Fault-Tolerant Quantum Computing with the Parity Code: Discrete and Bosonic Concatenations

We present a fault-tolerant quantum computing architecture based on a code concatenation of the parity code. In addition to the recently explored concatenation with bosonic (i.e. continuous-variable) qubits, we propose and study a concatenation with discrete-variable stabilizer codes. The parity code can be understood as a low-density-parity-check (LDPC) code tailored specifically to obtain any desired logical connectivity from nearest neighbor physical interactions. On the bosonic side, the proposed scheme enables codes with better encoding rate 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. On the discrete-variable side, the proposed scheme enables Calderbank-Shor-Steane (CSS) codes with less physical qubits overhead compared to the surface code but they do not form a quantum LDPC family.