This special issue of Mathematical Structures in Computer Science grew out of the 2nd International Workshop on Quantum Programming Languages (QPL 2004), which was held July 12–13, 2004 in Turku, Finland. The purpose of the workshop was to bring together researchers working on mathematical formalisms and programming languages for quantum computing. It was the second in a series of workshops aimed at addressing a growing interest in logical tools, languages, and semantical methods for analysing quantum computation.

We define a language CQP (Communicating Quantum Processes) for modelling systems that combine quantum and classical communication and computation. CQP combines the communication primitives of the pi-calculus with primitives for measurement and transformation of the quantum state; in particular, quantum bits (qubits) can be transmitted from process to process along communication channels. CQP has a static type system, which classifies channels, distinguishes between quantum and classical data, and controls the use of quantum states. We formally define the syntax, operational semantics and type system of CQP, prove that the semantics preserves typing, and prove that typing guarantees that each qubit is owned by a unique process within a system. We also define a typechecking algorithm and prove that it is sound and complete with respect to the type system. We illustrate CQP by defining models of several quantum communication systems, and outline our plans for using CQP as the foundation for formal analysis and verification of combined quantum and classical systems.

Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components, as well as communications between these components. Moreover, to model concurrent and distributed quantum computations and quantum communication protocols, communications over quantum channels that move qubits physically from one place to another must also be taken into account.

Inspired by classical process algebras, which provide a framework for modelling cooperating computations, a process algebraic notation is defined. This notation provides a homogeneous style for formal descriptions of concurrent and distributed computations comprising both quantum and classical parts. Based upon an operational semantics that makes sure that quantum objects, operations and communications operate according to the postulates of quantum mechanics, an equivalence is defined among process states considered as having the same behaviour. This equivalence is a probabilistic branching bisimulation. From this relation, an equivalence on processes is defined. However, it is not a congruence because it is not preserved by parallel composition.

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming, we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example, we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilisers.

We show that the model of quantum computation based on density matrices and superoperators can be decomposed into a pure classical (functional) part and an effectful part modelling probabilities and measurement. The effectful part can be modelled using a generalisation of monads called arrows. We express the resulting executable model of quantum computing in the Haskell programming language using its special syntax for arrow computations. However, the embedding in Haskell is not perfect: a faithful model of quantum computing requires type capabilities that are not directly expressible in Haskell.

We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.

The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.

In this paper we develop a functional programming language for quantum computers by extending the simply-typed lambda calculus with quantum types and operations. The design of this language adheres to the ‘quantum data, classical control’ paradigm, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.