Noise Models and Noisy Simulation

A noise model is a map with a declared domain#

The teaching domain is a $2\times2$ density matrix $\rho$: Hermitian, positive semidefinite, trace one. A physical quantum channel must be trace preserving and completely positive [Nielsen and Chuang]. These conditions prevent a simulator from emitting negative probabilities or changing total probability while still drawing an attractive plot.

The bit-flip channel is $\mathcal E_p(\rho)=(1-p)\rho+pX\rho X$, with dimensionless p per channel application. This is a pedagogical Pauli channel, not a named device mechanism. Decoherence models are useful only when their assumptions remain visible [Preskill notes]. The Lindblad and Gorini–Kossakowski–Sudarshan results characterize generators of continuous-time Markovian completely positive dynamical semigroups; citing them does not make every processor’s noise Markovian [Lindblad] [Gorini, Kossakowski, and Sudarshan].

The Kraus form makes the validity check executable. Set $K_0=\sqrt{1-p}I$ and $K_1=\sqrt p X$, then compute $\mathcal E_p(\rho)=\sum_k K_k\rho K_k^\dagger$. The completeness relation $\sum_k K_k^\dagger K_k=I$ proves trace preservation for $0\le p\le1$; the operator-sum construction supplies complete positivity. Reject p outside that interval rather than clipping it. Clipping conceals a caller error and changes a supposedly recorded experiment.

Test positivity with eigenvalues, not diagonal entries alone. A Hermitian matrix can have nonnegative diagonal elements and still possess a negative eigenvalue because its coherence is too large. For a $2\times2$ state, checking trace, Hermiticity, and both eigenvalues to a declared numerical tolerance is inexpensive. The tolerance belongs to floating-point validation; it is not permission to accept a materially unphysical state.

Bit flips move populations#

Apply the channel to $\lvert0\rangle\!\langle0\rvert$. The result is diagonal: population 1−p on $\lvert 0 \rangle$ and p on $\lvert 1 \rangle$. A Z-basis measurement sees it directly. At p=0, recover the input; at p=1, recover $\lvert 1 \rangle$; at p=0.1, exact probabilities are 0.9 and 0.1. Assert trace one, Hermiticity, nonnegative diagonal probabilities, and analytic entries at tolerance 10⁻¹².

On an arbitrary state $\rho=\begin{bmatrix}a&c\\c^*&1-a\end{bmatrix}$, the bit-flip channel gives diagonal entry $a'= (1-p)a+p(1-a)$ and exchanges the off-diagonal elements inside the weighted sum. At p=1/2, populations become equal for every input, but coherence need not vanish. This is why “bit flip at fifty percent” is not the completely depolarizing channel: some X-axis information survives.

Use $\lvert 0 \rangle$ and $\lvert + \rangle$ as complementary probes. The first exposes population transfer in Z. The second is an +1 eigenstate of X, so an X error changes it only by the same state and the X-basis outcome remains deterministic. A diagnostic that reports identical degradation for both probes has likely implemented a depolarizing shortcut instead of the declared bit-flip map.

Dephasing hides until the basis changes#

For $\lvert+\rangle\!\langle+\rvert$, computational-basis populations begin at one half and stay there under dephasing. The off-diagonal coherence shrinks. A Z histogram therefore cannot distinguish ideal $\lvert + \rangle$ from a fully dephased mixture. Rotate with H and measure Z—equivalently measure X—and the lost coherence becomes a changed outcome probability.

This is why “we measured the same bit distribution” is not a channel-validation claim. State preparation and measurement basis decide which channel signature is identifiable.

The fixture uses the phase-flip convention $\mathcal Z_p(\rho)=(1-p)\rho+pZ\rho Z$. For $\lvert + \rangle$, the coherence is multiplied by $1-2p$, and the X-basis minus probability is p. At p=1/2 the state is fully dephased; at p=1 the state is $\lvert - \rangle$, which is coherent rather than mixed. Calling p=1 “maximum dephasing” would therefore be wrong under this parameterization.

A phase-damping model is often parameterized so coherence decays monotonically toward zero instead. The two channels may coincide at selected parameter values but their parameters do not share a universal meaning. Store the channel name, Kraus operators or defining equation, application scope, and parameter convention. A bare field named dephasing: 0.1 is not reproducible.

The two-basis diagnosis is now analytic. With p=0.1 and $\lvert + \rangle$ input, Z probabilities remain (0.5,0.5), while after H the Z probability of one is 0.1. For a bit-flip channel on $\lvert 0 \rangle$, Z probability of one is 0.1, while an X measurement is balanced. The same decimal produces different pairs of signatures; the pair, not the decimal, identifies the implemented mechanism.

Shot noise and channel noise leave different records#

The exact post-channel matrix and probabilities belong in one record. Sampled counts, shots, pseudorandom generator, seed, and acceptance interval belong in another. Increasing shots narrows sampling uncertainty; it does not remove channel error. Conversely, setting p=0 does not force a finite sample to contain exact expected proportions. Hardware benchmark interpretation must separate sampling, preparation, gate, and measurement contributions [benchmarking review]; randomized benchmarking supports its own bounded decay-model claim rather than a general application claim [Magesan, Gambetta, and Emerson].

