respectively. The active space decomposition is illustrated for the
electrostatic (Sec. D.1), exchange (Sec. D.2), and electrostatic-exchange
(Sec. D.3) operators below. Throughout this appendix, we will use Einstein
notation exclusively with the core orbital indices ($i/j$) to reduce the
verbosity of the equations. We will also ignore the symmetrization of the
tensor elements but will discuss the symmetrized version of the active space
expressions near the end. Real orbitals are also assumed throughout.

###  D.1 Electrostatics

We first consider the full-space electrostatic operator,

$\displaystyle\hat{V}$
$\displaystyle=\sum_{\textbf{{p}}\textbf{{q}}}{v}^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}}\hat{E}_{\textsc{p}_{1}\textsc{p}_{2}}\hat{E}_{\textsc{q}_{1}\textsc{q}_{2}}=\sum_{\begin{subarray}{c}\textbf{{p}}\textbf{{q}}\\\
\sigma\tau\end{subarray}}{v}^{p_{1}p_{2}}_{q_{1}q_{2}}\hat{E}^{\sigma}_{p_{1}p_{2}}\hat{E}^{\tau}_{q_{1}q_{2}}$
(142)

The active space operator is derived by tracing out the inactive degrees of
freedom with a Hartree-Fock-like wavefunction in the inactive space. As a
result, we require decomposing the summation of each of the indices in terms
of core $i/j$ and active space $t/u$ indices,

$\displaystyle\sum_{p_{1}p_{2}\in A}\sum_{q_{1}q_{2}\in B}$
$\displaystyle=\sum_{i_{1}i_{2}\in A}\sum_{j_{1}j_{2}\in
B}+\sum_{t_{1}t_{2}\in A}\sum_{j_{1}j_{2}\in B}+\sum_{i_{1}i_{2}\in
A}\sum_{u_{1}u_{2}\in B}+\sum_{t_{1}t_{2}\in A}\sum_{u_{1}u_{2}\in B}.$ (143)

The virtual orbitals are assumed to remain unoccupied therefore they will not
contribute to the active space renormalization procedure. By tracing out the
core orbitals, the active space electrostatic operator may be written as:

$\displaystyle\hat{V}_{\mathrm{active}}$
$\displaystyle=v^{i_{1}i_{2}}_{j_{1}j_{2}}D^{A}_{i_{1}i_{2}}D^{B}_{j_{1}j_{2}}+\sum_{t_{1}t_{2}\in
A}\\!\\!v^{t_{1}t_{2}}_{j_{1}j_{2}}\hat{E}^{+}_{t_{1}t_{2}}D^{B}_{j_{1}j_{2}}+\sum_{u_{1}u_{2}\in
B}\\!\\!v^{i_{1}i_{2}}_{u_{1}u_{2}}D^{A}_{i_{1}i_{2}}\hat{E}^{+}_{u_{1}u_{2}}+\sum_{\begin{subarray}{c}t_{1}t_{2}\in
A\\\ u_{1}u_{2}\in
B\end{subarray}}\\!\\!v^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}}.$
(144)

defined with respect to the core-orbital-based reduced density matrix,
$D_{i_{1}i_{2}}^{(A)}$/$D_{j_{1}j_{2}}^{B}$, for monomers A/B. Assuming that
the core orbitals are doubly occupied in a closed-shell configuration,
$D^{X}_{nn^{\prime}}=2\delta_{nn^{\prime}}$, we obtain the following active
space electrostatic operator:

$\hat{V}_{\mathrm{active}}=4v^{ii}_{jj}+2\sum_{\mathbf{t}}v^{t_{1}t_{2}}_{jj}\hat{E}^{+}_{t_{1}t_{2}}+2\sum_{\mathbf{u}}v^{ii}_{u_{1}u_{2}}\hat{E}^{+}_{u_{1}u_{2}}+\sum_{\mathbf{t}\mathbf{u}}v^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}},$
(145)

Note that it is possible to combine all of the zero-body and one-body terms
into a spin-independent four-index tensor:

$\displaystyle\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}=v^{t_{1}t_{2}}_{u_{1}u_{2}}+2v^{t_{1}t_{2}}_{jj}\delta_{u_{1}u_{2}}/\tilde{\eta}_{B}+2v^{ii}_{u_{1}u_{2}}\delta_{t_{1}t_{2}}/\tilde{\eta}_{A}+4v^{ii}_{jj}\delta_{t_{1}t_{2}}\delta_{u_{1}u_{2}}/(\tilde{\eta}_{A}\tilde{\eta}_{B}),$
(146)

where $\tilde{\eta}_{A}$ and $\tilde{\eta}_{B}$ correspond to the total number
of electrons in the active spaces of monomer A and B respectively. Using this
formulation, we obtain the final form of the active-space electrostatic
operator:

$\hat{V}_{\mathrm{active}}=\sum_{\mathbf{t}\mathbf{u}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}}.$
(147)

###  D.2 Exchange

The full space exchange term is given by Eq. (68) which, following the same
steps as above, results in the following expression for the active space
exchange operator:

$\hat{P}_{\mathrm{active}}=-2S^{i}_{j}S_{i}^{j}-\sum_{\mathbf{t}}S^{t_{1}}_{j}S_{t_{2}}^{j}\hat{E}^{+}_{t_{1}t_{2}}-\sum_{\mathbf{u}}S_{i}^{u_{1}}S^{i}_{u_{2}}\hat{E}^{+}_{u_{1}u_{2}}-\sum_{\mathbf{t}\mathbf{u}}S_{u_{2}}^{t_{1}}S_{t_{2}}^{u_{1}}\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}}.$
(148)

###  D.3 Electrostatic-Exchange

The derivation of the active space operators for the electrostatic-exchange
term is much more involved. We proceed by deriving the active space operators
for each of the four terms separately. For notational clarily, we also define
$\hat{e}_{\mathbf{t}}^{\sigma_{1}\sigma_{2}}=(\hat{E}^{\sigma_{1}}_{t_{1}t_{2}}\hat{E}^{\sigma_{2}}_{t_{3}t_{4}}-\delta_{t_{2}t_{3}}^{\sigma_{1}\sigma_{2}}\hat{E}^{\sigma_{2}}_{t_{1}t_{4}})$
and
$\hat{e}_{\mathbf{u}}^{\sigma_{1}\sigma_{2}}=(\hat{E}^{\sigma_{1}}_{u_{1}u_{2}}\hat{E}^{\sigma_{2}}_{u_{3}u_{4}}-\delta_{u_{2}u_{3}}^{\sigma_{1}\sigma_{2}}\hat{E}^{\sigma_{2}}_{u_{1}u_{4}})$.

####  D.3.1 $\widehat{VP}_{1}$

The first term of the full space electrostatic-exchange operator is given by
Eq. (94) which, following the same steps as above, results in the following
expression for the active space operator:

$\displaystyle\widehat{VP}_{\mathrm{1,active}}$
$\displaystyle=-2v^{ij}_{ji}-\sum_{\mathbf{u}}v^{iu_{2}}_{u_{1}i}\hat{E}_{u_{1}u_{2}}^{+}-\sum_{\mathbf{t}}v^{t_{1}j}_{jt_{2}}\hat{E}_{t_{1}t_{2}}^{+}-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}v^{t_{1}u_{2}}_{u_{1}t_{2}}\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}}.$
(149)

####  D.3.2 $\widehat{VP}_{2}$

For the second term, Eq. (82), we find the following active space expression:

