Depolarizing Noise

Real quantum hardware never executes a gate perfectly. One of the most widely studied noise models is the depolarizing channel, which blends a target state with the maximally mixed state $I/2$. It captures the combined effect of random bit flips, phase flips, and simultaneous flip-and-phase errors.

The channel and its Kraus decomposition

For a single qubit with error rate $p \in [0, 1]$, the depolarizing channel acts on a density matrix $\rho$ as

The four terms can be read off immediately: with probability $1-p$ nothing happens; with probability $p/3$ each, an $X$, $Y$, or $Z$ error is applied. The four operators

are the Kraus operators of the channel. They satisfy the completeness relation $\sum_k K_k^\dagger K_k = I$, guaranteeing that the channel is trace-preserving.

Effect on measurement probabilities

Starting from the pure state $|0\rangle\langle 0|$ and applying the depolarizing channel with rate $p$, the resulting density matrix is

Spelling out the arithmetic: $X|0\rangle = |1\rangle$, $Y|0\rangle = i|1\rangle$, and $Z|0\rangle = |0\rangle$, so only the $Z$ error leaves the state in $|0\rangle\langle 0|$ while both $X$ and $Y$ errors rotate it to $|1\rangle\langle 1|$. The probability of measuring $0$ is therefore

At $p = 0$ (no noise) this gives $P(0) = 1$, as expected. At $p = 3/4$ it gives $P(0) = 1/2$ — the outcome is a fair coin flip, identical to what you would get from the maximally mixed state. For $p = 1$ the formula gives $P(0) = 1/3$, consistent with the $X$ and $Y$ errors each contributing $1/3$ probability of ending in $|1\rangle$.

Bloch-sphere picture

On the Bloch sphere the depolarizing channel is an isotropic contraction toward the origin. The Bloch vector $\vec{r}$ shrinks by the factor $(1 - 4p/3)$:

At $p = 3/4$ the Bloch vector vanishes entirely, confirming the maximally mixed state. This contraction is symmetric in all three directions, which is why the channel is called depolarizing — it removes polarization uniformly.

Try it

The depolarizing channel with $p = 3/4$ is applied to $|0\rangle$. Using the formula $P(0) = 1 - 2p/3$, compute and return the probability of measuring $|0\rangle$.

Sign in on the full site to ask questions and join the discussion.