Error Models

To design and benchmark codes we need tractable mathematical models of noise. The workhorse is the Pauli error model, in which errors are random Pauli operators drawn from a probability distribution. Thanks to the discretization theorem, this captures the worst of physical noise while remaining simple enough to analyse and simulate.

The depolarizing channel

The most-used single-qubit model is the depolarizing channel, which applies $X$ , $Y$ , or $Z$ each with probability $p/3$ and leaves the qubit alone with probability $1-p$ :

\mathcal{D}_p(\rho) = (1-p)\,\rho + \frac{p}{3}\big(X\rho X + Y\rho Y + Z\rho Z\big).

It is the maximally symmetric noise — no Pauli is preferred — and equivalently shrinks the Bloch vector uniformly toward the centre, $\vec{r} \mapsto (1-\tfrac{4p}{3})\,\vec{r}$ . For an $n$ -qubit register one usually assumes the channel acts independently and identically on each qubit, so a weight- $w$ Pauli error occurs with probability $\sim (p/3)^w (1-p)^{n-w}$ times a multiplicity.

Biased and structured noise

Real hardware is rarely symmetric. The dephasing channel applies only $Z$ ,

\mathcal{Z}_p(\rho) = (1-p)\,\rho + p\, Z\rho Z,

modelling the common case where phase coherence decays much faster than population (the $T_2 \ll T_1$ regime). Many platforms have biased noise with $Z$ errors far more likely than $X$ ; tailoring a code to a bias (e.g. the XZZX surface code) can dramatically raise thresholds. The general Pauli channel assigns an independent probability $p_P$ to each Pauli string $P$ .

Coherent vs. incoherent errors

A subtlety: a small coherent over-rotation $R_z(\theta)$ is not the same as an incoherent dephasing of equal average fidelity. Coherent errors can add in amplitude across many gates, accumulating as $\theta$ rather than $\theta^2$ , and are harder to suppress. Randomized compiling and Pauli twirling deliberately convert coherent errors into stochastic Pauli errors so the simple models — and the thresholds derived from them — actually apply.

Amplitude damping: a non-Pauli model

Energy loss $|1\rangle \to |0\rangle$ is described by amplitude damping, with Kraus operators

K_0 = \begin{pmatrix}1 & 0\\ 0 & \sqrt{1-\gamma}\end{pmatrix}, \qquad K_1 = \begin{pmatrix}0 & \sqrt{\gamma}\\ 0 & 0\end{pmatrix},

where $\gamma = 1 - e^{-t/T_1}$ is the decay probability over time $t$ . This is not a Pauli channel ( $K_1$ is not a Pauli), but each $K_i$ still expands in the Pauli basis, so a distance-3 code that corrects weight-1 Paulis suppresses amplitude damping to leading order — the bridge to the approximate-QEC discussion later in the module.

The takeaway

The standard noise models are Pauli channels — depolarizing (symmetric), dephasing, and biased — in which errors are random Paulis applied i.i.d. per qubit. Twirling justifies treating general noise as Pauli noise for the purpose of code design, while genuinely non-Pauli processes like amplitude damping are still controlled to leading order through their Pauli expansion.