# What is the variance associated to the average gate fidelity?

Let $$U$$ be a unitary and $$\mathcal E$$ a map on a space of dimension $$D$$, and say we want to assess how "close" $$U$$ is to $$\mathcal E$$.

A standard way to do this is using the average fidelity $$\overline{F}(\mathcal E, U)$$, defined as $$\overline{F}(\mathcal E, U)\equiv\int d\psi \langle\psi\rvert U^\dagger \mathcal E(|\psi\rangle\langle\psi|) U|\psi\rangle.\tag1$$ It is a standard result (see e.g. quant-ph/0205035) that (1) can be computed as $$\overline{F}(\mathcal E, U) = \frac{1}{D+1}\left[ 1 + \frac{1}{D} \langle i|U^\dagger \mathcal E(|i\rangle\langle j|)U|j\rangle \right]. \tag2$$

In the special case of $$\mathcal E$$ also being a unitary evolution (that is, $$\mathcal E(\rho)=V\rho V^\dagger$$ for some unitary $$V$$), the formula simplifies to $$\overline{F}(V, U) = \frac{1}{D+1}\left[ 1 + \frac{1}{D} \lvert\operatorname{tr}(U^\dagger V)\rvert^2 \right]. \tag3$$ These expressions are very useful as they allow to compute the average fidelity without actually computing an average over the states.

I wasn't however able to find discussed the matter of what the variance of this quantity actually is. In other words, if I were to try to estimate (1) using a sample size of $$N$$ states, how likely am I to get a value close enough to the true average given by (2)?

To more clearly understand the question, here is a plot of the average fidelity between two unitary gates, estimated with various sample sizes and compared with the true value given by (3):

I want to know how big do I need to choose the sample size in order to have a good enough estimate with a good enough confidence.

Is there any closed expression for this variance? Is there anything known about it (will it depend on $$U$$ and $$\mathcal E$$, how does it scale with system dimension, etc.)?

As far as I am aware, there is no simple expression for the variance of the average fidelity. However, the sample variance can be computed easily. Suppose you prepared Haar random states and collected the data $\{f_i\}$ then the sample variance is $$s_{N-1}^2 = \frac{1}{N-1}\sum_i (f_i - \bar{f})^2$$ where $$\bar{f} = \frac{1}{N}\sum_i f_i.$$

It is straightforward to determine how many samples you need. Alternatively, you could try bootstrapping or using Bayesian inference.

I should mention one more fact. $$\text{Var}(\bar{F}) \leq O(\frac{1}{N})$$ where D is the dimension of your Hilbert space. This means that if your Hilbert space is large one random state is enough to determine the average fidelity with high probability!

• thanks. Can you provide a reference for the bound on the variance of the average fidelity?
– glS
Sep 11, 2018 at 10:04
• Page 84 of uwspace.uwaterloo.ca/bitstream/handle/10012/6832/…. There are quite a few other results about the gate fidelity in there you may find useful.
– ABW
Sep 12, 2018 at 15:46