Fault-tolerant quantum computing with the parity code and biased-noise qubits

We present a fault-tolerant universal quantum computing architecture based on a code concatenation of biased-noise qubits and the parity architecture. The parity architecture can be understood as a low-density parity-check code tailored specifically to obtain any desired logical connectivity from nearest-neighbor physical interactions. The code layout can be dynamically adjusted to algorithmic requirements on the fly. This allows for implementations with any desired code distance with a universal set of fault-tolerant gates. In addition to the previously explored toolsets for concatenated cat codes, our approach features parallelizable interactions between arbitrary sets of qubits by directly addressing the parity qubits in the code. The proposed scheme supports codes with lower physical qubit overhead than the repetition code at the same code distances, while requiring only weight-3 and weight-4 stabilizers and nearest-neighbor connectivity on a two-dimensional square lattice.