  We present a fault-tolerant universal quantum computing architecture based on
a code concatenation of noise-biased qubits and the parity architecture. The
parity architecture can be understood as a LDPC 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 reaching arbitrary code distances and
thereby exponential suppression of errors with a universal set of
fault-tolerant gates. In addition to the previously explored tool-sets 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 enables codes with less physical qubit overhead
compared to the repetition code with the same code distances, while requiring
only weight-3 and weight-4 stabilizers and nearest neighbor 2D square-lattice
connectivity.