$\displaystyle\widehat{VP}_{\mathrm{2,active}}=$
$\displaystyle-(4{v}^{i_{1}i_{1}}_{j_{1}i_{2}}S^{i_{2}}_{j_{1}}-2{v}^{i_{2}i_{1}}_{j_{1}i_{2}}S^{i_{1}}_{j_{1}})-\sum_{\mathbf{u}}(2{v}^{i_{1}i_{1}}_{u_{1}i_{2}}S^{i_{2}}_{u_{2}}-{v}^{i_{2}i_{1}}_{u_{1}i_{2}}S^{i_{1}}_{u_{2}})\hat{E}_{u_{1}u_{2}}^{+}$
$\displaystyle-\sum_{\mathbf{t}}(2{v}^{t_{1}t_{2}}_{j_{1}i_{1}}S^{i_{1}}_{j_{1}}+2{v}^{i_{1}i_{1}}_{j_{1}t_{2}}S^{t_{1}}_{j_{1}}-{v}^{t_{1}i}_{jt_{2}}S^{i_{1}}_{j_{1}}-{v}^{i_{1}t_{2}}_{j_{1}i_{1}}S^{t_{1}}_{j_{1}})\hat{E}_{t_{1}t_{2}}^{+}-\sum_{\mathbf{t}}{v}^{t_{1}t_{2}}_{j_{1}t_{4}}S^{t_{3}}_{j_{1}}\hat{E}^{+}_{t_{1}t_{2}}$
$\displaystyle-\sum_{\mathbf{t}\mathbf{u}}v^{t_{1}t_{2}}_{u_{1}i_{1}}S^{i_{1}}_{u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}}-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}(2v^{i_{1}i_{1}}_{u_{1}t_{2}}S^{t_{1}}_{u_{2}}-v^{t_{1}i_{1}}_{u_{1}t_{2}}S^{i_{1}}_{u_{2}}-v^{i_{1}t_{2}}_{u_{1}i_{1}}S^{t_{1}}_{u_{2}})\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}{v}^{t_{1}t_{2}}_{u_{1}t_{4}}S^{t_{3}}_{u_{2}}\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{t}}\hat{E}^{\sigma_{2}}_{u_{1}u_{2}}.$
(150)

####  D.3.3 $\widehat{VP}_{3}$

Next, we find the following expression for the active space version of the
third term, Eq. (87):

$\displaystyle\widehat{VP}_{\mathrm{3,active}}=$
$\displaystyle-(4v^{i_{1}j_{2}}_{j_{1}j_{2}}S_{i}^{j_{2}}-2v^{i_{1}j_{2}}_{j_{2}j}S_{i_{1}}^{j_{1}})-\sum_{\mathbf{t}}(v^{t_{1}j_{2}}_{j_{1}j_{2}}S_{t_{2}}^{j_{2}}-v^{t_{1}j_{2}}_{j_{2}j}S_{t_{2}}^{j_{1}})\hat{E}^{+}_{t_{1}t_{2}}$
$\displaystyle-\sum_{\mathbf{u}}(2v^{i_{1}j_{1}}_{u_{1}u_{2}}S_{i_{1}}^{j_{1}}+2v^{i_{1}u_{2}}_{j_{1}j_{2}}S_{i_{1}}^{u_{1}}-v^{i_{1}u_{2}}_{u_{1}j_{1}}S_{i_{1}}^{j_{1}}-v^{i_{1}j_{1}}_{ju_{1}}S_{i_{1}}^{u_{2}})\hat{E}^{+}_{u_{1}u_{2}}-\sum_{\mathbf{u}}v^{i_{1}u_{4}}_{u_{1}u_{2}}S_{i_{1}}^{u_{3}}\hat{E}^{+}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\mathbf{t}\mathbf{u}}v^{t_{1}j_{1}}_{u_{1}u_{2}}S_{t_{2}}^{j_{1}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}}-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}(2v^{t_{1}u_{2}}_{j_{1}j_{2}}S_{t_{2}}^{u_{1}}-v^{t_{1}u_{2}}_{u_{1}j_{1}}S_{t_{2}}^{j_{1}}-v^{t_{1}j_{1}}_{ju_{1}}S_{t_{2}}^{u_{2}})\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}v^{t_{1}u_{4}}_{u_{1}u_{2}}S_{t_{2}}^{u_{3}}\hat{E}^{\sigma_{1}}_{t_{1}t_{2}}\hat{e}^{\sigma_{2}\sigma_{1}}_{\mathbf{u}}.$
(151)

####  D.3.4 $\widehat{VP}_{4}$

Finally, we find the following active space operator for the fourth term, Eq.
(93):

$\displaystyle\widehat{VP}_{\mathrm{4,active}}=$
$\displaystyle-(8v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{i_{2}}_{j_{2}}S_{i_{2}}^{j_{2}}-4v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{j_{2}}S^{i_{1}}_{j_{1}}-4v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{j_{2}}+2v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{j_{2}})$
$\displaystyle-\sum_{\mathbf{u}}(4v^{i_{1}i_{1}}_{u_{1}u_{2}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{j_{1}}-2v^{i_{1}i_{2}}_{u_{1}u_{2}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{j_{1}}+4v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{i_{2}}_{u_{2}}S^{i_{2}}_{u_{1}}-2v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{u_{2}}S^{i_{1}}_{u_{1}}$
$\displaystyle-2v^{i_{1}i_{1}}_{u_{1}j_{1}}S^{i_{2}}_{u_{2}}S^{i_{2}}_{j_{1}}+v^{i_{1}i_{2}}_{u_{1}j_{1}}S^{i_{2}}_{u_{2}}S^{i_{1}}_{j_{1}}-2v^{i_{1}i_{1}}_{j_{1}u_{1}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{u_{2}}+v^{i_{1}i_{2}}_{j_{1}u_{1}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{u_{2}})\hat{E}^{+}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}(2v^{i_{1}i_{1}}_{u_{1}u_{2}}S^{i_{2}}_{u_{4}}S^{i_{2}}_{u_{3}}-v^{i_{1}i_{2}}_{u_{1}u_{2}}S^{i_{2}}_{u_{4}}S^{i_{1}}_{u_{3}})\hat{e}^{+}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\mathbf{t}}(4v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{i_{1}}_{j_{2}}S^{i_{1}}_{j_{2}}-2v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{i_{1}}_{j_{1}}S^{i_{1}}_{j_{2}}+4v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{t_{1}}_{j_{2}}S^{t_{2}}_{j_{2}}-2v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{t_{1}}_{j_{1}}S^{t_{2}}_{j_{2}}$
$\displaystyle-2v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{j_{2}}S^{t_{2}}_{j_{2}}+v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{j_{2}}-2v^{it_{1}}_{j_{1}j_{2}}S^{t_{2}}_{j_{2}}S^{i_{1}}_{j_{2}}+v^{it_{1}}_{j_{1}j_{2}}S^{t_{2}}_{j_{1}}S^{i_{1}}_{j_{2}})\hat{E}^{+}_{t_{1}t_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\\\
\sigma_{1}\sigma_{2}\end{subarray}}(2v^{t_{1}t_{2}}_{j_{1}j_{2}}S_{t_{3}j_{2}}S_{t_{4}j_{2}}-v^{t_{1}t_{2}}_{j_{1}j_{2}}S_{t_{3}j}S_{t_{4}j_{2}})\hat{e}^{\sigma_{1}\sigma_{2}}_{t_{1}t_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\end{subarray}}(2v^{t_{1}t_{2}}_{u_{1}u_{2}}S^{i_{1}}_{j_{1}}S^{i_{1}}_{j_{1}}+2v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{i_{1}}_{u_{1}}S^{i_{1}}_{u_{2}}-v^{t_{1}t_{2}}_{u_{1}j_{1}}S^{i_{1}}_{u_{2}}S^{i_{1}}_{j_{1}}-v^{t_{1}t_{2}}_{j_{1}u_{1}}S^{i_{1}}_{j_{1}}S^{i_{1}}_{u_{2}}+2v^{i_{1}i_{1}}_{u_{1}u_{2}}S^{t_{1}}_{j_{1}}S^{t_{2}}_{j_{1}}$
$\displaystyle-v^{t_{1}i_{1}}_{u_{1}u_{2}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{j_{1}}-v^{i_{1}t_{2}}_{u_{1}u_{2}}S^{t_{1}}_{j_{1}}S^{i_{1}}_{j_{1}})\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}(4v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{t_{1}}_{u_{2}}S^{t_{2}}_{u_{1}}-2v^{i_{1}i_{1}}_{u_{1}j_{1}}S^{t_{1}}_{u_{2}}S^{t_{2}}_{j}-2v^{i_{1}i_{1}}_{ju_{2}}S^{t_{1}}_{j}S^{t_{2}}_{u_{1}}-2v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{u_{2}}S^{t_{2}}_{u_{1}}-2v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{u_{2}}S^{t_{2}}_{u_{1}}$
$\displaystyle+v^{t_{1}i_{1}}_{u_{1}j_{1}}S^{i_{1}}_{u_{2}}S^{t_{2}}_{j_{1}}+v^{t_{1}i_{1}}_{j_{1}u_{1}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{u_{2}}+v^{i_{1}t_{2}}_{u_{1}j_{1}}S^{t_{1}}_{u_{2}}S^{i_{1}}_{j_{1}}+v^{i_{1}t_{2}}_{j_{1}u_{2}}S^{t_{1}}_{j_{1}}S^{i_{1}}_{u_{1}})\hat{E}^{\sigma}_{\mathbf{t}}\hat{E}^{\sigma}_{\mathbf{u}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}(2v^{i_{1}i_{1}}_{u_{1}u_{2}}S^{t_{1}}_{u_{4}}S^{t_{2}}_{u_{3}}-v^{t_{1}i_{1}}_{u_{1}u_{2}}S^{i_{1}}_{u_{4}}S^{t_{2}}_{u_{3}}-v^{it_{1}}_{u_{1}u_{2}}S^{t_{2}}_{u_{4}}S^{i_{1}}_{u_{3}})\hat{E}^{\sigma_{2}}_{\mathbf{t}}\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{u}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\sigma_{3}\end{subarray}}v^{t_{1}t_{2}}_{u_{1}u_{2}}S^{i_{1}}_{u_{4}}S^{i_{2}}_{u_{3}}\hat{E}^{\sigma_{1}}_{t_{1}t_{2}}\hat{e}^{\sigma_{2}\sigma_{3}}_{u_{1}u_{2}}-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\sigma_{3}\end{subarray}}v^{t_{1}t_{2}}_{u_{1}u_{2}}S^{t_{3}}_{j_{1}}S^{t_{4}}_{j_{1}}\hat{e}^{\sigma_{1}\sigma_{2}}_{t_{1}t_{2}}\hat{E}^{\sigma_{3}}_{u_{1}u_{2}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}(2v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{t_{3}}_{u_{2}}S^{t_{4}}_{u_{1}}-v^{t_{1}t_{2}}_{u_{1}j_{1}}S^{t_{3}}_{u_{2}}S^{t_{4}}_{j_{1}}-v^{t_{1}t_{2}}_{j_{1}u_{1}}S^{t_{3}}_{j_{1}}S^{t_{4}}_{u_{2}})\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{t}}\hat{E}^{\sigma_{2}}_{\mathbf{u}}$
$\displaystyle-\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\sigma_{3}\end{subarray}}v^{t_{1}t_{2}}_{u_{1}u_{2}}S^{t_{3}}_{u_{4}}S^{t_{4}}_{u_{3}}\hat{e}^{\sigma_{1}\sigma_{3}}_{\mathbf{t}}\hat{e}^{\sigma_{2}\sigma_{3}}_{\mathbf{u}}$
(152)

