Concatenated Steane Code with Flags

March 18, 2024

The Steane code is one of the most studied quantum error-correcting codes, but it still has aspects worth exploring. When concatenated with itself, the 7-qubit Steane code becomes a 49-qubit code, where instead of the seven physical qubits, 7 Steane code logical qubits are connected. We show that these 49 qubits can be embedded into the 2D plane, making this construction more viable for quantum computing architectures limited to planar configurations. We furthermore show that it is possible to use this code with as few as two extra ancillary qubits to preserve the distance of this code under circuit-level noise. This required the careful ordering of the gates during the error correction cycle and a lookup table-based decoder, which we developed in a recently published work. Under our circuit-level noise model, surprisingly, the Steane code, even though it requires the extraction of the eigenvalue of weight-12 Pauli operators, outperforms the 6.6.6 color code, which only requires the measurement of weight-6 Pauli operators.  We also prove that the 4.8.8 code needs more than two ancilla qubits to preserve the underlying distance.