse, a new additive trace is
introduced on the restriction of this category to isometries, which trace is
carried over to directed quantum Turing machines as monoidal automata. The
Joyal-Street-Verity Int construction is then used to extend this structure to a
reversible bidirectional one.Comment: In Proceedings DCM 2012, arXiv:1403.757
Reverse Derivative Categories
The reverse derivative is a fundamental operation in machine learning and
automatic differentiation. This paper gives a direct axiomatization of a
category with a reverse derivative operation, in a similar style to that given
by Cartesian differential categories for a forward derivative. Intriguingly, a
category with a reverse derivative also has a forward derivative, but the
converse is not true. In fact, we show explicitly what a forward derivative is
missing: a reverse derivative is equivalent to a forward derivative with a
dagger structure on its subcategory of linear maps. Furthermore, we show that
these linear maps form an additively enriched category with dagger biproducts.Comment: Extended version of paper to appear at CSL 202
De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure
Elaborating on our joint work with Abramsky in quant-ph/0402130 we further
unravel the linear structure of Hilbert spaces into several constituents. Some
prove to be very crucial for particular features of quantum theory while others
obstruct the passage to a formalism which is not saturated with physically
insignificant global phases.
First we show that the bulk of the required linear structure is purely
multiplicative, and arises from the strongly compact closed tensor which,
besides providing a variety of notions such as scalars, trace, unitarity,
self-adjointness and bipartite projectors, also provides Hilbert-Schmidt norm,
Hilbert-Schmidt inner-product, and in particular, the preparation-state
agreement axiom which enables the passage from a formalism of the vector space
kind to a rather projective one, as it was intended in the (in)famous Birkhoff
& von Neumann paper.
Next we consider additive types which distribute over the tensor, from which
measurements can be build, and the correctness proofs of the protocols
discussed in quant-ph/0402130 carry over to the resulting weaker setting. A
full probabilistic calculus is obtained when the trace is moreover linear and
satisfies the \em diagonal axiom, which brings us to a second main result,
characterization of the necessary and sufficient additive structure of a both
qualitatively and quantitatively effective categorical quantum formalism
without redundant global phases. Along the way we show that if in a category a
(additive) monoidal tensor distributes over a strongly compact closed tensor,
then this category is always enriched in commutative monoids.Comment: Essential simplification of the definitions of orthostructure and
ortho-Bornian structure: the key new insights is captured by the definitions
in terms of commutative diagrams on pages 13 and 14, which state that if in a
category a (additive) monoidal tensor distributes over a strongly compact
closed tensor, then this category is always enriched in commutative monoid
Way of the dagger
A dagger category is a category equipped with a functorial way of reversing morphisms,
i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories
with additional structure have been studied under different names in categorical
quantum mechanics, algebraic field theory and homological algebra, amongst others.
In this thesis we study the dagger in its own right and show how basic category theory
adapts to dagger categories.
We develop a notion of a dagger limit that we show is suitable in the following
ways: it subsumes special cases known from the literature; dagger limits are unique
up to unitary isomorphism; a wide class of dagger limits can be built from a small
selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to
a diagonal functor; dagger limits can be built from ordinary limits in the presence of
polar decomposition; dagger limits commute with dagger colimits in many cases.
Using cofree dagger categories, the theory of dagger limits can be leveraged to
provide an enrichment-free understanding of limit-colimit coincidences in ordinary
category theory. We formalize the concept of an ambilimit, and show that it captures
known cases. As a special case, we show how to define biproducts up to isomorphism
in an arbitrary category without assuming any enrichment. Moreover, the limit-colimit
coincidence from domain theory can be generalized to the unenriched setting and we
show that, under suitable assumptions, a wide class of endofunctors has canonical fixed
points.
The theory of monads on dagger categories works best when all structure respects
the dagger: the monad and adjunctions should preserve the dagger, and the monad and
its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an
adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-
Eilenberg-Moore algebras, which again have a dagger.
We use dagger categories to study reversible computing. Specifically, we model reversible
effects by adapting Hughes’ arrows to dagger arrows and inverse arrows. This
captures several fundamental reversible effects, including serialization and mutable
store computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse arrows
satisfy certain additional properties. These semantics inform the design of functional
reversible programs supporting side-effects