Quantum channel models

QMetro++ package implements several commonly used quantum channels:

\[\Lambda_\theta(\rho) = \sum_k K_{\theta,k} \rho K_{\theta,k}^\dagger,\]

acting on qubit systems. All of them encode the parameter \(\theta\) via a unitary

(30)\[U_\theta= e^{-\frac{i}{2}\theta\sigma_z},\]

where \(\sigma_z\) is the Pauli z matrix and the parameter \(\theta\) is assumed to be in the neighborhood of \(0\).

The signal can be applied either before or after the noise action defined by the Kraus operators \(\{K_k\}_k\):

noise first: \(K_{\theta,k} = U_{\theta}K_k\).
noise second: \(K_{\theta,k} = K_k U_{\theta}\),

Depolarization

Depolarization is defined by the Kraus operators

\[K_0 = \sqrt{p}\, \mathbb{1}, \quad K_i =\sqrt{\frac{1-p}{3}} \, \sigma_i \quad \mathrm{for} \quad i=x,y,z,\]

for some \(p \in [0, 1]\). This models the noise \(\rho \rightarrow p\rho + \frac{1-p}{2}\mathbb{1}\).

Alternative Kraus representation is given by

\[K_0 = \sqrt{\frac{1+3\eta}{4}}\, \mathbb{1}, \quad K_i =\sqrt{\frac{1-\eta}{4}} \, \sigma_i \quad \mathrm{for} \quad i=x,y,z,\]

for some \(\eta = (4p-1)/3 \in [-1/3, 1]\). Then the noise can be understood as a uniform contraction of the Bloch ball by a factor \(\eta\).

Parallel amplitude damping

Parallel amplitude damping channel is defined by the Kraus operators

\[\begin{split}K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-p} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{p} \\ 0 & 0 \end{pmatrix},\end{split}\]

for some \(p \in [0,1]\). This models the decay \(|1\rangle \rightarrow |0\rangle\).

Parallel dephasing

Parallel dephasing channel is defined by the Kraus operators

\[K_0 = \sqrt{p}\, \mathbb{1}, \quad K_1 = \sqrt{1-p} \, \sigma_z,\]

for some \(p \in [0,1]\). This models the dephasing noise that shrinks the Bloch sphere along the x and y directions by a factor of \(2p-1\).

Alternative Kraus representation is given by rotations around the z-axis (see (30)):

\[K_+=\frac{1}{\sqrt{2}} U_\epsilon, \quad K_-=\frac{1}{\sqrt{2}} U_{-\epsilon},\]

where \(p = \cos^2(\epsilon/2)\).

Perpendicular amplitude damping

Perpendicular amplitude damping channel is defined by the Kraus operators

\[K_0 = | - \rangle \langle - | +\sqrt{p} | + \rangle \langle + |, \quad K_1 = \sqrt{1-p} | - \rangle \langle + |,\]

for some \(p \in [0,1]\). This models the decay \(| + \rangle \rightarrow | - \rangle\).

Perpendicular dephasing

Perpendicular dephasing channel is defined by the Kraus operators

\[K_0 = \sqrt{p}\, \mathbb{1}, \quad K_1 = \sqrt{1-p} \, \sigma_x,\]

for some \(p \in [0,1]\). This models the dephasing noise that shrinks the Bloch sphere along the y and z directions by a factor \(2p-1\).