Syndrome measurement strategies for the [[7, 1, 3]] code

Quantum error correction requires the measurement of error syndromes to properly locate and identify errors. Here we compare three syndrome measurement strategies for the [[7,1,3]] quantum error correction code: Shor states, Steane states, and one ancilla qubit. The first two of these strategies are fault tolerant while the third is not. For each strategy we compare the fidelities of applying 50 logical gates with quantum error correction applied at different intervals. We then compare the fidelities for the different syndrome measurement strategies. Our simulations show that the optimal syndrome measurment strategy depends on the details of the error environment. The simulations thus allow a quantum computer programmer to weigh computational accuracy versus resource consumption for a particular error environment. In addition, we show that applying syndrome measurment 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 show that the non-fault tolerant syndrome measurement strategy gives comparable accuracy results with those that are fault tolerant.