We introduce a flow condition on open graph states (graph states with inputs and outputs) which guarantees globally deterministic behavior of a class of measurement patterns defined over them. Dependent Pauli corrections are derived for all such patterns, which equalize all computation branches, and only depend on the underlying entanglement graph and its choice of inputs and outputs. The class of patterns having flow is stable under composition and tensorization, and has unitary embeddings as realizations. The restricted class of patterns having both flow and reverse flow supports an operation of adjunction, and has all and only unitaries as realizations.

• Received 22 January 2006

DOI:https://doi.org/10.1103/PhysRevA.74.052310