10 research outputs found
Compositional reversible computation
Reversible computing is motivated by both pragmatic and foundational considerations arising from a variety of disciplines. We take a particular path through the development of reversible computation, emphasizing compositional reversible computation. We start from a historical perspective, by reviewing those approaches that developed reversible extensions of lambda-calculi, Turing machines, and communicating process calculi. These approaches share a common challenge: computations made reversible in this way do not naturally compose locally.We then turn our attention to computational models that eschew the detour via existing irreversible models. Building on an original analysis by Landauer, the insights of Bennett, Fredkin, and Toffoli introduced a fresh approach to reversible computing in which reversibility is elevated to the status of the main design principle. These initial models are expressed using low-level bit manipulations, however.Abstracting from the low-level of the Bennett-Fredkin-Toffoli models and pursuing more intrinsic, typed, and algebraic models, naturally leads to rig categories as the canonical model for compositional reversible programming. The categorical model reveals connections to type isomorphisms, symmetries, permutations, groups, and univalent universes. This, in turn, paves the way for extensions to reversible programming based on monads and arrows. These extensions are shown to recover conventional irreversible programming, a variety of reversible computational effects, and more interestingly both pure (measurement-free) and measurement-based quantum programming
Formalizing of Category Theory in Agda
The generality and pervasiness of category theory in modern mathematics makes
it a frequent and useful target of formalization. It is however quite
challenging to formalize, for a variety of reasons. Agda currently (i.e. in
2020) does not have a standard, working formalization of category theory. We
document our work on solving this dilemma. The formalization revealed a number
of potential design choices, and we present, motivate and explain the ones we
picked. In particular, we find that alternative definitions or alternative
proofs from those found in standard textbooks can be advantageous, as well as
"fit" Agda's type theory more smoothly. Some definitions regarded as equivalent
in standard textbooks turn out to make different "universe level" assumptions,
with some being more polymorphic than others. We also pay close attention to
engineering issues so that the library integrates well with Agda's own standard
library, as well as being compatible with as many of supported type theories in
Agda as possible
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Lambda: The Ultimate Sublanguage (experience report)
We describe our experience teaching an advanced typed functional programming course based around the use of System FÏ as a programming language.</jats:p
Quantum Information Effects
We study the two dual quantum information effects to manipulate the amount of
information in quantum computation: hiding and allocation. The resulting
type-and-effect system is fully expressive for irreversible quantum computing,
including measurement. We provide universal categorical constructions that
semantically interpret this arrow metalanguage with choice, starting with any
rig groupoid interpreting the reversible base language. Several properties of
quantum measurement follow in general, and we translate quantum flow charts
into our language. The semantic constructions turn the category of unitaries
between Hilbert spaces into the category of completely positive
trace-preserving maps, and they turn the category of bijections between finite
sets into the category of functions with chosen garbage. Thus they capture the
fundamental theorems of classical and quantum reversible computing of Toffoli
and Stinespring.Comment: 32 pages, including 10 page appendi
With a Few Square Roots, Quantum Computing is as Easy as {\Pi}
Rig groupoids provide a semantic model of \PiLang, a universal classical
reversible programming language over finite types. We prove that extending rig
groupoids with just two maps and three equations about them results in a model
of quantum computing that is computationally universal and equationally sound
and complete for a variety of gate sets. The first map corresponds to an
root of the identity morphism on the unit . The second map
corresponds to a square root of the symmetry on . As square roots are
generally not unique and can sometimes even be trivial, the maps are
constrained to satisfy a nondegeneracy axiom, which we relate to the Euler
decomposition of the Hadamard gate. The semantic construction is turned into an
extension of \PiLang, called \SPiLang, that is a computationally universal
quantum programming language equipped with an equational theory that is sound
and complete with respect to the Clifford gate set, the standard gate set of
Clifford+T restricted to qubits, and the computationally universal
Gaussian Clifford+T gate set
With a Few Square Roots, Quantum Computing Is as Easy as Pi
Rig groupoids provide a semantic model of Î , a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Î , called âÎ , that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to â€2 qubits, and the computationally universal Gaussian Clifford+T gate set