Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes. (arXiv:1803.09758v2 [quant-ph] UPDATED)
Flag qubits have recently been proposed in syndrome extraction circuits to
detect high-weight errors arising from fewer faults. The use of flag qubits
allows the construction of fault-tolerant protocols with the fewest number of
ancillas known to-date. In this work, we prove some critical properties of CSS
codes constructed from classical cyclic codes that enable the construction of a
flag fault-tolerant error correction scheme. We then develop fault-tolerant
protocols as well as a family of circuits for flag fault-tolerant error
correction and operator measurement, requiring only four ancilla qubits and
applicable to distance-three CSS codes constructed from classical cyclic codes.
The measurement protocol can be further used for logical Clifford gate
implementation via quantum gate teleportation. We also provide examples of
cyclic CSS codes with large encoding rates.