Combining all four terms, the active space electrostatic-exchange operator may
be written as:

$\displaystyle\widehat{VP}_{\mathrm{active}}$
$\displaystyle=VP_{0,\mathrm{active}}+\widehat{VP}_{A,\mathrm{active}}+\widehat{VP}_{B,\mathrm{active}}+\widehat{VP}_{1m,\mathrm{active}}$
$\displaystyle+\widehat{VP}_{1\ell,\mathrm{active}}+\widehat{VP}_{2,\mathrm{active}}+\widehat{VP}_{3,\mathrm{active}}+\widehat{VP}_{4,\mathrm{active}}$
(153)

where

$\displaystyle\widehat{VP}_{A,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\\\
\sigma\end{subarray}}\tilde{f}_{t_{1}t_{2}}^{(A)}\hat{E}^{\sigma}_{t_{1}t_{2}}+\sum_{\begin{subarray}{c}\mathbf{t}\\\
\sigma\tau\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{t_{3}t_{4}}\,\hat{e}_{\mathbf{t}}^{\sigma\tau},$
(154) $\displaystyle\widehat{VP}_{B,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{u}\\\
\sigma\end{subarray}}\tilde{f}_{u_{1}u_{2}}^{(B)}\hat{E}^{\sigma}_{u_{1}u_{2}}+\sum_{\begin{subarray}{c}\mathbf{u}\\\
\sigma\tau\end{subarray}}\tilde{v}^{u_{1}u_{2}}_{u_{3}u_{4}}\hat{e}_{\mathbf{u}}^{\sigma\tau},$
(155) $\displaystyle\widehat{VP}_{1m,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}},$
(156) $\displaystyle\widehat{VP}_{1\ell,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\end{subarray}}\tilde{v}^{t_{1}u_{2}}_{u_{1}t_{2}}\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}},$
(157) $\displaystyle\widehat{VP}_{2,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\tau\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{u_{1}t_{4}}S^{t_{3}}_{u_{2}}\,\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{t}}\hat{E}^{\sigma_{2}}_{\mathbf{u}},$
(158) $\displaystyle\widehat{VP}_{3,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}\tilde{v}^{t_{1}u_{4}}_{u_{1}u_{2}}S^{t_{2}}_{u_{3}}\hat{E}^{\sigma_{2}}_{\mathbf{t}}\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{u}},$
(159) $\displaystyle\widehat{VP}_{4,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\sigma_{3}\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}\tilde{p}^{t_{3}t_{4}}_{u_{3}u_{4}}\;\hat{e}^{\sigma_{1}\sigma_{2}}_{\mathbf{t}}\hat{e}^{\sigma_{3}\sigma_{2}}_{\mathbf{u}},$
(160)

with renormalized active space tensor coefficients defined as:

