Algorithmic Analysis of Termination Problems for Quantum Programs

摘要：           We introduce the notion of linear ranking super-martingale (LRSM) for quantum programs (with nondeterministic choices, namely
                       angelic and demonic choices). Several termination theorems are established showing that the existence of the LRSMs of a quantum
                      program implies its termination. Thus, the termination problems of quantum programs is reduced to realisability and synthesis of
                      LRSMs. We further show that the realisability and synthesis problem of LRSMs for quantum programs can be reduced to an SDP
                      (Semi-Definite Programming) problem, which can be settled with the existing SDP solvers. The techniques developed in this paper
                     are used to analyse the termination of several example quantum programs, including quantum random walks and quantum Bernoulli
                     factory for random number generation. This work is essentially a generalization of constraint-based approach to the corresponding
                     problems for probabilistic programs developed in the recent literature by adding two novel ideas: (1) employing the fundamental
                     Gleason’s theorem in quantum mechanics to guide the choices of templates; and (2) a generalized Farkas’ lemma in terms of
                     observables (Hermitian operators) in quantum physics.