XYZ Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime

PRX Quantum, vol. 6

Abstract

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 2 + 1 -dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a Z 3 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 XYZ ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of approximately equal to 0.18 % , comparable with other Floquet codes and 2 + 1 -dimensional color codes.

Authors 6

1. Julio C. Magdalena de la Fuente corresponding

   Freie Universität Berlin

   Affiliation as printed

   Freie Universität Berlin

2. Josias Old Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Forschungszentrum Jülich

   RWTH Aachen University

3. Alex Townsend-Teague

   Freie Universität Berlin

   Affiliation as printed

   Freie Universität Berlin

4. Manuel Rispler Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Forschungszentrum Jülich

   RWTH Aachen University

5. Jens Eisert

   Helmholtz-Zentrum Berlin für Materialien und Energie · Freie Universität Berlin

   Affiliation as printed

   Freie Universität Berlin

   Helmholtz-Zentrum Berlin für Materialien und Energie

6. Markus Michael Müller Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Forschungszentrum Jülich

   RWTH Aachen University