$\displaystyle VP_{0,\mathrm{active}}=$
$\displaystyle-2v^{i_{1}j_{1}}_{j_{1}i_{1}}-(4{v}^{i_{1}i_{1}}_{j_{1}i_{2}}S^{i_{2}}_{j_{1}}-2{v}^{i_{2}i_{1}}_{j_{1}i_{2}}S^{i_{1}}_{j_{1}})-(4v^{i_{1}j_{2}}_{j_{1}j_{2}}S^{i_{1}}_{j_{2}}-2v^{i_{1}j_{2}}_{j_{2}j}S^{i_{1}}_{j_{1}})$
(161)
$\displaystyle-(8v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{i_{2}}_{j_{2}}S^{i_{2}}_{j_{2}}-4v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{j_{2}}S^{i_{1}}_{j_{2}}-4v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{j_{2}}+2v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{j_{2}})$
(162) $\displaystyle\tilde{f}_{t_{1}t_{2}}^{(A)}=$
$\displaystyle-v^{j_{1}t_{2}}_{t_{1}j_{1}}-(2v^{t_{1}t_{2}}_{j_{1}i_{1}}S^{i_{1}}_{j_{1}}+2{v}^{i_{1}i_{1}}_{j_{1}t_{2}}S^{t_{1}}_{j_{1}}-{v}^{t_{1}i}_{jt_{2}}S^{i_{1}}_{j_{1}}-{v}^{i_{1}t_{2}}_{j_{1}i_{1}}S^{t_{1}}_{j_{1}})-(v^{t_{1}j_{2}}_{j_{1}j_{2}}S^{t_{2}}_{j_{2}}-v^{t_{1}j_{2}}_{j_{2}j}S^{t_{2}}_{j_{1}})$
(163)
$\displaystyle-(4v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{i_{1}}_{j_{2}}S^{i_{1}}_{j_{2}}-2v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{i_{1}}_{j_{1}}S^{i_{1}}_{j_{2}}-2v^{i_{1}i_{1}}_{j_{1}j_{2}}S^{t_{1}}_{j_{1}}S^{t_{2}}_{j_{2}}-2v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{j_{2}}S^{t_{2}}_{j_{2}}+v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{j_{2}}-2v^{it_{1}}_{j_{1}j_{2}}S^{t_{2}}_{j_{2}}S^{i_{1}}_{j_{2}}+v^{it_{1}}_{j_{1}j_{2}}S^{t_{2}}_{j_{1}}S^{i_{1}}_{j_{2}})$
$\displaystyle\tilde{f}_{u_{1}u_{2}}^{(B)}=$
$\displaystyle-v^{i_{1}u_{2}}_{u_{1}i_{1}}-(2{v}^{i_{1}i_{1}}_{u_{1}i_{2}}S^{i_{2}}_{u_{2}}-{v}^{i_{2}i_{1}}_{u_{1}i_{2}}S^{i_{1}}_{u_{2}})-(2v^{i_{1}j_{1}}_{u_{1}u_{2}}S^{i_{1}}_{j_{1}}+2v^{i_{1}u_{2}}_{j_{1}j_{2}}S^{i_{1}}_{u_{1}}-v^{i_{1}u_{2}}_{u_{1}j_{1}}S^{i_{1}}_{j_{1}}-v^{i_{1}j_{1}}_{ju_{1}}S^{i_{1}}_{u_{2}})$
(164)
$\displaystyle-(4v^{i_{1}i_{1}}_{u_{1}u_{2}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{j_{1}}-2v^{i_{1}i_{2}}_{u_{1}u_{2}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{j_{1}}-2v^{i_{1}i_{2}}_{j_{1}j_{2}}S^{i_{2}}_{u_{2}}S^{i_{1}}_{u_{1}}-2v^{i_{1}i_{1}}_{u_{1}j_{1}}S^{i_{2}}_{u_{2}}S^{i_{2}}_{j_{1}}+v^{i_{1}i_{2}}_{u_{1}j_{1}}S^{i_{2}}_{u_{2}}S^{i_{1}}_{j_{1}}-2v^{i_{1}i_{1}}_{j_{1}u_{1}}S^{i_{2}}_{j_{1}}S^{i_{2}}_{u_{2}}+v^{i_{1}i_{2}}_{j_{1}u_{1}}S^{i_{2}}_{j_{1}}S^{i_{1}}_{u_{2}})$
$\displaystyle\tilde{v}^{t_{1}t_{2}}_{t_{3}t_{4}}=$
$\displaystyle-{v}^{t_{1}t_{2}}_{j_{1}t_{4}}S^{t_{3}}_{j_{1}}+v^{t_{1}t_{2}}_{j_{1}j_{2}}S^{t_{3}}_{j_{1}}S^{t_{4}}_{j_{2}}$
(165) $\displaystyle\tilde{v}^{u_{1}u_{2}}_{u_{3}u_{4}}=$
$\displaystyle-v^{i_{1}u_{4}}_{u_{1}u_{2}}S^{i_{1}}_{u_{3}}+v^{i_{1}i_{2}}_{u_{1}u_{2}}S^{i_{2}}_{u_{4}}S^{i_{1}}_{u_{3}}$
(166) $\displaystyle\tilde{v}^{t_{1}u_{2}}_{u_{1}t_{2}}=$
$\displaystyle-v^{t_{1}u_{2}}_{u_{1}t_{2}}-(2v^{i_{1}i_{1}}_{u_{1}t_{2}}S^{t_{1}}_{u_{2}}-v^{t_{1}i_{1}}_{u_{1}t_{2}}S^{i_{1}}_{u_{2}}-v^{i_{1}t_{2}}_{u_{1}i_{1}}S^{t_{1}}_{u_{2}})-(2v^{t_{1}u_{2}}_{j_{1}j_{2}}S^{t_{2}}_{u_{1}}-v^{t_{1}u_{2}}_{u_{1}j_{1}}S^{t_{2}}_{j_{1}}-v^{t_{1}j_{1}}_{ju_{1}}S^{t_{2}}_{u_{2}})$
(167)
$\displaystyle+(4v^{i_{1}i_{1}}_{u_{1}j_{1}}S^{t_{1}}_{u_{2}}S^{t_{2}}_{j}+4v^{t_{1}i_{1}}_{j_{1}j_{2}}S^{i_{1}}_{u_{2}}S^{t_{2}}_{u_{1}}-2v^{t_{1}i_{1}}_{u_{1}j_{1}}S^{i_{1}}_{u_{2}}S^{t_{2}}_{j_{1}}-2v^{t_{1}i_{1}}_{j_{1}u_{1}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{u_{2}})$
$\displaystyle\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}=$
$\displaystyle-v^{t_{1}t_{2}}_{u_{1}i_{1}}S^{i_{1}}_{u_{2}}-v^{t_{1}j_{1}}_{u_{1}u_{2}}S^{t_{2}}_{j_{1}}-(2v^{t_{1}t_{2}}_{u_{1}u_{2}}S^{i_{1}}_{j_{1}}S^{i_{1}}_{j_{1}}-2v^{t_{1}t_{2}}_{u_{1}j_{1}}S^{i_{1}}_{u_{2}}S^{i_{1}}_{j_{1}}-2v^{t_{1}i_{1}}_{u_{1}u_{2}}S^{i_{1}}_{j_{1}}S^{t_{2}}_{j_{1}})$
(168) $\displaystyle\tilde{v}^{t_{1}t_{2}}_{u_{1}t_{4}}=$
$\displaystyle-(v^{t_{1}t_{2}}_{u_{1}t_{4}}-2v^{t_{1}t_{2}}_{u_{1}j_{1}}S^{t_{4}}_{j_{1}})$
(169) $\displaystyle\tilde{v}^{t_{1}u_{4}}_{u_{1}u_{2}}=$
$\displaystyle-({v}^{t_{1}u_{4}}_{u_{1}u_{2}}-2v^{t_{1}i_{1}}_{u_{1}u_{2}}S^{i_{1}}_{u_{4}})$
(170) $\displaystyle\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}=$
$\displaystyle+v^{t_{1}t_{2}}_{u_{1}u_{2}}+2v^{t_{1}t_{2}}_{j_{1}j_{2}}\delta_{u_{3}u_{4}}/\tilde{\eta}_{B}+2v^{i_{1}i_{1}}_{u_{1}u_{2}}\delta_{t_{3}t_{4}}/\tilde{\eta}_{A}+4v^{i_{1}i_{1}}_{j_{1}j_{2}}\delta_{t_{3}t_{4}}\delta_{u_{3}u_{4}}/(\tilde{\eta}_{A}\tilde{\eta}_{B})$
(171) $\displaystyle\tilde{p}^{t_{3}t_{4}}_{u_{3}u_{4}}=$
$\displaystyle-S^{t_{3}}_{u_{4}}S^{t_{4}}_{u_{3}}-S^{t_{3}}_{j_{1}}S^{t_{4}}_{j_{1}}\delta_{u_{3}u_{4}}/\tilde{\eta}_{B\sigma}-S^{i_{1}}_{u_{3}}S^{i_{1}}_{u_{4}}\delta_{t_{3}t_{4}}/\tilde{\eta}_{A\sigma}.$
(172)

Converting to chemist notation, we obtain:

