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-König’s parallel quantum algorithm solving a linear algebra problem, which gives for the first time an unconditional proof of a computational quantum advantage.
See also the published version [Ying2021a]