Routed quantum circuits

Augustin Vanrietvelde1,2,3, Hlér Kristjánsson1,3, and Jonathan Barrett1

1Quantum Group, Department of Computer Science, University of Oxford
2Department of Physics, Imperial College London
3HKU-Oxford Joint Laboratory for Quantum Information and Computation

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Abstract

We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by $\textit{routed linear maps}$ and $\textit{routed quantum circuits}$. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

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Cited by

[1] Robin Lorenz and Jonathan Barrett, "Causal and compositional structure of unitary transformations", arXiv:2001.07774, Quantum 5, 511 (2021).

[2] Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum circuits with classical versus quantum control of causal order", arXiv:2101.08796.

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