$\displaystyle\widehat{VP}_{A,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\\\
\sigma\end{subarray}}(\tilde{f}_{t_{1}t_{2}}^{(A)}-\sum_{t}\tilde{v}^{t_{1}t}_{tt_{2}})\hat{E}^{\sigma}_{t_{1}t_{2}}+\sum_{\begin{subarray}{c}\mathbf{t}\\\
\sigma\tau\end{subarray}}\tilde{\nu}^{t_{1}t_{2}}_{t_{3}t_{4}}\,\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\tau}_{t_{3}t_{4}},$
(173) $\displaystyle\widehat{VP}_{B,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{u}\\\
\sigma\end{subarray}}(\tilde{f}_{u_{1}u_{2}}^{(B)}-\sum_{u}\tilde{v}^{u_{1}u}_{uu_{2}})\hat{E}^{\sigma}_{u_{1}u_{2}}+\sum_{\begin{subarray}{c}\mathbf{u}\\\
\sigma\tau\end{subarray}}\tilde{\nu}^{u_{1}u_{2}}_{u_{3}u_{4}}\hat{E}^{\sigma}_{u_{1}u_{2}}\hat{E}^{\tau}_{u_{3}u_{4}},$
(174) $\displaystyle\widehat{VP}_{1m,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\end{subarray}}\tilde{\nu}^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{E}^{+}_{t_{1}t_{2}}\hat{E}^{+}_{u_{1}u_{2}},$
(175) $\displaystyle\widehat{VP}_{1\ell,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\end{subarray}}\tilde{\nu}^{t_{1}u_{2}}_{u_{1}t_{2}}\hat{E}^{\sigma}_{t_{1}t_{2}}\hat{E}^{\sigma}_{u_{1}u_{2}},$
(176) $\displaystyle\widehat{VP}_{2,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}\tilde{\nu}^{t_{1}t_{2}}_{u_{1}t_{4}}S^{t_{3}}_{u_{2}}\,\hat{E}^{\sigma_{1}}_{t_{1}t_{2}}\hat{E}^{\sigma_{2}}_{t_{3}t_{4}}\hat{E}^{\sigma_{2}}_{u_{1}u_{2}},$
(177) $\displaystyle\widehat{VP}_{3,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\end{subarray}}\tilde{\nu}^{t_{1}u_{4}}_{u_{1}u_{2}}S^{t_{2}}_{u_{3}}\hat{E}^{\sigma_{2}}_{t_{1}t_{2}}\hat{E}^{\sigma_{1}}_{u_{1}u_{2}}\hat{E}^{\sigma_{2}}_{u_{3}u_{4}},$
(178) $\displaystyle\widehat{VP}_{4,\mathrm{active}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma_{1}\sigma_{2}\sigma_{3}\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}\tilde{p}^{t_{3}t_{4}}_{u_{3}u_{4}}\;\hat{E}^{\sigma_{1}}_{t_{1}t_{2}}\hat{E}^{\sigma_{2}}_{t_{3}t_{4}}\hat{E}^{\sigma_{3}}_{u_{1}u_{2}}\hat{E}^{\sigma_{2}}_{u_{3}u_{4}},$
(179)

where the active space tensor coefficients in the chemist notation are given
by,

$\displaystyle\tilde{\nu}^{t_{1}t_{2}}_{u_{1}u_{2}}$
$\displaystyle=\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}-\sum_{t_{3}}\tilde{v}^{t_{1}t_{3}}_{u_{1}u_{2}}S^{t_{3}}_{j_{1}}S^{t_{2}}_{j_{1}}-\sum_{u_{3}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{3}}S^{i_{1}}_{u_{3}}S^{i_{1}}_{u_{2}},$
(180) $\displaystyle\tilde{\nu}^{t_{1}u_{2}}_{u_{1}t_{2}}$
$\displaystyle=\tilde{v}^{t_{1}u_{2}}_{u_{1}t_{2}}+\sum_{t_{3}u_{3}}\tilde{v}^{t_{1}t_{3}}_{u_{1}u_{3}}S^{t_{3}}_{u_{2}}S^{t_{2}}_{u_{2}}-\sum_{u_{3}}\tilde{v}_{u_{1}u_{3}}^{t_{1}u_{2}}S^{t_{2}}_{u_{3}}-\sum_{t_{3}}\tilde{v}^{t_{1}t_{3}}_{u_{1}t_{2}}S^{t_{3}}_{u_{2}},$
(181) $\displaystyle\tilde{\nu}^{t_{1}t_{2}}_{u_{1}t_{4}}$
$\displaystyle=\tilde{v}^{t_{1}t_{2}}_{u_{1}t_{4}}-\sum_{u_{3}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{3}}S^{t_{4}}_{u_{3}}\,,$
(182) $\displaystyle\tilde{\nu}_{u_{1}u_{2}}^{t_{1}u_{4}}$
$\displaystyle=\tilde{v}_{u_{1}u_{2}}^{t_{1}u_{4}}-\sum_{t_{3}}\tilde{v}^{t_{1}t_{3}}_{u_{1}u_{2}}S^{t_{3}}_{u_{4}}\,.$
(183)

The intra-monomer tensors under the chemist notation remain equivalent, i.e
$\tilde{\nu}^{t_{1}t_{2}}_{t_{3}t_{4}}=\tilde{v}^{t_{1}t_{2}}_{t_{3}t_{4}}$
and
$\tilde{\nu}^{u_{1}u_{2}}_{u_{3}u_{4}}=\tilde{v}^{u_{1}u_{2}}_{u_{3}u_{4}}$.

##  Appendix E SAPT-EVE algorithm

To ensure a first order interaction energy estimate to chemical accuracy, the
target accuracy of the three observables
$\hat{F}=\\{\hat{V},\hat{P},\widehat{VP}\\}$ must be optimally chosen to
simultaneously satisfy the desired accuracy and minimize computational costs.
That is, we must find a target precision $\varepsilon_{F}$ for every
observable $\hat{F}$, such that they satisfy one over all precision
$\varepsilon_{\mathrm{targ}}$ while minimizing the cost of the three prime
phase estimation algorithms. The resource cost minimization can be formulated
as the following optimization problem,

$\displaystyle\min_{\varepsilon_{V},\varepsilon_{VP},\varepsilon_{P}}\left(\frac{\lambda_{V}}{\varepsilon_{V}}+\frac{\lambda_{VP}}{\varepsilon_{VP}}+\frac{\lambda_{P}}{\varepsilon_{P}}\right)$
(184)
$\displaystyle\text{s.t.}\quad(1+\lambda_{P})\varepsilon_{V}+\varepsilon_{VP}+\lambda_{V}\varepsilon_{P}=\varepsilon_{\mathrm{targ}}\,.$
(185)

which can be solved with the Lagrange method yielding

$\displaystyle\left\\{\vphantom{\frac{\sum_{1}}{\sum_{1}}}\varepsilon_{V},\,\varepsilon_{V\\!P},\,\varepsilon_{P}\right\\}=\frac{\varepsilon_{\mathrm{targ}}}{\sqrt{(1+\lambda_{P})\lambda_{V}}+\sqrt{\lambda_{V\\!P}}+\sqrt{\lambda_{V}\lambda_{P}}}\left\\{\vphantom{\frac{\sum_{1}}{\sum_{1}}}\sqrt{\frac{\lambda_{V}}{1+\lambda_{P}}},\,\sqrt{\lambda_{V\\!P}},\,\sqrt{\frac{\lambda_{P}}{\lambda_{V}}}\right\\}\,.$
(186)

This solution is only optimal if given no further information about the
expected values of $\langle\hat{V}\rangle$, $\langle\hat{P}\rangle$ and
$\langle\widehat{VP}\rangle$. We have also upper bounded expectation values
$\langle F\rangle$ with their respective $\ell_{1}$ norms $\lambda_{F}$ in Eq.
(185), which is a very loose bound. In practice, one will certainly be able to
relax the target accuracies by bootstrapping Eq. (185) with low accuracy
estimates $\langle F\rangle_{\text{low}}$, replacing
$\lambda_{F}\mapsto\langle F\rangle_{\text{low}}$.

