# Noise channels, measurement data, and mitigation {#channels}

**Learning goals.** Evolve density matrices with Kraus operators, distinguish noise conventions, estimate observables, and identify mitigation's bias and sampling costs.

## Open-system dynamics

A closed joint system evolves unitarily, but a subsystem interacting with an unobserved environment generally does not. For an initially uncorrelated system and environment, tracing out the environment gives a completely positive trace-preserving map:

$$\mathcal E(\rho)=\sum_jK_j\rho K_j^\dagger,\qquad
\sum_jK_j^\dagger K_j=I.$$

The second equation preserves the trace. Complete positivity means positivity remains even when the system is part of a larger entangled state. Merely mapping isolated positive matrices to positive matrices is not enough.

Kraus representations are not unique. The operators can be interpreted through an environment measurement in a chosen dilation, but an arbitrary Kraus index should not be treated as a directly observed physical error. Conditioning on an actual record and averaging over unobserved records are different operations.

## Three channels with explicit conventions

The phase-flip channel is

$$\mathcal E_Z(\rho)=(1-p)\rho+pZ\rho Z.$$

It leaves populations unchanged and multiplies off-diagonal entries by $1-2p$. Thus complete dephasing occurs at $p=1/2$, not at $p=1$. At $p=1$ it is a deterministic Z gate. A different convention writes coherence decay as $1-\lambda$; these parameters must not be interchanged.

We define depolarization by

$$\mathcal D_p(\rho)=(1-p)\rho+pI/2,\quad 0\le p\le1.$$

It shrinks the Bloch vector by $1-p$. Equivalently,

$$\mathcal D_p(\rho)=(1-3p/4)\rho+
\frac p4(X\rho X+Y\rho Y+Z\rho Z).$$

Here the probability of a nonidentity Pauli in this representation is $3p/4$. Some packages call that total probability p instead. Always inspect the implemented equation.

Amplitude damping models relaxation from $|1\rangle$ toward $|0\rangle$:

$$K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix},
\qquad K_1=\begin{pmatrix}0&\sqrt\gamma\\0&0\end{pmatrix}.$$

For $\rho=\begin{pmatrix}a&c\\c^*&b\end{pmatrix}$,

$$\mathcal A_\gamma(\rho)=
\begin{pmatrix}a+\gamma b&\sqrt{1-\gamma}\,c\\
\sqrt{1-\gamma}\,c^*&(1-\gamma)b\end{pmatrix}.$$

The trace remains $a+b=1$. The map is not unital: it sends $I/2$ toward a state biased to $|0\rangle$. In an exponential relaxation model, $\gamma=1-e^{-t/T_1}$. Coherence time $T_2$ also includes dephasing; under the usual independent Markovian model,
$1/T_2=1/(2T_1)+1/T_\phi$. These are model-specific time constants, not interchangeable gate fidelities.

**Worked example.** Apply damping with $\gamma=1/4$ to $|+\rangle$:

$$\rho'=\begin{pmatrix}5/8&\sqrt3/4\\\sqrt3/4&3/8\end{pmatrix}.$$

Its Bloch vector is $(\sqrt3/2,0,1/4)$, with squared length $13/16$ and purity $29/32$. Both the population shift and coherence reduction matter.

::: {.lab #lab-noise}
**Laboratory L11 — Density-matrix noise.** Choose a pure input, optionally mix it with $I/2$, then apply bit flip, phase flip, depolarization, or amplitude damping. Inspect the full output matrix, Bloch vector, eigenvalues, and purity.
:::

## Noise is not always an independent coin flip

A coherent overrotation applies $R_z(\epsilon)$ every time. After L repetitions, it is $R_z(L\epsilon)$, so errors may add coherently. An independent stochastic phase-flip channel combines differently. Crosstalk, leakage out of the qubit subspace, slow drift, loss, and correlated events require richer models.

Replacing coherent noise by a Pauli model can simplify simulation but change the prediction. Pauli twirling is a physical or mathematical averaging operation with specific assumptions, not proof that the original noise was already stochastic.

