Injection is all you need: parallel universal logic in high-rate concatenated self-dual codes

We propose a quantum architecture that achieves addressable and parallel universal fault-tolerant logical in high-rate codes. The codes are obtained from the flexible concatenation of small-distance high-rate self-dual CSS codes. Addressable and parallel universal logic on the resulting K logical qubits is injected by fault-tolerantly preparing and then consuming diverse logical K-qubit CSS stabilizer states and logical H K global magic states. Flag-at-origin is used to prepare the stabilizer resource states while a concatenated cultivation protocol prepares the global magic states. These resource states are consumed by Steane-QEC or Knill-QEC teleportation gadgets, which simultaneously perform quantum error correction. Any quantum algorithm with T-depth D T can be executed by consuming 12D T or fewer resource states. We benchmark the fault-tolerant preparation of the resource states for several two-level concatenations of the [[k+4,k,2]] H codes and the Hamming [[15,7,3]] code. In particular, we find that the concatenated cultivation offers at least one order of magnitude spacetime reduction compared to cultivation in the surface code.