# What is the definition of physical gate error rate?

The fidelity of two quantum states $$\rho$$ and $$\sigma$$ is a well-defined (up to discussions about a square):
$$F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.$$

## Infidelity based error rate

For a noisy quantum channel $$\mathcal{E}$$ trying to implementing an ideal quantum gate $$U$$ we can define the gate fidelity $$F(\mathcal{E}) = \text{min}_{\rho}F( \mathcal{E}(\rho), U\rho U^\dagger).$$

On measure of the error rate is the infidelity defined by $$1-F(\mathcal{E})$$.

But it seems to me that there are several competing definitions of the (single physical qubit gate) error rate.

## Diamond norm based error rate

The starting point for another definition of the error rate $$\vert \vert A \vert \vert_1 = \text{Tr}(\sqrt{A A^\dagger})$$ is the trace norm.

Another norm is the diamond norm: $$\vert \vert A \vert \vert_\Diamond = \max_{X \leq 1} \vert \vert (A \otimes 1)X \vert \vert_1$$ For the noisy channel $$\mathcal{E}$$ and the ideal channel $$\mathcal{E}_U$$ defined by $$\mathcal{E}_U(\rho)= U \rho U^\dagger.$$

We can define a diamond norm based error rate of the channel $$\mathcal{E}$$ by
$$d_\Diamond( \mathcal{E} \circ \mathcal{E}_U^{-1}, 1)$$

My question later is how the error rate is defined and I have a suspension that the error rate may sometimes be defined along the following lines:

### Measurement based error rate

I suspect that another definition of the error rate of the channel $$\mathcal{E}$$ is the following: If we run a noisy (single qubit) channel followed by a measurement in a specified basis I could also define an error rate based on difference to the measurement statistics.

### Error correction based error rate

If I measure error-syndromes repeatedly I can look at the rate of errors appearing there and define my error rate from them.

## Actual question

In experimental papers such as Suppressing quantum errors by scaling a surface code logical qubit, where they report error rate, what is their definition?

In the paper How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits they write:

"Physical gate error rate: The probability that executing a physical gate will introduce Pauli errors onto targeted qubits."

Why is it sensible to restrict oneself only to Pauli errors?

## 2 Answers

Here is a partial answer:

Error rate is not a well defined quantity.

In experimental papers, the numbers reported are most often measured using a technique called randomized benchmarking (RB) which gives a (bad, but easy to measure) proxy of the average gate infidelity.

There are some theoretical arguments that the average gate infidelity is in itself not a good metric for errors (lack of invariance under tensorization), so in the theoretical litterature for example the worst case fidelity is more common.

In the simulation litterature ( https://journals.aps.org/pra/pdf/10.1103/PhysRevA.86.032324 )on surface code the error rate is more often read off from the channels.

Finally, in the mathematical litterature where the threshold theorem is proven the error rate used is the distance from the channel to be implemented in diamond norm (https://arxiv.org/abs/quant-ph/9906129).

Roughly speaking: The experimental metrics (RB) gives better numbers than the ones used for simulation which is again better than the metrics needed for mathematical guranteees. There are examples where the metrics differ very dramatically ("Indeed, we have an explicit example (Example 3) of a two-qubit gate with fidelity 99% but an error rate slightly under 13%.” (https://iopscience.iop.org/article/10.1088/1367-2630/18/1/012002) ) , but it is claimed (Haah (https://www.youtube.com/watch?v=l4smz_J8f1E)) that for the concrete setups considered metrics at least agree up to an order of magnitude.

I only tried to dig into this for the super conducting platform, and even there they have different metrics depending on single qubit, two qubit gates and idle data qubits and it is a topic of discussion https://www.nature.com/articles/s41586-022-05434-1?fromPaywallRec=false#Sec9, p. 23 in supplementary information.

Here is another partial answer:

Infidelity does not tell us the error rate of a physical gate. It is a poor definition for speaking about the physical error rate of the gates. Fidelity is something average, and if your noise is purely Pauli noise, then there is a formula through the fidelity to error rate. Even in this case, infidelity does not mean error rate, but at least we have a connection to error rate. In terms of Quantum Error Correction (QEC), we analyze the gates in terms of stochastic Pauli channels. To find error rates in the context of QEC, we need a whole process. First, you need to have your process matrices of your system, which can be obtained via Randomized Benchmarking or Gate Set Tomography or Process Tomography. Then you analyze these matrices in terms of stochastic Pauli channels. There you can get a number which can be called the physical error rate of your gate in terms of stochastic Pauli channels. Once again, fidelity does not tell you the error rate or type of error that you have in your system. No idea how this metric, why, and how it started to be used as an error rate. It is an average metric which can tell us something very briefly: i.e if you have 99% fidelity one day and you do some improvements for your system and then measure the fidelity, if you find your average gate fidelity is 99.9 this time, then you can tell that you are improving, but you can't tell anything more than that. There are a couple of papers that explain this in great detail. Here are some of them:

https://arxiv.org/abs/2306.08849

https://arxiv.org/abs/1501.04932

https://arxiv.org/abs/2009.07301