Detector Error Model

• \(r\) generators and \(|T|\) time snapshots in total, so the syndrome vector's length \(N\) is \(\equiv r|T|\).

  Dimension of syndrome space is \(2^N\), i.e., \(N\) detectors.

• Fault event: faults can cause the detector to flip between "fire" and "no fire" states. (each detector starts in "no fire" state)

  • Fault \(\chi_{1}\) occurs with probability \(p_{1}\), and will cause detector \(D_{1}\) to flip.
  • Fault \(\chi_{12}\) occurs with probability \(p_{12}\), and will cause detector \(D_{1}\) and \(D_{2}\) to flip.
  • Fault \(\chi_{236}\) occurs with probability \(p_{236}\), and will cause detector \(D_{2}\), \(D_{3}\), and \(D_{6}\) to flip.

  How many \(\chi\) faults are there? We can at most identify \(2^N\) faults.

• The conditional probability of detector \(D_{1}\) being in the "fire" state at the end, with all other detectors in the "no fire" state, is given by:

  \[ \rm{Pr}(D_{1} = 1 | D_{j} = 0, j \neq 1) = p_{1} + p_{2} p_{12} + p_{3} p_{13} + \cdots \]

  which contain first-order, second-order, and higher-order combination of \(p\)-terms.

• Estimation task:

  Given the measured \(\{\mathbb{E} [D] \}\), can we infer all the \(p\)-terms?