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Say I have a physical Hamiltonian $\mathcal{H}$ which is $D$-dimensional and I encode it into a larger matrix $M$ which is $\Delta$-dimensional. In the cases I care about $\Delta$ is strictly greater than $D$. Therefore there are $(\Delta-D)$ non-physical ``junk states'' in the basis of $M$. Also suppose that I have an ansatz for VQE which is only capable of producing physical states for any configuration of the ansatz parameters.

For VQE: is it strictly required that the entire matrix $M$ is Hermitian or just that the embedded physical Hamiltonian $\mathcal{H}$ is Hermitian?

For example: If the encoded physical basis states are $|00>$, $|10>$ and $|11>$, and the output of the ansatz circuit is always some superposition of those three, could $M$ be something like

\begin{equation} M=\begin{pmatrix} a & \alpha & b & 0 \\ 0 & \beta & 0 & 0 \\ b & 0 & c & d \\ 0 & \gamma & d & f \\ \end{pmatrix}\quad\text{where}\quad \mathcal{H}= \begin{pmatrix} a & b & 0 \\ b & c & d \\ 0 & d & f \\ \end{pmatrix}. \end{equation}

I've come up with several examples where $M\neq M^\dagger$ gives the correct answers with VQE but can't find anything on why that works. Any citations on this would be extremely appreciated.

As a side note: this non-Hermitian-ness in $M$ comes from the encoded creation operators not being the adjoint of the encoded annihilation operators unless you restrict yourself to the space spanned by the physical basis states.

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