PBS-calculus: A Graphical Language for Quantum-Controlled Computations

We introduce the PBS-calculus to represent and reason on quantum computations involving polarising beam splitters (PBS for short). PBS-diagrams can be used to represent various schemes including quantum-controlled computations, which are known to have multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is equipped with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we show that any PBS-diagram involving only unitary matrices can be transformed into a diagram without feedback loop.