Yichen Xu

We introduce a framework called spacetime concatenation for fault-tolerant compilation of syn- drome extraction circuits of stabilizer codes. This framework enables efficient compilation of syn-drome extraction circuits into dynamical codes through structured gadget layouts and encoding matrices, facilitating low-weight measurements while preserving logical information. This frame-work uses conditions that are sufficient for fault-tolerance of the dynamical code, including not measuring logical operators and preserving the spacetime distance. This framework goes beyond dynamical weight-reduction methods, which do not explicitly produce Floquet codes with nontrivial logical automorphisms. Using this fault-tolerant framework, we reproduce known Floquet codes such as the hon-eycomb Floquet code and the ruby lattice Floquet color code with nontrivial automorphisms and a fault-tolerant planar variant of the Floquet toric code. We also construct new explicit examples of dynamical codes, including a dynamical bivariate bicycle code and a dynamical Haah code. Be-yond constructing examples, we write a restricted equivalence relation which can be used to discuss classification and resource trade-offs of dynamical codes. Lastly, we demonstrate the adaptability of our framework to arbitrary qubit layouts and fabrication defects, making it well-suited for a hardware-first design approach.