Quantum Relational Hoare Logic Judgements

Abstract

With the development of quantum and post-quantum cryptography, it becomes necessary to verify security proofs of protocols. For this, quantum Relational Hoare Logic was developed. The main object of this logic is judgments that demonstrate the various connections between two quantum programs. Their are for possible ways to define judgements: with quantum predicates or operators, with separability requirement or without it. The subject of study in this thesis is the question of whether it is possible for such a judgment to answer the question whether it holds or not. The chosen approach is to use the linear and convex conical structure of the studied objects to apply semidefinite and cone programming. This allowed to apply results related to the optimization of linear functions over a set of separable operators. This provide an efficient algorithm for the two definitions without the separability constraint. For the other two types of judgements this approach gives a reformulation of the problem in terms of cone programming and polynomials non-negativity. This allows to algorithmically check connections between quantum programs and continue the studies with a new point of view on the judgements solvability.