I have mainly seen two ways of studying noise in quantum algorithm simulation.

The first one is to suppose that your quantum gate can be implemented with a probability of success of $\mathcal{F}$ (being this the fidelity of the gate). If it is not implemented, then the identity is applied to the state of your system. So this is also a quantum channel.

The second one is to include a specific noise model of your system by including the quantum channels that your QPU may physically show.

Do each of these models exclude the other one?

My thought is that this fidelity $\mathcal{F}$ is obtained experimentally so it is already accounting for all the quantum channels your QPU has in an approximated way so the inclusion of these two models would be overestimating noise. Is this right?

Thank you all!


1 Answer 1


First of all, it should be intuitively clear that if you combine two noise models with similar "strength" that this results in a "noisier" model. The noise model should fit to your experimental data, e.g. the fidelity that you or your collaborators have measured.

IMO, the first fidelity has nothing to do with the second. Given a unitary $U$, the channel that you describe has the form $$ \mathcal E (\rho) = p U\rho U^\dagger + (1-p)\rho. $$ Let us denote the channel defined by $U$ as $\mathcal U$. Using some standard identities, the (average gate) fidelity of $\mathcal E$ with $U$ is $$ F_\mathrm{avg}(\mathcal E,\mathcal U) = p F_\mathrm{avg}(\mathcal U,\mathcal U) + (1-p) F_\mathrm{avg}(\mathrm{id},\mathcal U) = p + (1-p) \frac{|\mathrm{tr} U|^2 - d}{d(d+1)}. $$ Note that this is equal to $p$ if and only if $U$ is (proportional to) the identity matrix. Of course, you may change the definition of $p$ accordingly.

The reason why the fidelity is not just the "success probability" in the above sense, is that any unitary is acting trivially on its eigenvectors, $$ U |\psi\rangle\langle\psi| U^\dagger = e^{i\varphi}|\psi\rangle\langle\psi|e^{-i\varphi} = |\psi\rangle\langle\psi|, $$ hence, it acts as the identity on at least a $d$-dimensional subspace of operator space.

As a final note: Given the fidelity of a gate, you do not know anyhing about the type of noise in your system. Ideally, you would come up with a noise model that is motivated by the physics of the platform that you are using. I'm not sure what the motivation behind the channel $\mathcal E$ is. To me, it looks more like some kind of "detector noise", i.e. either the detector clicks, or it doesn't. For gates, this seems weird. In the end, you are hitting your system with a laser, RF pulse, whatever, so you're definitely doing something in any case. If you do not know anything about the system, you should rather use depolarizing noise.


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