Error Models
Error models describe how noise acts on qubits, from simple independent Pauli flips to full circuit-level noise, setting what a code must correct.
Why models matter
A code's performance is only meaningful relative to a noise model. The same code can have very different thresholds depending on whether one assumes idealized single-qubit flips or realistic noise on every gate, measurement, and idle step. Choosing an honest model is essential to evaluating whether error correction will help on a given machine.
A ladder of models
Models range from simple to realistic. The code-capacity model applies independent Pauli errors to data qubits only and assumes perfect syndrome measurement; it is the easiest to analyze and gives the most optimistic thresholds. The phenomenological model adds measurement errors, so syndromes must be repeated in time. The circuit-level model is the realistic one: every gate, reset, idle, and measurement can fail, and two-qubit gates inject correlated errors on both qubits. Thresholds quoted for real machines should be circuit-level.
- Code-capacity: data errors only, perfect measurement (optimistic).
- Phenomenological: adds measurement errors, needs repeated rounds.
- Circuit-level: every operation faulty, gates spread errors (realistic).
- Independent Pauli noise is a common but imperfect approximation.
Common channels
Within these models, specific channels describe the physics. The depolarizing channel applies X, Y, Z with equal probability and is the standard benchmark. Biased noise, where Z errors dominate because dephasing is faster than relaxation, motivates codes tailored to that bias. Amplitude damping models energy loss (T1). Leakage, where a qubit escapes the computational subspace, and correlated or non-Markovian noise are harder to model but real. Because of the discretization of errors, a code that corrects the Pauli basis handles any of these single-qubit channels.
A useful simplification is Pauli twirling, which converts complicated coherent noise into an effective Pauli channel that codes and decoders are designed to handle. Realistic evaluation of any code in this category rests on stating its noise model explicitly.