<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mi>XYZ</mml:mi></mml:math> Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime

Analyzing and developing new quantum error-correcting (QEC) 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 functioning of the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ . Based on the projective symmetries of the building blocks of the tensor network we develop a framework of . 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-based 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>-dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:msub><d:mrow><d:mi mathvariant="double-struck">Z</d:mi></d:mrow><d:mn>3</d:mn></d:msub></d:math> automorphism to the logical Pauli group every three time steps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:mi>XYZ</h:mi></h:math> ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of approximately equal to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:mn>0.18</k:mn><k:mi mathvariant="normal">%</k:mi></k:math>, comparable with other Floquet codes and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><o:mn>2</o:mn><o:mo>+</o:mo><o:mn>1</o:mn></o:math>-dimensional color codes.