Fault tolerance and threshold assumptions

A controlled-NOT propagates an X fault on its control to both control and target, whereas X on the target remains on the target. A Z fault on the target propagates backward to both qubits, whereas Z on the control stays there. These identities follow by conjugating Paulis through the ideal gate. They describe faults placed before the gate and moved to an equivalent location after it; reversing that convention without care produces incorrect propagation diagrams.

An ancilla reused to collect a parity can therefore pass one fault to several data qubits. A distance-three data code corrects one data error, but it need not tolerate one arbitrary fault in an unprotected extraction circuit. Verified ancillas, flagged circuits, appropriate interaction ordering, repeated checks, and other constructions address this risk under their particular fault models. The relevant property concerns the whole gadget, including preexisting errors, rather than the nominal weight of the parity it measures.

Suppose a verified distance-three construction makes every gadget with at most one fault behave correctly in the required sense. In an independent stochastic model, failure then needs at least a harmful pair of faults. Counting these pairs and bounding higher-order terms can yield a recurrence in which the next encoded error strength is at most A times the square of the previous one. A includes circuit structure; it cannot be inferred from data distance alone.

This illustrative recurrence contracts whenever its input lies below one over A. Iterating gives rapid suppression, but only if the same valid bound applies at every level. Overlapping gadgets, propagation between blocks, state preparation, measurement, and idle locations must be treated consistently in a real proof. Coherent local-noise theorems may instead bound operator amplitudes, not stochastic probabilities. Substituting one number into the other kind of bound without translation changes the physical statement.

A computation with N logical locations can fail at any of them. If each location has error probability at most the chosen encoded bound, the union bound limits total failure by their sum; independence of logical failures is not required for that step. A small per-gate error is therefore meaningful only relative to circuit size and an acceptable overall failure probability. Increasing code size or concatenation level consumes additional qubits, gates, measurements, and classical processing.

Threshold statements also assume access to adequately prepared ancillas, fault handling at the relevant spatial scale, and decoding or feed-forward within the architecture's timing budget. Long-range correlated disturbances, uncontrolled leakage, or drift can violate a model based on local independent faults. Some broader noise models admit different theorems, but this requires their stated bounds. A hardware report must therefore specify the noise model, circuit family, decoder, and logical task before comparing its rates with any threshold estimate.