For every observable $\hat{F}$ we have to adjust the polynomial degrees of
SAPT-EVE’s inner phase estimation routines $\mathsf{iQPE_{A}}$ and
$\mathsf{iQPE_{B}}$ according to the target errors $\varepsilon_{F}$. While we
have demonstrated this procedure in Ref. [24] for QSP-EVE, it is not
immediately clear how the discretization errors of two inner phase estimation
routines would be dealt with in SAPT-EVE. Let us quickly summarize the
estimation procedure behind QSP-EVE and SAPT-EVE. The iterate of Figure 2 is
the product of two reflections $\mathcal{R}_{\pi}$ and $\mathcal{R}_{\tau}$,
each of which has a structure
$\mathcal{R}_{\mathfrak{X}}=1-2\hat{\mathfrak{X}}$ with some arbitrary-rank
projectors $\hat{\mathfrak{X}}=\hat{\pi}$, $\hat{\tau}$. The product
$\mathcal{R}_{\tau}\mathcal{R}_{\pi}$ now has the eigenvalues $\exp(\pm
i2\arccos w_{k})$, where $w_{k}$ are the singular values of the product of
projectors $\hat{\pi}\cdot\hat{\tau}$. The idea behind SAPT-EVE is now that
one of the singular values is a function of $\langle\hat{F}\rangle$. In a
world without discretization errors, the circuits we employ would fix the
reference states exactly:

$\displaystyle\mathsf{iQPE}^{\dagger}_{q}\,\mathsf{Refl}^{\phantom{\dagger}}_{q}\,\mathsf{iQPE}^{\phantom{\dagger}}_{q}\approx
1-2|Q_{q}\rangle\\!\langle
Q_{q}|_{\mathsf{sim}\,q,\mathsf{enc}[H_{q}]}\otimes|\boldsymbol{0}\rangle\\!\langle\boldsymbol{0}|_{\mathsf{phase}\,q}\,-\dots$
(187)

for $q=\mathsf{A},\mathsf{B}$ labeling routines associated with the two
subsystems, and $|Q_{\mathsf{A}}\rangle$, $|Q_{\mathsf{B}}\rangle$ are their
respective qubitization ground states. Following the state on the right-hand
side of Eq. (187) are other terms orthogonal to the all zero state
$|\boldsymbol{0}\rangle$ on the $\mathsf{phase}$ register of the corresponding
monomer $q=\mathsf{A},\mathsf{B}$. Considering the leading order contribution
of the discretization errors, we introduce the qubitization states
$|\mathcal{E}_{q}\rangle$ associated with the first excited states in the
respective subsystem $q$. Note that $|Q_{\mathsf{A}}\rangle$ and
$|\mathcal{E}_{\mathsf{A}}\rangle$ are associated with Hamiltonian $H_{A}$,
they are generally formed with respect to different eigenenergies than
$|Q_{\mathsf{B}}\rangle$ and $|\mathcal{E}_{\mathsf{B}}\rangle$, which are
associated with $H_{B}$. For both subsystems we introduce the projectors
$\varrho^{\mathsf{A}}(E)_{\mathsf{phase\,A}}$ and
$\varrho^{\mathsf{B}}(E)_{\mathsf{phase\,B}}$ associated with either excited
state $E=\mathcal{E}$ or ground state $E=Q$. We find

$\displaystyle\hat{\pi}=\sum_{E=Q,\mathcal{E}}\left(\bigotimes_{q=\mathsf{A},\mathsf{B}}|E_{q}\rangle\\!\langle
E_{q}|_{\mathsf{sim}\,q,\mathsf{enc}[H_{q}]}\otimes\varrho^{q}(E)_{\mathsf{phase}\,q}\right)\otimes|\boldsymbol{0}\rangle\\!\langle\boldsymbol{0}|_{\mathsf{enc}[F]}\,,$
(188)
$\displaystyle\hat{\tau}=\frac{1}{2}\left(1-\mathcal{B}[\hat{F}]\right)_{\mathsf{sim\,A,\,sim\,B,\,enc}[F]}\otimes\bigotimes_{q=\mathsf{A},\mathsf{B}}|\boldsymbol{0}\rangle\\!\langle\boldsymbol{0}|_{\mathsf{enc}[H_{q}]}\,.$
(189)

To investigate the product $\hat{\pi}\cdot\hat{\tau}$, we consider the
singular value decomposition

$\displaystyle\varrho^{q}(Q)\cdot\varrho^{q}(\mathcal{E})=\sum_{j=1}^{r}\Omega^{(q)}|\xi_{Q,q,j}\rangle\\!\langle\xi_{\mathcal{E},q,j}|\,$
(190)

where $\Omega^{(q)}$ are singular values, $|\xi_{Q,q,j}\rangle$ and
$|\xi_{\mathcal{E},q,j}\rangle$ are singular vectors and where the sum over
$j$ is in the range from $1$ to $r$, the rank of
$\varrho^{q}(Q)\cdot\varrho^{q}(\mathcal{E})$. We will now learn the singular
values $w_{k}$ of $\hat{\pi}\cdot\hat{\tau}$ by computing the eigenvalues
$(w_{k})^{2}$ of $\hat{\pi}\cdot\hat{\tau}\cdot\hat{\pi}$, which has a block-
diagonal form due to the singular value decomposition in Eq. (190):

$\displaystyle\hat{\pi}\cdot\hat{\tau}\cdot\hat{\pi}=\bigoplus_{i,j}\frac{1}{8}\left[\begin{matrix}1-F_{00}&-F_{01}\Omega^{(\mathsf{B})}_{j}&-F_{02}\Omega^{(\mathsf{A})}_{i}&-F_{03}\Omega^{(\mathsf{A})}_{i}\Omega^{(\mathsf{B})}_{j}\\\\[4.0pt]
-F_{10}\Omega^{(\mathsf{B})}_{j}&1-F_{11}&-F_{12}\Omega^{(\mathsf{A})}_{i}\Omega^{(\mathsf{B})}_{j}&-F_{13}\Omega^{(\mathsf{A})}_{i}\\\\[4.0pt]
-F_{20}\Omega^{(\mathsf{A})}_{i}&-F_{21}\Omega^{(\mathsf{A})}_{i}\Omega^{(\mathsf{B})}_{j}&1-F_{22}&-F_{23}\Omega^{(\mathsf{B})}_{j}\\\\[4.0pt]
-F_{30}\Omega^{(\mathsf{A})}_{i}\Omega^{(\mathsf{B})}_{j}&-F_{31}\Omega^{(\mathsf{A})}_{i}&-F_{32}\Omega^{(\mathsf{B})}_{j}&1-F_{33}\end{matrix}\right]_{i_{1}j_{1}},$
(191)

where $F_{lm}$ are some matrix elements of the observable $\hat{F}$, and the
block matrices $[\cdots]_{i_{1}j_{1}}$ denote operators with respect to the
bases states

