Decoding and measurement-error models

Encode a check eigenvalue by a binary bit s, with zero for positive and one for negative. Let its measured bit be $m_t=s_t\oplus e_t$, where $e_t$ denotes a readout flip. Compare consecutive measurements through their exclusive-or. If the underlying check is initially stable, one isolated wrong measurement creates two consecutive changes: one on entering the wrong value and another on returning. A persistent data error instead changes the true syndrome until another data event or correction occurs.

The first comparison needs an initial value or boundary assumption, and the last needs a final measurement or an explicit temporal boundary. Ignoring these endpoints can turn a valid history into an apparent unmatched event. In a spatial code, data faults also connect neighboring checks. The decoder therefore works in a graph with spatial and temporal directions: different edge types represent different hypothesized faults, rather than literal trajectories of a measured quantum particle.

For independent binary fault locations with probabilities p, a candidate history has a product of fault and no-fault probabilities. Relative to the no-fault baseline, each selected location contributes odds $p/(1-p)$. Taking negative logarithms converts their product into a sum of edge weights. For probabilities below one half, these weights are positive, making minimum-weight search natural. Different data and measurement probabilities imply different weights; the shortest geometric chain need not be most probable.

Choosing the single most likely history is not always the same as choosing the most likely logical class. Many distinct histories can differ by stabilizers while producing the same logical action. A maximum-likelihood logical decoder sums probabilities within each class before choosing. Minimum-weight matching is an important approximation for suitable graphlike noise models, not a universal Bayesian solver. Correlated multi-qubit faults and biased or coherent noise can require different models, weights, or inference algorithms.

Repeated readout can suppress an isolated measurement error when the underlying parity is fixed during the votes. Three independent flips with the same probability produce an incorrect majority only if at least two occur. That calculation is useful for a stationary calibration check. During active computation, however, real data errors may change the parity between rounds, and blindly voting across those changes can erase the signal the decoder needs.

Correlated readout faults can also defeat the apparent improvement: if a common disturbance flips all three outcomes together, their majority is wrong whenever that disturbance occurs. A faster decoder is useful only if its assumed graph captures relevant faults and its processing keeps pace with syndrome production. Recovery may be tracked as a Pauli frame instead of immediately applied, but later gates and measurements must interpret that frame consistently. Classical inference never reconstructs or measures the unknown logical amplitudes.