22 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
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 game semantics of game theory
We use a reformulation of compositional game theory to reunite game theory
with game semantics, by viewing an open game as the System and its choice of
contexts as the Environment. Specifically, the system is jointly controlled by
noncooperative players, each independently optimising a real-valued
payoff. The goal of the system is to play a Nash equilibrium, and the goal of
the environment is to prevent it. The key to this is the realisation that
lenses (from functional programming) form a dialectica category, which have an
existing game-semantic interpretation.
In the second half of this paper, we apply these ideas to build a compact
closed category of `computable open games' by replacing the underlying
dialectica category with a wave-style geometry of interaction category,
specifically the Int-construction applied to the cartesian monoidal category of
directed-complete partial orders
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
Semantics for a Quantum Programming Language by Operator Algebras
This paper presents a novel semantics for a quantum programming language by
operator algebras, which are known to give a formulation for quantum theory
that is alternative to the one by Hilbert spaces. We show that the opposite
category of the category of W*-algebras and normal completely positive
subunital maps is an elementary quantum flow chart category in the sense of
Selinger. As a consequence, it gives a denotational semantics for Selinger's
first-order functional quantum programming language QPL. The use of operator
algebras allows us to accommodate infinite structures and to handle classical
and quantum computations in a unified way.Comment: In Proceedings QPL 2014, arXiv:1412.810
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
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
On the Relation of Interaction Semantics to Continuations and Defunctionalization
In game semantics and related approaches to programming language semantics,
programs are modelled by interaction dialogues. Such models have recently been
used in the design of new compilation methods, e.g. for hardware synthesis or
for programming with sublinear space. This paper relates such semantically
motivated non-standard compilation methods to more standard techniques in the
compilation of functional programming languages, namely continuation passing
and defunctionalization. We first show for the linear {\lambda}-calculus that
interpretation in a model of computation by interaction can be described as a
call-by-name CPS-translation followed by a defunctionalization procedure that
takes into account control-flow information. We then establish a relation
between these two compilation methods for the simply-typed {\lambda}-calculus
and end by considering recursion