Approximating the average of a rational function with respect to the Haar measure

Question

Suppose I have a state $|\psi\rangle = U|0\rangle$ where $U$ is a $d$-dimensional unitary sampled uniformly with respect the Haar measure. I'm interested in computing or approximating analytically an average quantity of the form

$$ \mathbb{E}_{U \sim U(d)} \left[ \frac{|\psi_i|^2}{|\psi_i|^2 + |\psi_j|^2} \right] = \int_{U(d)} \frac{|U_{i0}|^2}{|U_{i0}|^2 + |U_{j0}|^2} \mu(dU) \tag{1} $$ for some $0 \leq i,j \leq d-1$ and $\mu(dU)$ denotes the Haar measure. What are possible ways to compute or approximate this?

I figured one possibility is to take the limit $d \rightarrow \infty$ and then use a Gaussian approximation for $\psi_j$ (as explained in the answer to this question, for example). However I am a bit concerned about singularities in this polynomial making it an ill-defined computation somehow.

Are there other approaches? Alternatively a reference for explicit computation of rational polynomials of the form of (1) would be appreciated.

Answer 1

Let $|\psi\rangle$ be a Haar-random unit vector on $\mathbb{C}^d$. It means that we can write: $$|\psi\rangle=\frac{1}{\sqrt{\sum\limits_{j=1}^dX_j^2+Y_j^2}}\sum_{j=1}^d\left(X_j+\mathrm{i}Y_j\right)|j\rangle\overset{\text{def}}{=}\frac1N\sum_{j=1}^d\left(X_j+\mathrm{i}Y_j\right)|j\rangle$$ with $X_i$ and $Y_j$ being sampled from a $\mathcal{N}(0, 1)$ distribution. In particular, our goal is to compute: $$E=\mathbb{E}\left[\frac{\frac{X_i^2}{N^2}+\frac{Y_i^2}{N^2}}{\frac{X_i^2}{N^2}+\frac{Y_i^2}{N^2}+\frac{X_j^2}{N^2}+\frac{Y_j^2}{N^2}}\right].$$ By linearity of the expectation: $$E=\mathbb{E}\left[\frac{X_i^2+Y_i^2}{X_i^2+Y_i^2+X_j^2+Y_j^2}\right].$$ Now, both $X_i^2+Y_i^2$ and $X_j^2+Y_j^2$ follow a $\chi^2(2)$ distribution. Since they are independent, it means that $\frac{X_i^2+Y_i^2}{X_i^2+Y_i^2+X_j^2+Y_j^2}$ follows a $\beta\left(1,1\right)$ distribution, which is simply the uniform distribution on $[0, 1]$. This allows us to conclude: $$E=\frac12.$$

Answer 2

The integrand in the above equation corresponds to an observable. Representing that observable in the Pauli basis can, in general, be performed by $$ \mathcal{O} \mapsto \frac{1}{2^n}\sum_{i}Tr(\mathcal{O}P_i^n)P_i^n$$

Where $P_i^n \in \{I, X, Y, Z\}^n$. Once you've found the set of $P_{\mathcal{O}} = \{P_i^n | Tr(P_i^n\mathcal{O}) \neq 0\}$, then you need to find a set of unitary transformations that map commuting subgroups of $P_G \subset P_{\mathcal{O}}$ to a set of single qubit Z measurements, i.e. $UP_GU^{\dagger} = \{ZI\cdots I,IZI\cdots I,\cdots, I\cdots Z\}$.

For the measure, this corresponds to state preparation choice. You can find circuits that are prepared by a Haar random unitary in qiskit, to perform the expectation value, simple append the state prep circuit with the above unitary transformation and repeatedly measure and gather statistics!