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A lambda calculus for quantum computation
 SIAM Journal of Computing
"... The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propos ..."
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Cited by 49 (1 self)
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The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
Quantum computation, categorical semantics and linear logic. quantph/0312174
, 2003
"... We develop a type theory and provide a denotational semantics for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic. In our semantics, variables inhabit certain Hilbert bundles, and computations are interpreted as the appropriate inner ..."
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Cited by 27 (1 self)
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We develop a type theory and provide a denotational semantics for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic. In our semantics, variables inhabit certain Hilbert bundles, and computations are interpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 12 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at
A Logic for Probability in Quantum Systems
 Computer Science Logic, volume 2803 of Lecture Notes in Computer Science
, 2003
"... Quantum computation deals with projective measurements and unitary transformations in finite dimensional Hilbert spaces. The paper presents a propositional logic designed to describe quantum computation at an operational level by supporting reasoning about the probabilities associated to such me ..."
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Cited by 11 (2 self)
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Quantum computation deals with projective measurements and unitary transformations in finite dimensional Hilbert spaces. The paper presents a propositional logic designed to describe quantum computation at an operational level by supporting reasoning about the probabilities associated to such measurements: measurement probabilities, and transition probabilities (a quantum analogue of conditional probabilities) . We present two axiomatizations, one for the logic as a whole and one for the fragment dealing just with measurement probabilities.
From Basic Logic to Quantum Logics With CutElimination
 International Journal of Theoretical Physics
, 1997
"... . The results presented in this paper were born in the framework of basic logic, a new logic aiming at the unification of several logical systems. The first result is a sequent formulation for orthologic which allows to use methods of proof theory in quantum logic. Such formulation admits a very sim ..."
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Cited by 9 (4 self)
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. The results presented in this paper were born in the framework of basic logic, a new logic aiming at the unification of several logical systems. The first result is a sequent formulation for orthologic which allows to use methods of proof theory in quantum logic. Such formulation admits a very simple procedure of cutelimination and hence, because of the subformula property, also a method of proof search and an effective decision procedure. By using the framework of basic logic, we also obtain a cutfree formulation for orthologic with implication, for linear orthologic and, more in general, for a wide range of new quantumlike logics. These logics meet some requirements expressed by physicists and computer scientists. In particular, we propose a good candidate for a linear quantum logic with implication. Key words: orthologic, basic logic, cutelimination, linear quantum logic. 1 Introduction A sequent calculus for quantum logic was introduced twenty years ago by M. Dummett in [Dum...
The Separated Extensional Chu Category
 Theory and Applications of Categories
, 1998
"... . This paper shows that, given a factorization system, E=M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also autonomous under weaker conditions than had been given previously ([Barr, 1991)]. In the process we find conditions u ..."
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Cited by 9 (0 self)
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. This paper shows that, given a factorization system, E=M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also autonomous under weaker conditions than had been given previously ([Barr, 1991)]. In the process we find conditions under which the intersection of a full reflective subcategory and its coreflective dual in a Chu category is autonomous. 1. Introduction 1.1. Chu categories. An appendix to [Barr, 1979] was an extract from the master's thesis of P.H. Chu that described what seemed at the time a toosimpletobeinteresting construction of autonomous categories [Chu, 1979]. In fact, this construction, now called the Chu construction has turned out to be surprisingly interesting, both as a way of providing models of Girard's linear logic [Seely, 1988], in theoretical computer science [Pratt, 1993a, 1993b, 1995] and as a general approach to duality [Barr and Kleisli, to apear] and [Schlapfer, 1998].. Given a...
Chu Spaces: Automata with quantum aspects
 In Proc. Workshop on Physics and Computation (PhysComp’94
, 1994
"... Chu spaces are a recently developed model of concurrent computation extending automata theory to express branching time and true concurrency. They exhibit in a primitive form the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information ..."
