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A model for noise in quantum systems such as a decohering qubit that has particularly nice symmetry properties. The depolarizing channel is a “worst-case scenario” channel. It assumes that we just completely lose the input qubit with some probability, i.e., it replaces the lost qubit with the maximally mixed state.

Depolarizing channel $\mathcal{D}$ is defined as

$$ \mathcal{D}(\rho) = \lambda \rho + (1 - \lambda)\frac{I}{d} $$

where $\rho$ is the density matrix of the input state, $\lambda \in [-\frac{1}{d^2 - 1}, 1]$ is a parameter and $d$ is the dimension of the Hilbert space of the system on which the channel acts. For an $n$-qubit system, this is equivalent to

$$ \mathcal{D}(\rho) = (1 - p) \rho + \frac{p}{4^n-1}\sum_{i=1}^{4^n-1} P_i \rho P_i^\dagger $$

where $P_i$ are non-identity $n$-fold tensor products of the Pauli operators and $p$ is the probability of a Pauli error occurring.