Reasoning about Parallel Quantum Programs

We initiate the study of parallel quantum programming by defining the operational and denotational semantics of parallel quantum programs. The technical contributions of this paper include: (1) find a series of useful proof rules for reasoning about correctness of parallel quantum programs; (2) prove a (relative) completeness of our proof rules for partial correctness of disjoint parallel quantum programs; and (3) prove a strong soundness theorem of the proof rules showing that partial correctness is well maintained at each step of transitions in the operational semantics of a general parallel quantum program (with shared variables). This is achieved by partially overcoming the following conceptual challenges that are never present in classical parallel programming: (i) the intertwining of nondeterminism caused by quantum measurements and introduced by parallelism; (ii) entanglement between component quantum programs; and (iii) combining quantum predicates in the overlap of state Hilbert spaces of component quantum programs with shared variables. Applications of the techniques developed in this paper are illustrated by a formal verification of Bravyi-Gosset-K\"onig's parallel quantum algorithm solving a linear algebra problem, which gives for the first time an unconditional proof of a computational quantum advantage.Comment: Added an application on formal verification of Bravyi-Gosset-K\"onig's algorith

Quantum Temporal Logic

In this paper, we introduce a model of quantum concurrent program, which can be used to model the behaviour of reactive quantum systems and to design quantum compilers. We investigate quantum temporal logic, QTL, for the specification of quantum concurrent systems by suggesting the time-dependence of events. QTL employs the projections on subspaces as atomic propositions, which was established in the Birkhoff and von Neumann's classic treatise on quantum logic. For deterministic functional quantum program, We prove a quantum B\"{o}hm-Jacopini theorem which states that any such program is equivalent to a Q-While program. The decidability of basic QTL formulae for general quantum concurrent program is studied