Noise Model Simulation

A circuit noise model specifies where errors act, which degrees of freedom they affect, and which experimental assumptions are being idealized. Matching one reported error probability does not uniquely determine the channel. This page develops numerical distinctions that matter when comparing circuit predictions.

Helpful background. Quantum-channel validation checks physical maps; density-matrix circuit simulation applies them.

Channels attached to gates

For ideal gate $\mathcal U(\rho)=U\rho U^\dagger$, an implementation may use $\mathcal E\circ\mathcal U$ or $\mathcal U\circ\mathcal E$. These agree only under special commutation conditions. Store the placement and target order explicitly.

A Pauli channel is

$$\mathcal E(\rho)=p_I\rho+p_XX\rho X+p_YY\rho Y+p_ZZ\rho Z, \qquad p_\alpha\geq0,\quad\sum_\alpha p_\alpha=1.$$

Its action on a Bloch component is diagonal; for example $x\mapsto(1-2p_Y-2p_Z)x$. This supplies a direct regression test of probabilities and Pauli labels.

“Depolarizing probability” has multiple conventions. The map $\rho\mapsto(1-p)\rho+pI/2$ corresponds to $p_X=p_Y=p_Z=p/4$, whereas choosing a nonidentity Pauli with total probability $q$ gives $q=3p/4$. Compare equations before comparing parameters.

Coherent versus stochastic error

An overrotation $V=e^{-i\epsilon Z/2}$ repeatedly applied $m$ times accumulates angle $m\epsilon$. Starting from $|+\rangle$, the return probability is

$$P_+(m)=\cos^2(m\epsilon/2).$$

Replacing the error by independently sampled $Z$ flips with probability $q$ gives

$$P_+(m)=\frac{1+(1-2q)^m}{2}.$$

Matching a single-gate infidelity cannot make these curves agree for all $m$. A useful validation circuit repeats an identity-equivalent sequence to expose coherent accumulation. Randomization may change that accumulation; it must be included as an operation, not assumed.

The comparison can be made exact. The average gate infidelity relative to the identity is $r_{\rm coh}=\tfrac23\sin^2(\epsilon/2)$ for the unitary overrotation and $r_{\rm flip}=2q/3$ for the stochastic $Z$ channel. Thus $q=\sin^2(\epsilon/2)$ matches their single-gate average infidelity, as well as their one-step return probability on $|+\rangle$. Nevertheless, for small errors the return losses scale differently:

$$1-P_+^{\rm coh}(m)\simeq\frac{m^2\epsilon^2}{4}, \qquad 1-P_+^{\rm flip}(m)\simeq\frac{m\epsilon^2}{4}.$$

The first expansion requires $m|\epsilon|\ll1$, and the second requires $m\epsilon^2\ll1$. This benchmark compares two specified channels with the same scalar calibration; it does not infer the coherent fraction from that scalar. For a qubit, averaging the state fidelities over the six Pauli eigenstates reproduces the uniform pure-state average and provides a small numerical check.

Likewise, idle noise should depend on duration. Under Markovian amplitude damping a duration $t$ uses $p(t)=1-e^{-t/T_1}$. Giving every gate the same probability despite unequal duration changes the physical model.

Readout, correlation, and leakage

A classical readout confusion matrix $C_{m|b}$ maps ideal outcome probabilities to reported ones:

$$q_m=\sum_b C_{m|b}p_b,\qquad \sum_m C_{m|b}=1.$$

Its columns are normalized in this convention. Readout confusion at the end does not alter the premeasurement quantum state; a quantum error before measurement may do so. For mid-circuit feedback, even a classical reporting error changes later control.

Correlated errors require joint channels or shared stochastic variables. Independent one-qubit maps cannot reproduce a common fluctuating field with the same single-qubit marginals. Leakage requires extra levels or an explicit erasure flag; silently projecting leaked population back into the qubit subspace is postselection.

For a discrete correlation test, apply $Z\otimes Z$ with probability $q$ and identity otherwise. Each qubit separately has a $Z$-flip probability $q$, but $|\Phi^+\rangle$ is unchanged by the joint channel. Independent flips with those same marginal probabilities instead give Bell fidelity $(1-q)^2+q^2$: exactly one flip takes $|\Phi^+\rangle$ to its orthogonal phase partner. At $q=1/2$, the fidelities are respectively one and one half. Single-qubit calibration data cannot distinguish these two joint models.

Validation and sensitivity

Check every map’s complete positivity and trace preservation, then test a circuit where error placement has an analytic consequence. For example, damping after an $X$ gate on $|0\rangle$ produces excited probability $1-p$, whereas damping before that gate produces one.

Separate parameter uncertainty from numerical error. Vary plausible coherent fractions, temporal correlations, readout rates, and gate durations rather than reporting a single curve as a device prediction. Fixed parameters define a reproducible model, not a calibration claim.

Exercise

A readout device reports 1 with probability $a$ when the true bit is 0, and reports 0 with probability $b$ when the true bit is 1. Find the reported excitation probability.

Solution

For true excited probability $p$, $q=a(1-p)+(1-b)p=a+(1-a-b)p$. If $a+b=1$, the output carries no information about $p$, so inversion is singular. Near that limit readout correction amplifies statistical error.

References

• Magesan, E., Gambetta, J. M., and Emerson, J., “Characterizing quantum gates via randomized benchmarking,” Physical Review A 85, 042311 (2012), arXiv:1109.6887 — error models and operational error measures.
• Watrous, J., The Theory of Quantum Information, Cambridge University Press (2018), author’s text — channels and distinguishability.