# Which error channel definitions should I use if I want to compare the performance of a QEC protocol for depolarizing errors vs dephasing errors?

Let's say you want to investigate if a quantum error correction or error mitigation protocol performs better under dephasing errors compared to depolarizing errors, or if a quantum algorithm is more robust against dephasing errors compared to depolarizing errors, or vice versa. Which one of the two following definitions for the error channels would provide the most fair comparison?

# Option 1

Define the depolarizing channel as $$\Lambda_\mathrm{dep}(\rho) = \left(1-\frac{3}{4}\epsilon\right)\rho + \frac{\epsilon}{4}\left(X\rho X + Y\rho Y + Z\rho Z\right) = (1-\epsilon)\rho + \frac{\epsilon}{2}I,$$ where $$X,Y,Z$$ are Pauli matrices and $$\epsilon\in[0,1]$$ is the error probability, and define the dephasing channel as $$\Lambda_\mathrm{Z}(\rho) = \left(1-\frac{\epsilon}{2}\right)\rho + \frac{\epsilon}{2} Z\rho Z.$$ Using this definition, when $$\epsilon=1$$, we get a fully mixed state for the depolarizing channel and a fully dephased state for the dephasing channel.

# Option 2

Define the depolarizing channel as $$\Lambda_\mathrm{dep}(\rho) = \left(1-\epsilon\right)\rho + \frac{\epsilon}{3}\left(X\rho X + Y\rho Y + Z\rho Z\right) = \left(1-\frac{4}{3}\epsilon\right)\rho + \frac{2}{3}\epsilon I,$$ and define the dephasing channel as $$\Lambda_\mathrm{Z}(\rho) = \left(1-\epsilon\right)\rho + \epsilon Z\rho Z.$$ In this case we can interpret the two channels as having the same probability of an error occurring. However, in this case the worst case upper bounds are different for the two channels: $$\epsilon_\mathrm{worst}=3/4$$ for the depolarizing channel, and $$\epsilon_\mathrm{worst}=1/2$$ for the dephasing channel.

So relating back to the opening of this question. If I want to simulate a QEC protocol to see if it performs better under dephasing errors compared to depolarizing errors, should I go with Option 1 or Option 2?

• In the literature people use option 2. Sep 21 at 22:06
• @JahanClaes That's not necessarily true. See Eq. (2.2) and (2.3) in this paper arxiv.org/abs/1509.02921v1 Sep 21 at 22:12
• Fair enough, I'm thinking mainly of literature on QEC with biased errors. See my answer Sep 21 at 22:27

## 1 Answer

In the literature on biased-noise error correction, people use option 2. See, e.g.:

We define $$p = p_x +p_y +p_z$$ to be the probability of any single-qubit error

in https://arxiv.org/abs/1708.08474, and similar statements in https://arxiv.org/abs/1812.08186, https://arxiv.org/abs/1907.02554, https://arxiv.org/abs/2009.07851, https://arxiv.org/abs/2201.10566, and probably many others

• Another argument to using option 2 would be that the average channel fidelity is the same for the two channels. Sep 22 at 17:23