# Ansatz state for finding the lowest eigenvalue of a $2^n\times 2^n$ real matrix using VQE

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??)

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...