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CHAPTER 08

# Noise channels, measurement data, and mitigation

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

## 8.1 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:

ℰ(ρ)=∑j K j ρ K j†,∑j K j†K j=I.\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.

## 8.2 Three channels with explicit conventions

The phase-flip channel is

ℰ Z(ρ)=(1−p)ρ+p Z ρ Z.\mathcal E_Z(\rho)=(1-p)\rho+pZ\rho Z.

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

We define depolarization by

𝒟 p(ρ)=(1−p)ρ+p I/2,0≤p≤1.\mathcal D_p(\rho)=(1-p)\rho+pI/2,\quad 0\le p\le1.

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

𝒟 p(ρ)=(1−3 p/4)ρ+p 4(X ρ X+Y ρ Y+Z ρ Z).\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 3 p/4 3p/4. Some packages call that total probability p instead. Always inspect the implemented equation.

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

K 0=(1 0 0 1−γ),K 1=(0 γ 0 0).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 ρ=(a c c*b)\rho=\begin{pmatrix}a&c\\c^*&b\end{pmatrix},

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

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

ρ′=(5/8 3/4 3/4 3/8).\rho'=\begin{pmatrix}5/8&\sqrt3/4\\\sqrt3/4&3/8\end{pmatrix}.

Its Bloch vector is (3/2,0,1/4)(\sqrt3/2,0,1/4), with squared length 13/16 13/16 and purity 29/32 29/32. Both the population shift and coherence reduction matter.

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

## 8.3 Noise is not always an independent coin flip

A coherent overrotation applies R z(ϵ)R_z(\epsilon) every time. After L repetitions, it is R z(L ϵ)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.

## 8.4 Estimating observables and states

A Pauli measurement yields values ±1\pm1. With N independent shots and sample mean Ê\hat E,

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

For H=∑j h j P j H=\sum_jh_jP_j, estimate each Pauli expectation and sum. If independently sampled, the energy estimator variance is ∑j h j 2 Var(Êj)\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 ρ̂=(I+x̂X+ŷY+ẑZ)/2\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.

## 8.5 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 [10].

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

E(λ)=E 0+c 1 λ+c 2 λ 2+⋯.E(\lambda)=E_0+c_1\lambda+c_2\lambda^2+\cdots.

Measurements at λ\lambda and 2 λ 2\lambda give

2 E(λ)−E(2 λ)=E 0−2 c 2 λ 2+⋯.2E(\lambda)-E(2\lambda) =E_0-2c_2\lambda^2+\cdots.

The leading linear bias cancels. But for independent estimates,

Var(2 Ê1−Ê2)=4 Var(Ê1)+Var(Ê2).\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 3 E(λ)−3 E(2 λ)+E(3 λ)3E(\lambda)-3E(2\lambda)+E(3\lambda) has still larger coefficient-squared cost. Extrapolations can lie outside [−1,1][-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.

Laboratory L12 — Bias and variance. Use the stated model E(λ)=e−λ 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.

## 8.6 Exercises

8.1. What does the phase-flip channel do to |+⟩|+\rangle at p=0,1/2,1 p=0,1/2,1?

Show solution / guidance

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

8.2. Verify K 0†K 0+K 1†K 1=I K_0^\dagger K_0+K_1^\dagger K_1=I for amplitude damping.

Show solution / guidance

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

8.3. Find the depolarized state and purity of an initially pure qubit for p=0.2 p=0.2.

Show solution / guidance

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

8.4. Two independent expectation estimates each have variance 0.001. Find the linear extrapolation variance.

Show solution / guidance

4(0.001)+0.001=0.005 4(0.001)+0.001=0.005, with standard deviation 0.005≈0.0707\sqrt{0.005}\approx0.0707. Reduced bias does not guarantee smaller total mean-squared error.

8.5. Why must a tomography estimate with (x,y,z)=(0.8,0.8,0.3)(x,y,z)=(0.8,0.8,0.3) be repaired or reconsidered?

Show solution / guidance

Its squared radius is 0.64+0.64+0.09=1.37>1 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.

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