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@ARTICLE{delaFuente:1038557,
      author       = {de la Fuente, Julio C. Magdalena and Old, Josias and
                      Townsend-Teague, Alex and Rispler, Manuel and Eisert, Jens
                      and Müller, Markus},
      title        = {{T}he {XYZ} ruby code: {M}aking a case for a three-colored
                      graphical calculus for quantum error correction in
                      spacetime},
      reportid     = {FZJ-2025-01540, arXiv:2407.08566},
      year         = {2025},
      note         = {59 pages, 26 figures},
      abstract     = {Analyzing and developing new quantum error-correcting
                      schemes is one of the most prominent tasks in quantum
                      computing research. In such efforts, introducing time
                      dynamics explicitly in both analysis and design of
                      error-correcting protocols constitutes an important
                      cornerstone. In this work, we present a graphical formalism
                      based on tensor networks to capture the logical action and
                      error-correcting capabilities of any Clifford circuit with
                      Pauli measurements. We showcase the formalism on new Floquet
                      codes derived from topological subsystem codes, which we
                      call XYZ ruby codes. Based on the projective symmetries of
                      the building blocks of the tensor network we develop a
                      framework of Pauli flows. Pauli flows allow for a graphical
                      understanding of all quantities entering an error correction
                      analysis of a circuit, including different types of QEC
                      experiments, such as memory and stability experiments. We
                      lay out how to derive a well-defined decoding problem from
                      the tensor network representation of a protocol and its
                      Pauli flows alone, independent of any stabilizer code or
                      fixed circuit. Importantly, this framework applies to all
                      Clifford protocols and encompasses both measurement- and
                      circuit-based approaches to fault tolerance. We apply our
                      method to our new family of dynamical codes which are in the
                      same topological phase as the 2+1d color code, making them a
                      promising candidate for low-overhead logical gates. In
                      contrast to its static counterpart, the dynamical protocol
                      applies a Z3 automorphism to the logical Pauli group every
                      three timesteps. We highlight some of its topological
                      properties and comment on the anyon physics behind a planar
                      layout. Lastly, we benchmark the performance of the XYZ ruby
                      code on a torus by performing both memory and stability
                      experiments and find competitive circuit-level noise
                      thresholds of $0.18\%,$ comparable with other Floquet codes
                      and 2+1d color codes.},
      cin          = {PGI-2},
      cid          = {I:(DE-Juel1)PGI-2-20110106},
      pnm          = {5221 - Advanced Solid-State Qubits and Qubit Systems
                      (POF4-522)},
      pid          = {G:(DE-HGF)POF4-5221},
      typ          = {PUB:(DE-HGF)25},
      eprint       = {2407.08566},
      howpublished = {arXiv:2407.08566},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:2407.08566;\%\%$},
      url          = {https://juser.fz-juelich.de/record/1038557},
}