Repeated Syndrome Extraction in 3D
Because measurements themselves fault, syndromes are extracted over many rounds and decoded as a three-dimensional spacetime history.
Why one round is not enough
A single round of syndrome extraction cannot be trusted, because the measurement of a stabilizer can itself return the wrong value. If a decoder acted on one faulty round, it would apply a correction that creates a real error where there was none. The remedy is to repeat syndrome extraction many times and compare rounds to separate true data errors from measurement faults.
The spacetime lattice
Stacking successive rounds turns the decoding problem into three dimensions: two of space (the code lattice) and one of time (the round index). A data error persists across rounds and shows up as a vertical line of consistent syndrome changes; a measurement fault appears as an isolated blip that flips a syndrome in one round and flips back the next. Decoders match defects in this 3D volume.
- Measurement faults create transient, single-round syndrome blips.
- Data errors create persistent syndrome changes across rounds.
- Decoding matches defects in a 2D-space-plus-1D-time volume.
- About d rounds are needed to protect a distance-d code's readout.
The number of rounds matters. To keep the effective distance equal to the code distance d, roughly d rounds of syndrome extraction are performed between logical operations, so that a chain of measurement faults long enough to fool the decoder is as unlikely as a data error chain of length d. This is why time is treated on the same footing as space.
The 3D picture is what sliding-window decoding operates on, decoding a moving slab of rounds. It also explains why the circuit-level threshold is lower than the code-capacity one: every round adds a layer of potentially faulty measurements to the volume the decoder must get right.