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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
A system of interaction and structure IV: The exponentials
 IN THE SECOND ROUND OF REVISION FOR MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2007
"... We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. T ..."
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Cited by 11 (6 self)
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We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a selfdual, noncommutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
A brief survey of quantum programming languages
 In Proceedings of the 7th International Symposium on Functional and Logic Programming
, 2004
"... Abstract. This article is a brief and subjective survey of quantum programming language research. 1 Quantum Computation Quantum computing is a relatively young subject. It has its beginnings in 1982, when Paul Benioff and Richard Feynman independently pointed out that a quantum mechanical system can ..."
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Cited by 11 (0 self)
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Abstract. This article is a brief and subjective survey of quantum programming language research. 1 Quantum Computation Quantum computing is a relatively young subject. It has its beginnings in 1982, when Paul Benioff and Richard Feynman independently pointed out that a quantum mechanical system can be used to perform computations [11, p.12]. Feynman’s interest in quantum computation was motivated by the fact that it is computationally very expensive to simulate quantum physical systems on classical computers. This is due to the fact that such simulation involves the manipulation is extremely large matrices (whose dimension is exponential in the size of the quantum system being simulated). Feynman conceived of quantum computers as a means of simulating nature much more efficiently. The evidence to this day is that quantum computers can indeed perform certain tasks more efficiently than classical computers. Perhaps the bestknown example is Shor’s factoring algorithm, by which a quantum computer can find
Geometry of Interaction IV: the Feedback Equation
, 2005
"... The first three papers on Geometry of Interaction [9, 10, 11] did establish the universality of the feedback equation as an explanation of logic; this equation corresponds to the fundamental operation of logic, namely cutelimination, i.e., logical consequence; this is also the oldest approach to lo ..."
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The first three papers on Geometry of Interaction [9, 10, 11] did establish the universality of the feedback equation as an explanation of logic; this equation corresponds to the fundamental operation of logic, namely cutelimination, i.e., logical consequence; this is also the oldest approach to logic, syllogistics! But the equation was essentially studied for those Hilbert space operators coming from actual logical proofs. In this paper, we take the opposite viewpoint, on the arguable basis that operator algebra is more primitive than logic: we study the general feedback equation of Geometry of Interaction, h(x⊕y) = x ′ ⊕σ(y), where h,σ are hermitian, �h � ≤ 1, and σ is a partial symmetry, σ 3 = σ. We show that the normal form which yields the solution σ�h�(x) = x ′ in the invertible case can be extended in a unique way to the general case, by various techniques, basically ordercontinuity and associativity. From this we expect a definite break with essentialism à la Tarski: an interpretation of logic which does not presuppose logic! 1
A system of interaction and structure V: The exponentials and splitting
, 2009
"... System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential compositio ..."
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Cited by 4 (3 self)
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System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deepinference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequentlike structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cutelimination theorem for NEL. 1
Probabilistic coherence spaces as a model of higherorder probabilistic computation
, 2011
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On probabilistic coherence spaces
, 2008
"... We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation ..."
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We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation of the semantics of probabilistic PCF closed terms of ground type.
A logical basis for quantum evolution and entanglement
, 2011
"... We reconsider discrete quantum casual dynamics where quantum systems are viewed as discrete structures, namely directed acyclic graphs. In such a graph, events are considered as vertices and edges depict propagation between events. Evolution is described as happening between a special family of spa ..."
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We reconsider discrete quantum casual dynamics where quantum systems are viewed as discrete structures, namely directed acyclic graphs. In such a graph, events are considered as vertices and edges depict propagation between events. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Such slices are not so large as to result in acausal influences, but large enough to capture nonlocal correlations. In our logical interpretation, edges are assigned logical formulas in a special logical system, called BV, an instance of a deep inference system. We demonstrate that BV, with its mix of commutative and noncommutative connectives, is precisely the right logic for such analysis. We show that the commutative tensor encodes (possible) entanglement, and the noncommutative seq encodes causal precedence. With this interpretation, the locative slices are precisely the derivable strings of formulas.
On Differential Interaction Nets and the Picalculus
 Preuves, Programmes et Systèmes
, 2006
"... We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, w ..."
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We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, we restrict our attention to a monadic version of the picalculus, so that the differential interaction net structures we consider need only to have exponential cells. We prove that the nets obtained by this translation satisfy an acyclicity criterion weaker than the standard Girard (or DanosRegnier) acyclicity criterion, and we compare the operational semantics of the picalculus, presented by means of an environment machine, and the reduction of differential interaction nets. Differential interaction net structures being of a logical nature, this work provides a CurryHoward interpretation of processes.
Deep Inference and Probabilistic Coherence Spaces
, 2009
"... This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor pr ..."
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This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We show that coherence spaces, with Retoré’s noncommutative tensor, is a model.We then consider Girard’s category of probabilistic coherence spaces and show that it contains a selfdual monoidal structure in addition to the ∗autonomous structure exhibited by Girard. This