Clifford Manipulations of Stabilizer States: A graphical rule book for Clifford unitaries and measurements on cluster states, and application to photonic quantum computing

Stabilizer states along with Clifford manipulations (unitary transformations and measurements) thereof -- despite being efficiently simulable on a classical computer -- are an important tool in quantum information processing, with applications to quantum computing, error correction and networking. Cluster states, defined on a graph, are a special class of stabilizer states that are central to measurement based quantum computing, all-photonic quantum repeaters, distributed quantum computing, and entanglement distribution in a network. All cluster states are local-Clifford equivalent to a stabilizer state. In this paper, we review the stabilizer framework, and extend it, by: incorporating general stabilizer measurements such as multi-qubit fusions, and providing an explicit procedure -- using Karnaugh maps from Boolean algebra -- for converting arbitrary stabilizer gates into tableau operations of the CHP formalism for efficient stabilizer manipulations. Using these tools, we develop a graphical rule-book and a MATLAB simulator with a graphical user interface for arbitrary stabilizer manipulations of cluster states, a user of which, e.g., for research in quantum networks, will not require any background in quantum information or the stabilizer framework. We extend our graphical rule-book to include dual-rail photonic-qubit cluster state manipulations with probabilistically-heralded linear-optical circuits for various rotated Bell measurements, i.e., fusions (including new `Type-I' fusions we propose, where only one of the two fused qubits is destructively measured), by incorporating graphical rules for their success and failure modes. Finally, we show how stabilizer descriptions of multi-qubit fusions can be mapped to linear optical circuits.