For a binary measurement with exact probability q and n independent shots, the count variance is $nq(1-q)$; the frequency standard deviation is $\sqrt{q(1-q)/n}$. At q=0.1 and n=1,000, that standard deviation is about 0.0095. A frequency of 0.112 may be ordinary shot variation even though it differs from the exact channel output. The acceptance rule must be chosen before seeing the count and must state whether a normal approximation, exact interval, or hypothesis test is used.

Repeat seeds do not provide independent experimental replicates if every other simulated input is identical, and more simulated shots cannot establish that a toy channel matches hardware. They estimate the sampler wrapped around the chosen model. Report exact-model error separately from Monte Carlo error so that a reader can see which one additional computation can reduce.

def bit_flip_zero(p): return ((1-p, 0.0), (0.0, p))
def dephase_plus(p): return ((0.5, 0.5*(1-2*p)), (0.5*(1-2*p), 0.5))
def trace(rho): return rho[0][0] + rho[1][1]
def x_minus_probability(rho): return 0.5*(1 - 2*rho[0][1])
def readout_confuse(p_one, e=.05): return e*(1-p_one) + (1-e)*p_one
rho_x = bit_flip_zero(.1); rho_z = dephase_plus(.1)
identity_case = bit_flip_zero(0.0); complete_dephase = dephase_plus(0.5)
assert abs(trace(rho_x)-1) < 1e-12 and abs(trace(rho_z)-1) < 1e-12
assert abs(rho_x[1][1]-.1) < 1e-12
assert abs(x_minus_probability(rho_z)-.1) < 1e-12
assert abs(readout_confuse(.1)-.14) < 1e-12
assert identity_case == ((1.0, 0.0), (0.0, 0.0)) and complete_dephase[0][1] == 0.0
print(f"PASS: 41 noise bitflip_population={rho_x[1][1]:.3f} dephase_coherence={rho_z[0][1]:.3f} complete_dephase={complete_dephase[0][1]:.1f}")

Readout confusion belongs after the state#

A symmetric confusion matrix flips the reported classical bit with probability e. If the true probability of one is q, the reported probability is e(1−q)+(1−e)q. The matrix acts after the chosen quantum measurement. Treating it as decoherence would incorrectly make it change with preparation basis and would prevent separate calibration.

Official SDK simulators expose model-specific configurations, not a universal hardware truth [Qiskit noise documentation] [Cirq noise documentation]. Preserve whether a configured error is a channel, gate attachment, or readout model.

For asymmetric readout, write $M=\begin{bmatrix}1-e_{0\to1}&e_{1\to0}\\e_{0\to1}&1-e_{1\to0}\end{bmatrix}$ acting on the true probability column. Each column sums to one. Calibrate the two error directions separately using prepared $\lvert 0 \rangle$ and $\lvert 1 \rangle$. Symmetry is a simplifying assumption, not a property to infer from one average percentage.

Inverting M can estimate pre-readout probabilities when the matrix is known and well conditioned, but inversion amplifies calibration and sampling error. It may produce negative components. Keep the raw counts, calibration record, inversion method, and uncertainty; never replace observations with clipped corrected values. Mitigation changes the estimator and its variance, not the underlying state.

Calibrated claims need more than p#

A device claim needs channel scope, gate dependence, qubit pair, timing, leakage policy, correlations, crosstalk, drift, timestamp, calibration protocol, and uncertainty. One-qubit stationary channels omit non-Markovian behavior and context. Label the artifact’s p values “teaching parameters.” Calibration-like decimal places do not make them calibrated.

Composition order can matter. A coherent rotation followed by dephasing need not equal dephasing followed by that rotation, and a gate-attached channel is applied a different number of times after transpilation changes. Record the compiled operation trace and the precise insertion rule: before gate, after gate, per duration, or once per idle interval. Otherwise two simulators can accept the same parameter file and implement different processes.

Finally, validate the simulator against limiting cases and composition identities before fitting anything. Identity at p=0, X conjugation at p=1, complete phase loss at the convention’s p=1/2, trace preservation after repeated applications, and agreement between explicit matrices and Kraus evaluation are stronger evidence than a smooth noise curve. Smoothness is easy to generate; the declared channel algebra is the contract.

Channel order should be an explicit test case. Apply dephasing to $\lvert + \rangle$, rotate with H, then add readout confusion; compare that with the deliberately wrong implementation that confuses labels before the H. The first path changes a classical distribution only after a quantum basis choice. The second effectively defines a different quantum experiment. Their disagreement is the evidence that the simulator has preserved layer boundaries rather than collapsing every imperfection into one probability.

For each mechanism, store an identifiability statement: which inputs and observables respond, which remain unchanged, and which alternative channel shares that signature. Bit-flip p=0.1 on $\lvert 0 \rangle$ is visible in Z; phase flip p=0.1 on $\lvert + \rangle$ is visible in X; symmetric readout affects both after measurement. A model library without this table encourages parameter fitting to non-identifying data.

A final fixture composes each channel twice and compares the result with the analytic effective parameter. This catches implementations that mutate the input matrix, apply the channel once per matrix entry, or renormalize after every step. Composition tests the map as an operation, not merely one attractive endpoint.