$\displaystyle|Q_{\mathsf{A}}\rangle_{\mathsf{sim\,A,\,enc}[H_{A}]}\otimes|\xi_{Q,\mathsf{A},i}\rangle_{\mathsf{phase\,A}}\otimes|Q_{\mathsf{B}}\rangle_{\mathsf{sim\,B,\,enc}[H_{B}]}\otimes|\xi_{Q,\mathsf{B},j}\rangle_{\mathsf{phase\,B}}\otimes|\boldsymbol{0}\rangle_{\mathsf{enc}[F]},$
(192)
$\displaystyle|Q_{\mathsf{A}}\rangle_{\mathsf{sim\,A,\,enc}[H_{A}]}\otimes|\xi_{Q,\mathsf{A},i}\rangle_{\mathsf{phase\,A}}\otimes|\mathcal{E}_{\mathsf{B}}\rangle_{\mathsf{sim\,B,\,enc}[H_{B}]}\otimes|\xi_{\mathcal{E},\mathsf{B},j}\rangle_{\mathsf{phase\,B}}\otimes|\boldsymbol{0}\rangle_{\mathsf{enc}[F]},$
(193)
$\displaystyle|\mathcal{E}_{\mathsf{A}}\rangle_{\mathsf{sim\,A,\,enc}[H_{A}]}\otimes|\xi_{\mathcal{E},\mathsf{A},i}\rangle_{\mathsf{phase\,A}}\otimes|Q_{\mathsf{B}}\rangle_{\mathsf{sim\,B,\,enc}[H_{B}]}\otimes|\xi_{Q,\mathsf{B},j}\rangle_{\mathsf{phase\,B}}\otimes|\boldsymbol{0}\rangle_{\mathsf{enc}[F]},$
(194)
$\displaystyle|\mathcal{E}_{\mathsf{A}}\rangle_{\mathsf{sim\,A,\,enc}[H_{A}]}\otimes|\xi_{\mathcal{E},\mathsf{A},i}\rangle_{\mathsf{phase\,A}}\otimes|\mathcal{E}_{\mathsf{B}}\rangle_{\mathsf{sim\,B,\,enc}[H_{B}]}\otimes|\xi_{\mathcal{E},\mathsf{B},j}\rangle_{\mathsf{phase\,B}}\otimes|\boldsymbol{0}\rangle_{\mathsf{enc}[F]}\,.$
(195)

For vanishing singular values, $\Omega_{j}^{(q)}=0$, we find that there is one
solution of Eq. (191) with eigenvalue $w_{\hat{k}}^{2}=(1-F_{00})/8$. Since
$F_{00}$ is the matrix element associated with the ground states of both
subsystems, $w_{\hat{k}}^{2}$ is the solution with the eigenphase $\pm
2\arccos\sqrt{(1-\langle F\rangle)/8}$. For at least one $\Omega_{j}^{(q)}\neq
0$, the block matrices $[\cdots]_{i_{1}j_{1}}$ must be diagonalized. The good
solution is the one whose eigenvalue $w_{\hat{k}}^{2}$ is closest to
$(1-F_{00})/8$. Assuming that the contamination of this solution with
contributions of excited states will increase, we replace individual
$\Omega_{j}^{(\mathsf{A})}$ and $\Omega_{j}^{(\mathsf{B})}$ in Eq. (191) with
one

$\displaystyle\Omega=\max_{j}\max_{q=\mathsf{A,B}}\Omega_{j}^{(q)}\,.$ (196)

Perturbation theory in $\Omega$ informs us that the deviation of the solution
from $(1-F_{00})/8$ is $O(\Omega)$, only if the diagonal elements of
$F_{j_{1}j_{2}}$ are all equal. Assuming that the structure of
$F_{i_{1}j_{1}}$ is as malicious as possible, we set all $F_{j_{1}j_{2}}$ to
be equal. Neglecting quadratic contributions $\Omega^{2}$ due to their size,
we get a deviation of $w_{\hat{k}}^{2}$ from $(1-F_{00})/8$ that is entirely
linear in $\Omega$. Note that for $\Omega=1$, we would bound
$0<w_{\hat{k}}^{2}<1/4$ due to $||\hat{F}||\leq 1$. This means that we can
bound the error of estimated observable $F_{\text{est}}$ and actual observable
$\langle\hat{F}\rangle$ by

$\displaystyle\left|F_{\text{est}}-\langle\hat{F}\rangle\right|\leq
2\Omega\,.$ (197)

We therefore need to fortify both $\mathsf{iQPE}_{q}$ routines equally well
against errors using quantum signal processing (QSP). Given the QSP routines,
the values of $\Omega^{(\mathsf{A})}_{i}$ and $\Omega^{(\mathsf{B})}_{j}$ can
be obtained with a numerical procedure outlined in [24]. The success
probability of this routine, if one prepares an initial state of

$\displaystyle\left(\bigotimes_{q=\mathsf{A,B}}|Q_{q}\rangle_{\mathsf{sim}\,q,\,\mathsf{enc}[H_{q}]}\otimes|\boldsymbol{0}\rangle_{\mathsf{phase}\,q}\right)\otimes|\boldsymbol{0}\rangle_{\mathsf{enc}[F]}$
(198)

is between
$\langle\boldsymbol{0}|\varrho^{\mathsf{A}}(Q)\otimes\varrho^{\mathsf{B}}(Q)|\boldsymbol{0}\rangle/4$
and
$\langle\boldsymbol{0}|\varrho^{\mathsf{A}}(Q)\otimes\varrho^{\mathsf{B}}(Q)|\boldsymbol{0}\rangle$.

##  Appendix F SAPT operator encoding

In the following, we present the _sparse_ and _tensor factorization_ encoding
methods that allow us to design a full algorithm for SAPT observable
estimation. To this end, we first re-write all of the SAPT operators in terms
of Majorana operators as they provide a clear and direct decomposition in
terms of self-inverse operators required for block encoding.

###  F.1 Majorana representation

In the Majorana representation, the fermionic operators
$\hat{\gamma}_{\textsc{p},0}$ and $\hat{\gamma}_{\textsc{p},1}$ for monomer A
are defined as:

$\displaystyle\hat{\gamma}_{\textsc{p},0}=\hat{a}_{\textsc{p}}+\hat{a}^{\dagger}_{\textsc{p}}\;\;,\;\;\hat{\gamma}_{\textsc{p},1}=-i(\hat{a}_{\textsc{p}}-\hat{a}^{\dagger}_{\textsc{p}}),$
(199)

satisfying the properties,

$\displaystyle\\{\hat{\gamma}_{\textsc{p},i}\;,\hat{\gamma}_{\textsc{p}^{\prime},j}\\}=2\delta_{\textsc{p}\textsc{p}^{\prime}}\delta_{ij}\mathbb{1}\;\;,\;\;\hat{\gamma}_{\textsc{p},i}^{\dagger}=\hat{\gamma}_{\textsc{p},i}\;\;,\;\;\hat{\gamma}_{\textsc{p},i}^{2}=\mathbb{1}.$
(200)

For monomer B, the Majorana operators $\hat{\gamma}_{\textsc{q},0}$ and
$\hat{\gamma}_{\textsc{q},1}$ are defined as:

$\displaystyle\hat{\gamma}_{\textsc{q},0}=\hat{b}_{\textsc{q}}+\hat{b}^{\dagger}_{\textsc{q}}\;\;,\;\;\hat{\gamma}_{\textsc{q},1}=-i(\hat{b}_{\textsc{q}}-\hat{b}^{\dagger}_{\textsc{q}}),$
(201)

while also satisfying the properties,

$\displaystyle\\{\hat{\gamma}_{\textsc{q},i}\;,\hat{\gamma}_{\textsc{q}^{\prime},j}\\}=2\delta_{\textsc{q}\textsc{q}^{\prime}}\delta_{ij}\mathbb{1}\;\;,\;\;\hat{\gamma}_{\textsc{q},i}^{\dagger}=\hat{\gamma}_{\textsc{q},i}\;\;,\;\;\hat{\gamma}_{\textsc{q},i}^{2}=\mathbb{1}.$
(202)

To find the representation of the SAPT operators in terms of the Majorana
operators defined above, one may simply use the inverse relations,
$\hat{a}^{\dagger}_{\textsc{p}}=\tfrac{1}{2}(\hat{\gamma}_{\textsc{p},0}-i\hat{\gamma}_{\textsc{p},1})$,
$\hat{a}_{\textsc{p}}=\tfrac{1}{2}(\hat{\gamma}_{\textsc{p},0}+i\hat{\gamma}_{\textsc{p},1})$
and
$\hat{b}^{\dagger}_{\textsc{q}}=\tfrac{1}{2}(\hat{\gamma}_{\textsc{q},0}-i\hat{\gamma}_{\textsc{q},1})$,
$\hat{b}_{\textsc{q}}=\tfrac{1}{2}(\hat{\gamma}_{\textsc{q},0}+i\hat{\gamma}_{\textsc{q},1})$.
In the following, we provide the final result in both the full space and
active space pictures after simplification. Throughout this section, we will
use the symmetrized SAPT operators defined in Eqs. (110)-(112).

