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I would like to find the lowest eigenvalue of a $2^n\times 2^n$ real matrix $H$ using the VQE procedure. The measurement part is simple — I just expand $H$ in a sum of all possible $n$-qubit Pauli terms.

But I have not yet come up with a nice way of preparing an arbitrary $n$-qubit state with real amplitudes. What would be a good ansatz operator $U(\theta_1,\ldots\theta_{2^n-1})$ to do the job?

(If I were solving the problem in the second-quantized formalism using $2^n$ qubits, I would simply use the Unitary Coupled Cluster with $(2^n - 1)$ single amplitudes — does it have an analogue in such a 'binary' encoding??)

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If you don't have any prior on your matrix, you can use the Strongly Entangling Circuit as a very generic ansatz for qubits.

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Giving a general $2^n \times 2^n $ Hermitian matrix $H$, it's not guarantee that you can decompose/expand $H$ in polynomial strings of $n$-qubit Pauli matrices. So I don't know if VQE is a good method to find lowest eigenvalues of a matrix... unless you know that you can sum $H$ in polynomial terms of Pauli matrices. Even then, in general, VQE only gonna give you a good answer if you have some intuition about the problem. If you pick a variational form that is hardware efficient, using polynomial number of gates, then you can't expect it explore the entire Hilbert space...

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