  Quantum error correction (QEC) entails the encoding of quantum information
into a QEC code space, measuring error syndromes to properly locate and
identify errors, and, if necessary, applying a proper recovery operation. Here
we compare three syndrome measurement protocols for the [[7,1,3]] QEC code:
Shor states, Steane states, and one ancilla qubit by simulating the
implementation of 50 logical gates with the syndrome measurements interspersed
between the gates at different intervals. We then compare the fidelities for
the different syndrome measurement types. Our simulations show that the optimal
syndrome measurement strategy is generally not to apply syndrome measurements
after every gate but depends on the details of the error environment. Our
simulations also allow a quantum computer programmer to weigh computational
accuracy versus resource consumption (time and number of qubits) for a
particular error environment. In addition, we show that applying syndrome
measurements that are unnecessary from the standpoint of quantum fault
tolerance may be helpful in achieving better accuracy or in lowering resource
consumption. Finally, our simulations demonstrate that the single-qubit
non-fault tolerant syndrome measurement strategy achieves comparable fidelity
to those that are fault tolerant.
