It has been noted in several sources (e.g. ref. 1) that one of the disadvantages of the Coupled Cluster method for the description of electron correlation is that, since $e^T$ in $E_{CC} = \langle0|e^{-T}He^T|0\rangle$ isn't unitary, the method isn't variational (the energy obtained by solving CC equations isn't a rigorous upper bound to the exact energy).

Could someone please explain to me the connection between non-unitarity of $e^T$ and the method not being variational.

It has been noted in several sources (e.g. J. Romero et al. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. arXiv:1701.02691 [quant-ph]) that one of the disadvantages of the Coupled Cluster method for the description of electron correlation is that, since $e^T$ in $$E_\mathrm{CC} = \langle0|e^{-T}He^T|0\rangle$$ isn't unitary, the method isn't variational (the energy obtained by solving CC equations isn't a rigorous upper bound to the exact energy).

Could someone please explain to me the connection between non-unitarity of $e^T$ and the method not being variational.