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The paper Iterative Qubit Coupled Cluster approach with efficient screening of generators describes a new screening procedure for generators of the QCC ansatz.

The paper states:

'In the absence of external magnetic fields, the electronic Hamiltonian is real. This results in real coefficients of $\hat{H}$ and an even number of $\hat{y}$ terms in $\hat{P}_{k}$s. Accounting for this, the energy gradient for $\hat{P}_{i}$ can be rewritten as:

$\frac{dE[\hat{P}_{i}]}{d\tau}=\sum_{k}C_{k}Im\langle\Omega_{min}|\hat{P}_{k}\hat{P}_{i}|\Omega_{min}\rangle$

For any $|\Omega_{min}\rangle$, non-vanishing contributions in the equation can be produced only by $\hat{P}_{k}\hat{P}_{i}$ with purely imaginary matrix elements, requiring $\hat{P}_{i}$ to have odd powers of $\hat{y}$ terms.'

So far this makes sense. The paper later states:

'Frequently, the optimized QMF state $|\Omega_{min}\rangle$ is an eigenstate of all $\{\hat{z}_{i}\}_{i=1}^{n}$ operators and, hence, any product of them:

$\prod_{i}\hat{z}_{i}|\Omega_{0}\rangle = \pm|\Omega_{min}\rangle$

If the lowest-energy QMF solution does not satisfy this condition, one can define a “purified” mean-field state $|\phi_{0}\rangle$ that satisfies the equation and has the maximum overlap with the lowest-energy QMF solution.'

Is there a proof for this statement? It seems to me that this is transforming an imaginary $\hat{P}_{k}$ (comprised of odd number of $\hat{y}$) into a real $\hat{P}_{k}$ comprised only of $\hat{z}$ operators.

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