A Hierarchy of Quantum Semantics

Several domains [1,4,9,12] can be used to define the semantics of quantum programs. Among them Abramsky [1] has introduced a semantics based on probabilistic power domains, whereas the one by Selinger [12] associates with each program a completely positive map. In this abstract, we mainly introduce a semantical domain based on admissible transformations, i.e. multi-sets of linear operators. In order to establish a comparison with existing domains, we introduce a simple quantum imperative language (QIL), equipped with three different denotational semantics, called pure, observable, and admissible. The pure semantics is a natural extension of probabilistic (classical) semantics and is similar to the semantics proposed by Abramsky [1]. The observable semantics, à la Selinger [12], associates with any program a superoperator over density matrices. Finally, we introduce an admissible semantics which associates with any program an admissible transformation. These semantics are not equivalent, but exact abstraction [5] or interpretation relations are established between them, leading to a hierarchy of quantum semantics. 1 Quantum Computing Basics The basic carrier of information in quantum computing is a 2-level quantum system (qubit), or more generally a register of n qubits. The state of a n-qubit register is a normalized vector of a Hilbert space C2 n . So, for a given basis A, a general state |φ〉 ∈ C|A| can be written as: