# Which observable $M$ provides the Absolute Average of a statevector?

My question should be fairly simple, though I did not find an answer to it here or anywhere else.

I have been working on an algorithm which, similarly to the HHL algorithm, provides a state $$|x\rangle$$ for which I want to extract some global quantity at the end. I now want to build a simple few-qubits implementation of such algorithm and for that I would like to find the $$\texttt{Absolute Average}$$ given by $$\frac{1}{N}\big|\sum_{i=1}^Nx_i\big|$$, which is clearly possible according to this tutorial to Qiskit's HHL implementation.

However, I am struggling to understand which observable $$M$$ I should apply in order to get this quantity from $$\langle x|M|x\rangle$$.

## 1 Answer

Let $$|+\rangle$$ denote the uniform superposition state with zero relative phases. Then

$$\langle +|x\rangle = \frac{1}{\sqrt{N}}\begin{bmatrix}1&1&\dots&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\\dots\\x_N\end{bmatrix} = \frac{1}{\sqrt{N}}\sum_{i=1}^Nx_i.$$

For $$M$$ defined as $$M:=|+\rangle\langle+|$$, we have

$$\langle x|M|x\rangle = \langle x|+\rangle\langle+|x\rangle = \frac{1}{N}\left|\sum_{i=1}^Nx_i\right|^2$$

so we can obtain the absolute average from the measurement of $$\langle x|M|x\rangle$$ using

$$\frac{1}{N}\left|\sum_{i=1}^Nx_i\right| = \sqrt{\frac{\langle x|M|x\rangle}{N}}.$$