Margherita Zorzi

Probabilistic Operational Semantics for the Lambda Calculus

Probabilistic operational semantics for a nondeterminis- tic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkins CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions. Mathematics Subject Classification. 68Q55, 03B70.

On a measurement-free quantum lambda calculus with classical control

We study a measurement-free, untyped λ-calculus with quantum data and classical control. This work arises from previous proposals by Selinger and Valiron, and Van Tonder. We focus on operational and expressiveness issues, rather than (denotational) semantics. We prove subject reduction and confluence, and a standardisation theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.

On quantum lambda calculi: a foundational perspective

In this paper, we propose an approach to quantum λ-calculi. The ‘quantum data-classical control’ paradigm is considered. Starting from a measurement-free untyped quantum λ-calculus called Q , we will study standard properties such as confluence and subject reduction, and some good quantum properties. We will focus on the expressive power, analysing the relationship with other quantum computational models. Successively, we will add an explicit measurement operator to Q . On the resulting calculus, called Q *, we will propose a complete study of reduction sequences regardless of their finiteness, proving confluence results. Moreover, since the stronger motivation behind quantum computing is the research of new results in computational complexity, we will also propose a calculus which captures the three classes of quantum polytime complexity, showing an ICC-like approach in the quantum setting.

Confluence Results for a Quantum Lambda Calculus with Measurements

A strong confluence result for Q*, a quantum @l-calculus with measurements, is proved. More precisely, confluence is shown to hold both for finite and infinite computations. The technique used in the confluence proof is syntactical but innovative. This makes Q* different from similar quantum lambda calculi, which are either measurement-free or provided with a reduction strategy.

Quantum State Transformations and Branching Distributed Temporal Logic

The Distributed Temporal Logic DTL allows one to reason about temporal properties of a distributed system from the local point of view of the systems agents, which are assumed to execute independently and to interact by means of event sharing. In this paper, we introduce the Quantum Branching Distributed Temporal Logic

Wave-Style Token Machines and Quantum Lambda Calculi

\textsf{QBDTL}

A Qualitative Modal Representation of Quantum Register Transformations

, a variant of DTL able to represent quantum state transformations in an abstract, qualitative way. In

Non-determinism, Non-termination and the Strong Normalization of System T

\textsf{QBDTL}

A branching distributed temporal logic for reasoning about entanglement-free quantum state transformations

, each agent represents a distinct quantum bit the unit of quantum information theory, which evolves by means of quantum transformations and possibly interacts with other agents, and n-ary quantum operators act as communication/synchronization points between agents. We endow

On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

\textsf{QBDTL}