## Iterative qubit coupled cluster (iQCC) ansatz (Efficient screening procedure)

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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.