3,926 research outputs found
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
Compositional theories for embedded languages
Embedded programming style allows to split the syntax in two parts,
representing respectively a host language H and a core language C embedded in
H. This formally models several situations in which a user writes code in a
main language and delegates some tasks to an ad hoc domain specific language.
Moreover, as showed in recent years, a particular case of the host-core
approach allows a flexible management of data linearity, which is particularly
useful in non-classical computational paradigms such as quantum computing. The
definition of a systematised type theory to capture and standardize common
properties of embedded languages is unexplored. The aim of this paper is to
present a flexible fragment of such a type theory, together with its
categorical semantics in terms of enriched categories, following previous
investigations. We present the calculus HC0 and we use the notion of internal
language of a category to relate the language to the class of its models,
showing the equivalence between the category of models and the one of theories.
This provides a stronger result w.r.t. standard soundness and completeness
since it involves not only the models but also morphisms between models. We
observe that the definition of the morphisms between models highlights further
advantages of the embedded languages and we discuss some concrete instances,
extensions and specializations of the syntax and the semantics.Comment: 20 page
Quantum Alternation: Prospects and Problems
We propose a notion of quantum control in a quantum programming language
which permits the superposition of finitely many quantum operations without
performing a measurement. This notion takes the form of a conditional construct
similar to the IF statement in classical programming languages. We show that
adding such a quantum IF statement to the QPL programming language simplifies
the presentation of several quantum algorithms. This motivates the possibility
of extending the denotational semantics of QPL to include this form of quantum
alternation. We give a denotational semantics for this extension of QPL based
on Kraus decompositions rather than on superoperators. Finally, we clarify the
relation between quantum alternation and recursion, and discuss the possibility
of lifting the semantics defined by Kraus operators to the superoperator
semantics defined by Selinger.Comment: In Proceedings QPL 2015, arXiv:1511.0118
The Quantum Monadology
The modern theory of functional programming languages uses monads for
encoding computational side-effects and side-contexts, beyond bare-bone program
logic. Even though quantum computing is intrinsically side-effectful (as in
quantum measurement) and context-dependent (as on mixed ancillary states),
little of this monadic paradigm has previously been brought to bear on quantum
programming languages.
Here we systematically analyze the (co)monads on categories of parameterized
module spectra which are induced by Grothendieck's "motivic yoga of operations"
-- for the present purpose specialized to HC-modules and further to set-indexed
complex vector spaces. Interpreting an indexed vector space as a collection of
alternative possible quantum state spaces parameterized by quantum measurement
results, as familiar from Proto-Quipper-semantics, we find that these
(co)monads provide a comprehensive natural language for functional quantum
programming with classical control and with "dynamic lifting" of quantum
measurement results back into classical contexts.
We close by indicating a domain-specific quantum programming language (QS)
expressing these monadic quantum effects in transparent do-notation, embeddable
into the recently constructed Linear Homotopy Type Theory (LHoTT) which
interprets into parameterized module spectra. Once embedded into LHoTT, this
should make for formally verifiable universal quantum programming with linear
quantum types, classical control, dynamic lifting, and notably also with
topological effects.Comment: 120 pages, various figure
Compositional frameworks for supermaps and causality
Quantum supermaps are transformations of quantum processes, and have found many applications in quantum foundations and quantum information theory in the past two decades, particularly in the study of causality. Whilst the concept of a supermap is a simple and intuitive one, the current state-of-the-art formalisations of supermaps cannot be applied to arbitrary Hilbert spaces or Operational Probabilistic Theories (OPTs). We review the standard approaches to defining supermaps in quantum theory, wherever possible highlighting the background compositional principles at play using diagrammatic languages and referring to their algebraic formalisation in the field of category theory. The core argument of this thesis is that a more principled and general approach to defining quantum supermaps exists, using a definition of locally-applicable transformation, which can be applied to any symmetric monoidal category. As a consequence this approach can be applied to all quantum processes on general quantum degrees of freedom and to all transformations in OPTs. We identify key compositional features for entire theories of supermaps and show that the supermaps of those theories are always operationally described by locally-applicable transformations. Two tests for a good construction of supermaps on symmetric monoidal categories are identified, recovery of standard physicists definitions for quantum supermaps when applied to categories of standard quantum processes, and existence of key compositional features. By the end of the thesis we find a way to strengthen locally-applicable transformations to construct the theory of polyslots, which passes both tests. Applications of this new general framework for the study of quantum causality and quantum information theory are identified as future potential research directions
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