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We introduce the PBS-calculus to represent and reason on quantum computationsinvolving coherent control of quantum operations. Coherent control, and inparticular indefinite causal order, is known to enable multiple computationaland communication advantages over classically ordered models like quantumcircuits. The PBS-calculus is inspired by quantum optics, in particular thepolarising beam splitter (PBS for short). We formalise the syntax and thesemantics of the PBS-diagrams, and we equip the language with an equationaltheory, which is proved to be sound and complete: two diagrams are representingthe same quantum evolution if and only if one can be transformed into the otherusing the rules of the PBS-calculus. Moreover, we show that the equationaltheory is minimal. Finally, we consider applications like the implementation ofcontrolled permutations and the unrolling of loops.

PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations