Syndrome measurement order for the [[7, 1, 3]] quantum error correction code

The [[7,1,3]] quantum error correction code uses sets of three syndrome measurements to separately detect bit-flip and phase-flip errors. To comply with the strictures of fault tolerance and thus stem the possible spread of errors to multiple qubits, each set of syndromes is repeated twice. Still, there remains flexibility in the order in which the sets of syndromes are implemented. Here we explore different orders noting that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on whether or not an error is detected. To put these results in proper context we first explore the effect on output state fidelity of detecting an error. We find that this fidelity is significantly lower than when no error is detected and may decrease when the single qubit error probabilities decrease.