## Estimating observables and states

A Pauli measurement yields values $\pm1$. With N independent shots and sample mean $\hat E$,

$$\operatorname{Var}(\hat E)=\frac{1-E^2}{N}.$$

For $H=\sum_jh_jP_j$, estimate each Pauli expectation and sum. If independently sampled, the energy estimator variance is
$\sum_jh_j^2\operatorname{Var}(\hat E_j)$. Commuting terms may share measurement settings, in which case covariance must be included.

Single-qubit tomography estimates x,y,z from separate basis samples and forms $\hat\rho=(I+\hat xX+\hat yY+\hat zZ)/2$. Finite data can yield a vector outside the Bloch ball, making the naive estimate nonphysical. Constrained estimation can enforce positivity, but introduces statistical choices that should be reported. Full generic n-qubit tomography requires exponentially many parameters; special assumptions can reduce the task.

## Error mitigation is an estimator strategy

Error correction protects encoded quantum information during a computation. Error mitigation combines noisy experiments and classical inference to estimate an ideal quantity. Mitigation need not output a protected quantum state and may incur severe sampling costs [@mitigation].

For zero-noise extrapolation, assume an observable varies smoothly with a controllable effective noise strength:

$$E(\lambda)=E_0+c_1\lambda+c_2\lambda^2+\cdots.$$

Measurements at $\lambda$ and $2\lambda$ give

$$2E(\lambda)-E(2\lambda)
=E_0-2c_2\lambda^2+\cdots.$$

The leading linear bias cancels. But for independent estimates,

$$\operatorname{Var}(2\hat E_1-\hat E_2)
=4\operatorname{Var}(\hat E_1)+\operatorname{Var}(\hat E_2).$$

With comparable input variances, this is five times one estimate's variance. A quadratic combination $3E(\lambda)-3E(2\lambda)+E(3\lambda)$ has still larger coefficient-squared cost. Extrapolations can lie outside $[-1,1]$; clipping changes the estimator and its bias.

Probabilistic error cancellation represents an inverse noise operation with signed weights. The absolute-weight sum drives sampling overhead, often growing rapidly with circuit size. Readout mitigation inverts or fits a calibrated classical assignment model and can amplify statistical errors when that model is ill-conditioned. None provides a free route to arbitrary-depth ideal computation.

::: {.lab #lab-mitigation}
**Laboratory L12 — Bias and variance.** Use the stated model $E(\lambda)=e^{-\lambda}$, sample three noise levels, and compare raw, linear, and quadratic estimates. Change noise, shots, and seed. The model's ideal value is known so bias can be measured.
:::

## Exercises

:::: {.exercise #ex08-1}
**8.1.** What does the phase-flip channel do to $|+\rangle$ at $p=0,1/2,1$?

::: {.solution}

It yields $|+\rangle\langle+|$, $I/2$, and $|-\rangle\langle-|$. Purity decreases and then increases; the parameter is a flip probability, not monotone decoherence strength over the whole interval.
:::

::::

:::: {.exercise #ex08-2}
**8.2.** Verify $K_0^\dagger K_0+K_1^\dagger K_1=I$ for amplitude damping.

::: {.solution}

The two terms are $\operatorname{diag}(1,1-\gamma)$ and $\operatorname{diag}(0,\gamma)$, whose sum is identity.
:::

::::

:::: {.exercise #ex08-3}
**8.3.** Find the depolarized state and purity of an initially pure qubit for $p=0.2$.

::: {.solution}

Its Bloch vector is shortened to length 0.8. Purity is $(1+0.64)/2=0.82$; eigenvalues are 0.9 and 0.1.
:::

::::

:::: {.exercise #ex08-4}
**8.4.** Two independent expectation estimates each have variance 0.001. Find the linear extrapolation variance.

::: {.solution}

$4(0.001)+0.001=0.005$, with standard deviation $\sqrt{0.005}\approx0.0707$. Reduced bias does not guarantee smaller total mean-squared error.
:::

::::

:::: {.exercise #ex08-5}
**8.5.** Why must a tomography estimate with $(x,y,z)=(0.8,0.8,0.3)$ be repaired or reconsidered?

::: {.solution}

Its squared radius is $0.64+0.64+0.09=1.37>1$. One eigenvalue is negative. Sampling fluctuations, inconsistent data, or model errors must be addressed rather than interpreting it as a physical state.
:::

::::

