  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.
