We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a pre-defined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation-the quantum switch-produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection, or an extra query to the input black boxes.

The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and we 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.

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and we 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.