Homotopic Concurrency in Hilbert Space, Quantum Higher-Dimensional Automata for Entangled Processes

Standard quantum circuit descriptions represent processes by linear wiring and sequential composition, while concurrency is typically reduced to commutation checks and tensor factorisation. This manuscript introduces Quantum Higher-Dimensional Automata, a geometric model in which executions are directed paths on a cubical complex and quantum coherence is carried by structured projective unitary data on cells. The construction defines a quantum cubical set whose vertices represent configurations and whose higher cells represent coherent multi-parameter evolutions that witness compatibility of concurrent actions. Within this framework, compatibility is promoted from an algebraic condition on operators to a directed homotopy invariant condition on executions, and geometric phase is interpreted as curvature on execution surfaces when smooth lifts exist. The model yields a homotopy based compatibility criterion and supplies obstruction based notions of deadlock grounded in phase holonomy on cubical loops. The aim is an operational semantics in which concrete families of quantum controls induce computable invariants on an explicit concurrency geometry.