As an addition to Nat's answer, it's worth mentioning that 'noise' is a specific concept^{1} in quantum computing. This answer will use Preskill's lecture notes as a basis.

In essence, noise is indeed considered to be something that could be described as 'thermal noise', although it should be noted that it is an *interaction* with a thermal environment *causing* noise, as opposed to noise in and of itself. Approximations are made that means this noise can be described using quantum channels, which is what *Nielsen & Chuang* seem to be referring to, as they discuss this in chapter 8.3 of that very textbook. The most common types of noise described in this manner are: depolarising, dephasing and amplitude damping, which will be very briefly explained below.

## In a bit more detail^{2}

Start with a system with Hilbert space $\mathcal{H}_S$, coupled to a (thermal) bath with Hilbert space $\mathcal{H}_B$.

Take the density matrix of the system and 'course grain' it into chunks of $\rho\left(t + n\,\delta t\right)$. Make the assumption that the interaction is Markovian, that is, the environment 'forgets' much quicker than the coarse graining time and that whatever you're trying to observe occurs over a time much longer than the coarse graining time.

Express the density matrix at $t+\delta t$ as a channel acting on the density matrix at time $t$: $\rho\left(t + \delta t\right) = \varepsilon_{\delta t}\left(\rho\left(t\right)\right)$.

Expand this to first order in $\delta t$ to get $\varepsilon_{\delta t} = \mathrm{I} + \delta t\,\mathcal{L}$. As a channel, it must be completely positive and trace preserving, so $\varepsilon_{\delta t}\left(\rho\left(t\right)\right) = \sum_aM_a\rho\left(t\right)M_a^\dagger$ and satisfies $\sum_aM_a^\dagger M_a = \mathrm{I}$.

This gives a non-unitary quantum channel described by the *Lindblad Master equation* $$\dot\rho = -i\left[H, \rho\right] + \sum_{a>0} \gamma_a\left(L_a\rho L_a^\dagger - \frac{1}{2}\lbrace L^\dagger_aL_a, \rho\rbrace\right),$$ where $\gamma_a$'s are always positive for Markovian evolution.

This can also be written as $H_{\mathrm{eff}} = H - \frac{i}{2}\sum_a\gamma_aL_a^{\dagger}L_a$, with an additional term, such that the evolution can be written as $$\dot\rho = -i\left[H_{\text{eff}}, \rho\right] + \sum_{a>0} \gamma_aL_a\rho L_a^\dagger.$$

This now looks equivalent to the Kraus operator representation of a channel, with Kraus operators $K_a \propto L_a$ (as well as an additional Kraus operator to satisfy $\left[H_{\text{eff}}, \rho\right]$). Any non-trivial Lindbladian can then be described as noise, although in reality, it is an approximation of evolution of an open system.

## Some common types of noise^{3}

Trying out various different forms of $L_a$ gives different behaviours
of the system, which give different possible noises, of which there are a few common ones (in the single qubit case, anyway):

**Dephasing**: Causes the system to decohere - this gets rid/reduces the entanglement (i.e. coherence) of the system, necessarily making it more mixed, unless already maximally mixed
$$\varepsilon\left(\rho\right) = \left(1-\frac{p}{2}\right)\rho + \frac{1}{2}\sigma_z\rho\sigma_z$$

**Depolarising**: Upon measuring, either a bit flip ($\sigma_x$), phase flip ($\sigma_z$), or both bit and phase ($\sigma_y$) will have occurred with some probability
$$\varepsilon\left(\rho\right) = \left(1-p\right)\rho + \frac{p}{3}\left(\sigma_x\rho\sigma_x + \sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z\right)$$

**Amplitude Damping**: Represents the system decaying from $\lvert 1\rangle$ to $\lvert 0\rangle$, such as when an atom emits a photon. Leads to a simple version of the coherence times $T_1$ (decay of $\lvert 1\rangle$ to $\lvert 0\rangle$) and $T_2$ (decay of the off-diagonal terms). Described by the Kraus operators $$M_0 = \begin{pmatrix}1 & 0 \\ 0 & \sqrt{1-p}\end{pmatrix} \text{ and } M_1 = \begin{pmatrix}0 & \sqrt{p} \\ 0 & 0\end{pmatrix},$$ giving $$\varepsilon\left(\rho\right) = M_0\rho M_0^\dagger + M_1\rho M_1^\dagger$$

^{1 Or rather, several very broad concepts resulting from the same fundamental idea}

^{2 I wouldn't go around calling this rigorous or anything}

^{3 Within this context, naturally}