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Cited by 8 (3 self)
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Chu spaces are a recently developed model of concurrent computation extending automata theory to express branching time and true concurrency. They exhibit in a primitive form the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information and time, automata and schedules, and states and events. Uncertainty arises when we define a measurement to be a morphism and notice that increasing structure in the observed object reduces clarity of observation. For a Chu space this uncertainty can be calculated numerically in an attractively simple way directly from its form factor to yield the usual Heisenberg uncertainty relation. Chu spaces correspond to wavefunctions as vectors of Hilbert space, whose inner product operation is realized for Chu spaces as right residuation and whose quantum logic becomes Girard's linear logic. 1 Introduction 1.1 Prospects for Chu Spaces The automaton model of this paper, Chu spaces, is an outgrowth ...
*Autonomous Categories: Once More Around The Track
 AND CHU CONSTRUCTIONS: COUSINS? 149
, 1999
"... . This represents a new and more comprehensive approach to the  autonomous categories constructed in the monograph [Barr, 1979]. The main tool in the new approach is the Chu construction. The main conclusion is that the category of separated extensional Chu objects for certain kinds of equationa ..."
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Cited by 6 (1 self)
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. This represents a new and more comprehensive approach to the  autonomous categories constructed in the monograph [Barr, 1979]. The main tool in the new approach is the Chu construction. The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories. 1. Introduction The monograph [Barr, 1979] was devoted to the investigation of autonomous categories. Most of the book was devoted to the discovery of autonomous categories as full subcategories of seven different categories of uniform or topological algebras over concrete categories that were either equational or reflective subcategories of equational categories. The base categories were: 1. vector spaces over a discrete field; 2. vector spaces over the real or complex numbers; 3. modules over a ring with a dualizing module; 4. abelian groups; 5. modules ove...
Time and Information in Sequential and Concurrent Computation
 In Proc. Theory and Practice of Parallel Programming
, 1994
"... Time can be understood as dual to information in extant models of both sequential and concurrent computation. The basis for this duality is phase space, coordinatized by time and information, whose axes are oriented respectively horizontally and vertically. We fit various basic phenomena of computat ..."
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Cited by 5 (1 self)
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Time can be understood as dual to information in extant models of both sequential and concurrent computation. The basis for this duality is phase space, coordinatized by time and information, whose axes are oriented respectively horizontally and vertically. We fit various basic phenomena of computation, and of behavior in general, to the phase space perspective. The extant twodimensional logics of sequential behavior, the van Glabbeek map of branching time and true concurrency, eventstate duality and scheduleautomaton duality, and Chu spaces, all fit the phase space perspective well, in every case confirming our choice of orientation. 1 Introduction Our recent research has emphasized a basic duality between time and information in concurrent computation. In this paper we return to our earlier work on sequential computation and point out that a very similar duality is present there also. Our main goal here will be to compare concurrent and sequential computation in terms of this dua...
Logic of Dynamics & Dynamics of Logic; Some Paradigm Examples
"... The development ofoperationalquantum logicpointsoutthatclassicalbooleanstructuresaretoo rigidtodescribe theactualand potentialpropertiesofquantum systems.Operationalquantum logic bearsupon basicaxiomswhich aremotivatedby empiricalfactsand assuch supportsthedynamicshiftfromclassicaltononclassicallog ..."
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Cited by 4 (3 self)
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The development ofoperationalquantum logicpointsoutthatclassicalbooleanstructuresaretoo rigidtodescribe theactualand potentialpropertiesofquantum systems.Operationalquantum logic bearsupon basicaxiomswhich aremotivatedby empiricalfactsand assuch supportsthedynamicshiftfromclassicaltononclassicallogic resultinginto a dynamicsoflogic.On theotherhand,a dynamic extensionofoperationalquantum logicallows ustoexpressdynamic 1 reasoninginthesensethatwecancapturehow actualpropertiespropagate, includingtheirtemporalcausalstructure.Itisinthissense thatpassingfromstaticoperationalquantum logictodynamicoperationalquantum logicresultsina truelogicofdynamicsthatprovides a unifledlogicaldescriptionofsystemswhich evolve orwhich aresubmittedtomeasurements. Whilefocusingon thequantalesemantics fordynamicoperationalquantum logic,we cananalyzethepointsof difierencewiththeexistingquantalesemanticsfor(non)commutative linearlogic.Linearlogicisheretobeconceivedasa resourcesensitive logiccapableofdealingwi...