Continuous Indirect detection of stabilizers for quantum error correction

Measuring high-weight operators is an important problem in quantum computation. The conventional procedure to measure a high-weight operator involves multiple pairwise unitary operations, which may require a large number of quantum gates. We provide an alternative method to passively detect the value of an operator. This approach involves joint interactions between the system and continuously-monitored ancillary qubits. The continuous measurement outcomes of the monitor qubits reveal information about the values of the stabilizer generators. This information can be retrieved using an estimator, which is driven by the measurement outcomes. We also show that there is a more efficient way to read out the outcomes directly from the time average of the signals. The interaction Hamiltonian can use only two-local operators, based on techniques similar to perturbative gadgets. We apply this indirect detection scheme to the four-qubit Bacon-Shor code, where the two stabilizers are indirectly monitored using four ancillary qubits. Since it is an error-detecting code, we show that non-Markovian errors, e.g., the Hamiltonian 1/f noise, can be suppressed by the indirect detection. This detection scheme could be implemented in near-term experimental systems and operate in real time.