Simple Construction of Qudit Floquet Codes on a Family of Lattices

Dynamical quantum error-correcting codes (QECC) offer wider possibilities in how one can protect logical quantum information from noise and perform fault-tolerant quantum computation compared to static QECCs. A family of dynamical QECCs called the ``Floquet codes'' consists of a periodic sequence of two-body measurements that enables error-correction on many-body systems, relaxing hardware implementation requirements and improving error-correction reliability. Existing results on Floquet codes has been focused on qubits, two-level quantum systems, with very little attention given on higher dimensional quantum systems, or qudits. We bridge this gap by proposing a simple, yet general construction of qudit Floquet codes based on a simple set of conditions on the sequence two-body measurements defining the code. Moreover, this construction applies to a large family of configurations of qudits on the vertices of a three-colorable lattice which connectivity represented by the edges. We show that this construction includes the existing constructions of both qubit and qudit Floquet codes as special cases. In addition, any qudit Floquet code obtained by our construction achieves a rate of encoded logical qudits over physical qudits approaching $\frac{1}{2}$ as the number of physical qudits in total and on the faces of the lattice grows larger, as opposed to vanishing rate in existing qudit Floquet code constructions.