Channel and Noise Model Table

A noise model specifies an input-output map over a stated interval and system space. It does not by itself identify a microscopic mechanism or certify that noise is independent across gates. This lookup collects exact completely positive trace-preserving (CPTP) models. Common Noise Models explains how to select a model, and Kraus, Choi, and Stinespring Representations explains the representations.

Qubit Kraus models

Write $\Phi(\rho)=\sum_k K_k\rho K_k^\dagger$, with $\sum_kK_k^\dagger K_k=I$. The computational basis is $(\lvert0\rangle,\lvert1\rangle)$, and the Bloch vector satisfies $\rho=(I+xX+yY+zZ)/2$. Kraus lists describe the unconditional map; a particular list is not unique.

Model	Kraus operators and interpretation
Unitary error	$K_0=e^{-i\theta Z/2}$ Real angle θ; coherent Z overrotation, not a random phase flip.
Bit flip	$K_0=\sqrt{1-p}\,I$ $K_1=\sqrt p\,X$ $0\leq p\leq1$; probability of an X event in this representation.
Phase flip	$K_0=\sqrt{1-p}\,I$ $K_1=\sqrt p\,Z$ $0\leq p\leq1$; off-diagonal entries multiply by $1-2p$.
Pauli channel	$K_P=\sqrt{p_P}\,P$ $P\in\{I,X,Y,Z\}$ $p_P\geq0$, $\sum_Pp_P=1$; effective stochastic Pauli model.
Depolarizing, Pauli convention	$K_0=\sqrt{1-p}\,I$ $K_j=\sqrt{p/3}\,\sigma_j$ $0\leq p\leq1$; total nonidentity Pauli probability is p.
Amplitude damping	$K_0=\operatorname{diag}(1,\sqrt{1-\gamma})$ $K_1=\sqrt\gamma\,\lvert0\rangle\langle1\rvert$ $0\leq\gamma\leq1$; relaxation toward the declared ground state.
Dephasing, coherence convention	$K_0=\sqrt{(1+\eta)/2}\,I$ $K_1=\sqrt{(1-\eta)/2}\,Z$ $-1\leq\eta\leq1$ is CPTP; monotone Markovian decay uses $0\leq\eta\leq1$.

These effective channels occur across platforms. Relaxation is relevant to excited-state population loss; dephasing is relevant to uncontrolled relative phase; a Pauli approximation can simplify stabilizer simulations. A platform label is insufficient to select parameters, temporal correlations, or a particular effective model. See Noise in Quantum Information.

Parameter translation and Bloch checks

The depolarizing row has the alternative representation

$$\Phi_p(\rho)=\left(1-\frac{4p}{3}\right)\rho +\frac{4p}{3}\frac I2.$$

If $q$ denotes replacement weight in $(1-q)\rho+qI/2$, then $q=4p/3$. Complete depolarization is $p=3/4$, or $q=1$. The full CPTP range allows $q\leq4/3$, but values above one are not a convex probabilistic replacement mixture. At $p=1$ the map shrinks and reverses the Bloch vector by $-1/3$; it is not complete depolarization. See Pauli Noise and Depolarizing Channels.

The coherence convention gives

$$(x,y,z)\longmapsto(\eta x,\eta y,z), \qquad p=\frac{1-\eta}{2}.$$

Thus phase-flip probability $p=1/2$ completely removes coherence, while $p=1$ is the reversible unitary Z. A probability increasing all the way to one is not a monotonic measure of dephasing strength.

Amplitude damping acts as

$$\begin{aligned} x&\longmapsto\sqrt{1-\gamma}\,x,\\ y&\longmapsto\sqrt{1-\gamma}\,y,\\ z&\longmapsto(1-\gamma)z+\gamma. \end{aligned}$$

The translation toward $z=1$ makes this channel nonunital. In an exponential zero-temperature relaxation model, $\gamma=1-e^{-t/T_1}$. With additional independent exponential pure dephasing, $\eta=e^{-t/T_\phi}$ multiplies transverse components, giving $T_2^{-1}=(2T_1)^{-1}+T_\phi^{-1}$. These time-constant relations require that model; they are not identities for arbitrary colored noise. See Dephasing and Amplitude Damping.

Erasure, leakage, and conditioning

A flagged erasure channel enlarges the output space:

$$\mathcal E_\epsilon(\rho)=(1-\epsilon)\rho +\epsilon\lvert e\rangle\langle e\rvert, \qquad0\leq\epsilon\leq1,$$

where $\lvert e\rangle$ is orthogonal to the entire input computational subspace. A Kraus list is $\sqrt{1-\epsilon}\,V$ together with $\sqrt\epsilon\lvert e\rangle\langle j\rvert$ for each input basis vector, where V embeds the input in the output. Leakage is not automatically flagged erasure: an experiment needs a reliable flag and must account for false flags and missed events. See Leakage and Crosstalk.

For a quick check, apply amplitude damping to $\lvert1\rangle\langle1\rvert$. The output is $(1-\gamma)\lvert1\rangle\langle1\rvert+\gamma\lvert0\rangle\langle0\rvert$, with unit trace. Keeping only the no-jump Kraus outcome yields a trace-nonincreasing branch; renormalizing it describes a conditional state, not the original unconditional channel.

References

• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010, doi:10.1017/CBO9780511976667 — standard qubit noise channels.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, author manuscript and errata — CPTP maps, representations, and channel distinctions.