Routed quantum circuits

TL;DR

This paper introduces routed quantum circuits, an extended framework that captures complex subsystem combinations in quantum theory, providing a consistent diagrammatic representation for pure and mixed states.

Contribution

It proposes routed linear maps and circuits, extending standard quantum circuit models to handle nontrivial subsystem blends with a sound semantics.

Findings

Enables diagrammatic representation of superpositions of quantum channels

Covers causal decompositions of unitaries

Includes extended circuit diagrams as a special case

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.