Game Semantics for Quantum Stores

This paper presents a game semantics for a simply-typed gimel-calculus equipped with quantum stores. The quantum stores are equipped with quantum operations as commands which give the language enough expressiveness to encode any quantum circuits. The language uses a notion of extended variable, similar to that seen in functional languages with pattern matching, but adapted to the needs of dealing with tensor products. These tensored variables are used to refer to quantum stores and to keep track of the size of the states which they contain. The game semantics is constructed from classical game semantics using intervention operators to encode the effects of the commands. A soundess result for the semantics is given.