Affiliation:
1. Chiba University, Chiba, Japan
2. Tohoku University, Sendai, Japan
Abstract
Selinger gave a superoperator model of a first-order quantum programming language and proved that it is fully definable and hence fully abstract. This paper proposes an extension of the superoperator model to higher-order programs based on modules over superoperators or, equivalently, enriched presheaves over the category of superoperators. The enriched presheaf category can be easily proved to be a model of intuitionistic linear logic with cofree exponential, from which one can cave out a model of classical linear logic by a kind of bi-orthogonality construction. Although the structures of an enriched presheaf category are usually rather complex, a morphism in the classical model can be expressed simply as a matrix of completely positive maps. The model inherits many desirable properties from the superoperator model. A conceptually interesting property is that our model has only a state whose “total probability” is bounded by 1, i.e. does not have a state where true and false each occur with probability 2/3. Another convenient property inherited from the superoperator model is a ωCPO-enrichment. Remarkably, our model has a sufficient structure to interpret arbitrary recursive types by the standard domain theoretic technique. We introduce
Quantum FPC
, a quantum λ-calculus with recursive types, and prove that our model is a fully abstract model of Quantum FPC.
Funder
Japan Society for the Promotion of Science
ROIS NII Open Collaborative Research
Publisher
Association for Computing Machinery (ACM)
Subject
Safety, Risk, Reliability and Quality,Software