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Let $n$ be the number of qubits we're using, and let $$\mathrm H=\sum_{i=1}^T\alpha_i\mathrm U_i|0\rangle\langle0|\mathrm U_i^\dagger$$ be an $n$-qubit hermitian observable where $T=O(\mathrm{poly}(n))$, $\alpha_i\in\mathbb R\!\setminus\!\{0\}$ and $\mathrm U_i$ a unitary that can be implemented as a circuit of size $O(\mathrm{poly}(n))$.

For my purposes, I would like to express $\mathrm H$ as an observable consisting of only a single term, without any ancillary qubits; thus, the most general desired form is $\mathrm H=\mathrm W\mathrm D\mathrm W^\dagger$ with $\mathrm W$ an $n$-qubit unitary and $\mathrm D$ a $2^n\times2^n$ real diagonal matrix.

Question: is there a construction of $\mathrm W$ and $\mathrm D$ that can be computed in time $O(\mathrm{poly}(n))$, possibly exploiting that $\mathrm H$ is of low rank (namely polynomial in $n$)? Clearly, $\mathrm W$ and $\mathrm D$ can be brute forced by diagonalising $\mathrm H$, but from what I have found even the fastest diagonalisation algorithms still run in time proportional to the dimensionality of $\mathrm H$, which is exponential.

Of course, if you know of any papers related to this question, sharing would be highly appreciated. Thanks!

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There are a few things you can say quite easily. Let $$ |\psi_i\rangle=U_i|0\rangle. $$ Now, let $|\phi\rangle$ be a state that is orthogonal to all of these states. Then $$ H|\phi\rangle=0. $$ This is most of your job done given that the Hilbert space is of dimension $2^n$, and your space spanned by the $|\psi_i\rangle$ is no more than size poly($n$).

So, now, I'd probably construct a matrix $$ H'=\sum_{i,j}|i\rangle\langle j| \langle\psi_i|H|\psi_j\rangle, $$ which only has polynomial size. I don't know how easily you can construct this - it might depend on exactly what you're assuming you know but, frankly, if you cannot compute these values, there's not a whole lot you will be able to do classically! If you can diagonalise $H'$ (which should be fine), you can diagonalise $H$.

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