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I would like to check I precisely understand the definition of what a detector does in Stim. Below is my understanding and I am looking for confirmation that I understood correctly. If not, please correct me.

My current understanding of how detector work by definition:

I call $\{x_i\}_i$ a family of qubit measurement outcomes. We have $x_i \in \{0,1\}$.

A detector takes as input a family of measurements $\{x_i \}_{i=1}^N$.

Then, one considers two circuits: one with the noise "off". In this case, the family of measurement outcomes will be $\{x^{\text{no noise}}_i \}_{i=1}^N$.

The other family is with the noise "on". In this case, the family of measurement outcomes will be $\{x^{\text{noise}}_i \}_{i=1}^N$.

If the sum (modulo 2) $x^{\text{no noise}}_1 \oplus x^{\text{no noise}}_2 \oplus ... \oplus x^{\text{no noise}}_N=x^{\text{noise}}_1 \oplus x^{\text{noise}}_2 \oplus ... \oplus x^{\text{noise}}_N$, then, the detector returns FALSE. If not, it returns TRUE.

To give an example, the command DETECTOR rec[-1] rec[-2] rec[-3] (i) computes the sum of the last 3 measurements for a noiseless circuit run and compares it to the sum of the measurements of the noisy circuit run. It returns FALSE if they agree, TRUE if they disagree.

Am I correct?

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Yes, this is a good definition of a detector. Note that the binary sum of a set of measurement outcomes needs to be deterministic to be able to use it as a detector. For example, a single measurement outcome from measuring a qubit in the state $|0>$ in the + basis can't be used as a detector. But if the measurement is done in the computational basis one can define a detector: $m_1 = 0$.

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    $\begingroup$ Great thanks. Just to be sure: in principle one can define a detector based on non-deterministic measurements (like the measurement you say). However it is just that the outcome of the detector will be useless for practical purposes: one could not really interpret the meaning of FALSE/TRUE because they could not be implied by the noise but rather by the inherent probabilistic nature of measurements. Would you agree? $\endgroup$ Mar 13 at 15:51
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    $\begingroup$ Yes I agree. But I would choose a definition of detector which does not allow for such a choice. $\endgroup$
    – Peter-Jan
    Mar 13 at 15:52
  • $\begingroup$ Technically stim allows non-deterministic detectors; but you need to pass allow_gauge_detectors=True to stim.Circuit.detector_error_model for it to work. What it does, essentially, is turn every anticommutation into a 50% error mechanism by using Gaussian elimination. But I recommend not using this feature. $\endgroup$ Mar 13 at 20:28

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