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

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!