297 research outputs found
Wave-Style Token Machines and Quantum Lambda Calculi
Particle-style token machines are a way to interpret proofs and programs,
when the latter are written following the principles of linear logic. In this
paper, we show that token machines also make sense when the programs at hand
are those of a simple quantum lambda-calculus with implicit qubits. This,
however, requires generalising the concept of a token machine to one in which
more than one particle travel around the term at the same time. The presence of
multiple tokens is intimately related to entanglement and allows us to give a
simple operational semantics to the calculus, coherently with the principles of
quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441
The dagger lambda calculus
We present a novel lambda calculus that casts the categorical approach to the
study of quantum protocols into the rich and well established tradition of type
theory. Our construction extends the linear typed lambda calculus with a linear
negation of "trivialised" De Morgan duality. Reduction is realised through
explicit substitution, based on a symmetric notion of binding of global scope,
with rules acting on the entire typing judgement instead of on a specific
subterm. Proofs of subject reduction, confluence, strong normalisation and
consistency are provided, and the language is shown to be an internal language
for dagger compact categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
QPCF: higher order languages and quantum circuits
qPCF is a paradigmatic quantum programming language that ex- tends PCF with
quantum circuits and a quantum co-processor. Quantum circuits are treated as
classical data that can be duplicated and manipulated in flexible ways by means
of a dependent type system. The co-processor is essentially a standard QRAM
device, albeit we avoid to store permanently quantum states in between two
co-processor's calls. Despite its quantum features, qPCF retains the classic
programming approach of PCF. We introduce qPCF syntax, typing rules, and its
operational semantics. We prove fundamental properties of the system, such as
Preservation and Progress Theorems. Moreover, we provide some higher-order
examples of circuit encoding
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
The Measurement Calculus
Measurement-based quantum computation has emerged from the physics community
as a new approach to quantum computation where the notion of measurement is the
main driving force of computation. This is in contrast with the more
traditional circuit model which is based on unitary operations. Among
measurement-based quantum computation methods, the recently introduced one-way
quantum computer stands out as fundamental.
We develop a rigorous mathematical model underlying the one-way quantum
computer and present a concrete syntax and operational semantics for programs,
which we call patterns, and an algebra of these patterns derived from a
denotational semantics. More importantly, we present a calculus for reasoning
locally and compositionally about these patterns.
We present a rewrite theory and prove a general standardization theorem which
allows all patterns to be put in a semantically equivalent standard form.
Standardization has far-reaching consequences: a new physical architecture
based on performing all the entanglement in the beginning, parallelization by
exposing the dependency structure of measurements and expressiveness theorems.
Furthermore we formalize several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. This allows us to transfer all the theory
we develop for the one-way model to these models. This shows that the framework
we have developed has a general impact on measurement-based computation and is
not just particular to the one-way quantum computer.Comment: 46 pages, 2 figures, Replacement of quant-ph/0412135v1, the new
version also include formalization of several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. To appear in Journal of AC
The Geometry of Synchronization (Long Version)
We graft synchronization onto Girard's Geometry of Interaction in its most
concrete form, namely token machines. This is realized by introducing
proof-nets for SMLL, an extension of multiplicative linear logic with a
specific construct modeling synchronization points, and of a multi-token
abstract machine model for it. Interestingly, the correctness criterion ensures
the absence of deadlocks along reduction and in the underlying machine, this
way linking logical and operational properties.Comment: 26 page
Confluence in Probabilistic Rewriting
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of uniqueness of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired uniqueness (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the classical case, such as Newman's lemma, to simplify proving confluence in concrete languages. Using these criteria, we obtain simple proofs of confluence for λ1, an affine probabilistic λ-calculus, and for Q*, a quantum programming language for which a related property has already been proven in the literature.Fil: DÃaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: MartÃnez, Guido. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentin
Lineal: A linear-algebraic lambda-calculus
International audienceWe provide a computational de nition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the lambda-calculus with the possibility to make arbitrary linear combinations of terms : alpha t + beta u. We describe how to \execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the eld of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus
Lineal: A linear-algebraic Lambda-calculus
We provide a computational definition of the notions of vector space and
bilinear functions. We use this result to introduce a minimal language
combining higher-order computation and linear algebra. This language extends
the Lambda-calculus with the possibility to make arbitrary linear combinations
of terms alpha.t + beta.u. We describe how to "execute" this language in terms
of a few rewrite rules, and justify them through the two fundamental
requirements that the language be a language of linear operators, and that it
be higher-order. We mention the perspectives of this work in the field of
quantum computation, whose circuits we show can be easily encoded in the
calculus. Finally, we prove the confluence of the entire calculus.Comment: The complementary note "On the critical pairs of a rewrite system for
vector spaces" is provided in the source files. Short version :
"Linear-algebraic Lambda-calculus : higher-order and confluence", Proceedings
of RTA 08, Hagenberg, July 2008. LNCS 5117, 17, (2008). Long version : LMC
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