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I want to replicate the depolarizing noise channel

enter image description here

for a 4 qubit circuit system, where p is the probability for an error. enter image description here I tried doing this: enter image description here

But I get the error WARNING: all-qubit error already exists for instruction "measure", composing with additional error.

I have looked at Qiskits documentation on the depolarizing_error function but it is not clear, or I don't understand the maths.

Could anyone please shed more light on this?

Edit: I might have found a solution, but I'm not sure if my logic is right.

For my 4 qubit system, I want to apply the depolarizing channel noise to the measurement of the qubits $q_{0}$ and $q_{2}$ in the z-direction:

enter image description here

Edit: I realized that the single-qubit depolarizing channel above acts separately on the two qubits: enter image description here

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The aim is to use Qiskits built in depolarizing quantum error channel function $\texttt{depolarizing}$$\_$$\texttt{error(param, num_qubits, standard_gates=True)}$ to create the desired depolarizing channel. It takes the depolarization parameter $\lambda$ and number of qubits $n$ to create the depolarizing channel $$ \begin{equation} \label{eqn:depo-chan-qiskit} \varepsilon(\rho) = (1-\lambda)\rho + \lambda Tr\left[\rho\right]\frac{I}{2^{n}} \text{.} \end{equation}$$

It can then be manipulated to create the single qubit depolarizing channel $$ \begin{equation} \label{eqn:depo-chan-1qubit} \varepsilon(\rho_{single-qubit}) = (1-P)\rho + \frac{P}{3}\left(X\rho X + Y\rho Y + Z\rho Z\right) \text{.} \end{equation}$$ This is shown by the following derivation. For a single qubit $n=1$ $$ \frac{I}{2^{1}} = \frac{1}{4} \left(I\rho I + X\rho X + Y\rho Y + Z\rho Z\right),$$ and $Tr[\rho] = 1.$

If we then chose $$\lambda=\frac{4^{n}P}{4^{n}-1},$$ where $P$ is the probability. At $n=1$ the depolarization paramater $$\lambda = \frac{4P}{3}$$ and by substituting these results into $\varepsilon(\rho)$ we get that

$$ \begin{align} \varepsilon(\rho) &= (1-\frac{4P}{3})\rho + \frac{4P}{3} \left(I\rho I + X\rho X + Y\rho Y + Z\rho Z\right) \frac{1}{4} \\ &= \left( 1 - \frac{4P}{3} + \frac{P}{3} \right)\rho + \frac{P}{3} \left(X\rho X + Y\rho Y + Z\rho Z\right) \\ &= (1-P)\rho + \frac{P}{3}\left(X\rho X + Y\rho Y + Z\rho Z\right) \text{.} \end{align} $$ enter image description here

Here is a link to my notebook explaining how it works https://github.com/MIGUEL-LO/Qiskit_depolarazation_channel/blob/master/depo_channel_using_depolarizing_error.ipynb

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