####  F.1.1 Full space picture

The all-electron (full space) electrostatic and exchange operators in the
Majorana representation are defined as:

$\displaystyle\hat{V}$
$\displaystyle=\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}}v^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{q}}+\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{p}_{1}\textsc{p}_{2}\end{subarray}}f^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}+\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{q}_{1}\textsc{q}_{2}\end{subarray}}f^{(B)}_{\textsc{p}_{1}\textsc{p}_{2}}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}$
(203)
$\displaystyle-\tfrac{1}{4}\sum_{\begin{subarray}{c}\textbf{{p}}\textbf{{q}}\end{subarray}}\text{sym}(v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
$\displaystyle\hat{P}$
$\displaystyle=-\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}}S^{\textsc{p}}_{\textsc{q}}S^{\textsc{p}}_{\textsc{q}}-\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{p}_{1}\textsc{p}_{2}\\\
\textsc{q}\end{subarray}}p^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}-\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{q}_{1}\textsc{q}_{2}\\\
\textsc{p}\end{subarray}}p^{(B)}_{\textsc{q}_{1}\textsc{q}_{2}}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}$
(204)
$\displaystyle+\tfrac{1}{4}\sum_{\begin{subarray}{c}\textbf{{p}}\textbf{{q}}\end{subarray}}\text{sym}(S^{\textsc{p}_{1}}_{\textsc{q}_{2}}S^{\textsc{p}_{2}}_{\textsc{q}_{1}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$

where we have defined
$f^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}=\sum_{\textsc{q}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{q}}$,
$f^{(B)}_{\textsc{p}_{1}\textsc{p}_{2}}=\sum_{\textsc{p}}v^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{q}_{2}}$
and
$p^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}=\sum_{\textsc{q}}S^{\textsc{p}_{1}}_{\textsc{q}}S^{\textsc{p}_{2}}_{\textsc{q}}$,
$p^{(B)}_{\textsc{q}_{1}\textsc{q}_{2}}=\sum_{\textsc{p}}S^{\textsc{p}}_{\textsc{q}_{2}}S^{\textsc{p}}_{\textsc{q}_{1}}$,
as the single-body coefficients of the electrostatic and exchange operators in
the Majorana representation. The electrostatic-exchange operator is written
as:

$\widehat{VP}^{(m)}_{\mathrm{s}}=\widehat{VP}_{A}^{(m)}+\widehat{VP}_{B}^{(m)}+\widehat{VP}_{1\mathrm{m}}^{(m)}+\widehat{VP}_{1\ell}^{(m)}+\widehat{VP}_{2}^{(m)}+\widehat{VP}_{3}^{(m)}+\widehat{VP}_{4}^{(m)},$
(205)

where

$\displaystyle\widehat{VP}_{A}^{(m)}$
$\displaystyle=-\tfrac{i}{4}\sum_{\textbf{{p}}}\text{sym}(\kappa_{\textsc{p}_{1}\textsc{p}_{2}}^{(A)})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}+\tfrac{1}{8}\sum_{\begin{subarray}{c}\textbf{{p}}\end{subarray}}\text{sym}(\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{p}_{3}\textsc{p}_{4}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{p}_{3},0}\hat{\gamma}_{\textsc{p}_{4},1},$
(206) $\displaystyle\widehat{VP}_{B}^{(m)}$
$\displaystyle=-\tfrac{i}{4}\sum_{\textbf{{q}}}\text{sym}(\kappa_{\textsc{q}_{1}\textsc{q}_{2}}^{(B)})\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}+\tfrac{1}{8}\sum_{\begin{subarray}{c}\textbf{{q}}\end{subarray}}\text{sym}(\Lambda^{\textsc{q}_{1}\textsc{q}_{2}}_{\textsc{q}_{3}\textsc{q}_{4}})\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}\hat{\gamma}_{\textsc{q}_{3},0}\hat{\gamma}_{\textsc{q}_{4},1},$
(207) $\displaystyle\widehat{VP}_{1\mathrm{m}}^{(m)}$
$\displaystyle=\tfrac{1}{8}\sum_{\begin{subarray}{c}\textbf{{p}},\textbf{{q}}\end{subarray}}\text{sym}(\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}})\;\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
(208) $\displaystyle\widehat{VP}_{1\ell}^{(m)}$
$\displaystyle=\tfrac{1}{4}\sum_{\begin{subarray}{c}\textbf{{p}},\textbf{{q}}\end{subarray}}\text{sym}(\Lambda^{\textsc{p}_{1}\textsc{q}_{2}}_{\textsc{q}_{1}\textsc{p}_{2}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
(209) $\displaystyle\widehat{VP}_{2}^{(m)}$
$\displaystyle=\tfrac{i}{8}\sum_{\begin{subarray}{c}\textbf{{p}},\textbf{{q}}\end{subarray}}\text{sym}(\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{p}_{4}}S^{\textsc{p}_{3}}_{\textsc{q}_{2}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{p}_{3},0}\hat{\gamma}_{\textsc{p}_{4},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
(210) $\displaystyle\widehat{VP}_{3}^{(m)}$
$\displaystyle=\tfrac{i}{8}\sum_{\begin{subarray}{c}\textbf{{p}},\textbf{{q}}\end{subarray}}\text{sym}(\Lambda^{\textsc{p}_{1}\textsc{q}_{4}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{q}_{3}}_{\textsc{p}_{2}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}\hat{\gamma}_{\textsc{q}_{3},0}\hat{\gamma}_{\textsc{q}_{4},1},$
(211) $\displaystyle\widehat{VP}_{4}^{(m)}$
$\displaystyle=\tfrac{1}{2}(\hat{V}^{\prime}\hat{P}^{\prime}+\hat{P}^{\prime}\hat{V}^{\prime}).$
(212)

Here, we have defined the modified electrostatic and exchange operators
$\hat{V}^{\prime}$ and $\hat{P}^{\prime}$,

$\displaystyle\hat{V}^{\prime}$
$\displaystyle=-\tfrac{1}{4}\sum_{\textbf{{p}}\textbf{{q}}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}}\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
(213) $\displaystyle\hat{P}^{\prime}$
$\displaystyle=-\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{p}_{1}\textsc{p}_{2}\end{subarray}}p^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}-\tfrac{i}{4}\sum_{\begin{subarray}{c}\textsc{q}_{1}\textsc{q}_{2}\end{subarray}}p^{(B)}_{\textsc{q}_{1}\textsc{q}_{2}}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1}+\tfrac{1}{4}\sum_{\begin{subarray}{c}\textbf{{p}}\textbf{{q}}\\\
\end{subarray}}\text{sym}(S^{\textsc{p}_{1}}_{\textsc{q}_{2}}S^{\textsc{p}_{2}}_{\textsc{q}_{1}})\hat{\gamma}_{\textsc{p}_{1},0}\hat{\gamma}_{\textsc{p}_{2},1}\hat{\gamma}_{\textsc{q}_{1},0}\hat{\gamma}_{\textsc{q}_{2},1},$
(214)

in order to reduce the $\ell_{1}$ norm contribution of the total
electrostatic-exchange operator. The renormalized tensor coefficients in the
Majorana representation are given by,

$\displaystyle{VP}_{0}^{(m)}$
$\displaystyle=-\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}}\text{sym}(\bar{\nu}^{\textsc{p}\textsc{q}}_{\textsc{q}\textsc{p}})-\tfrac{1}{8}\sum_{\textsc{p}\textsc{q}\textsc{q}^{\prime}}\text{sym}(\nu^{\textsc{p}\textsc{q}^{\prime}}_{\textsc{q}\textsc{q}}S^{\textsc{p}}_{\textsc{q}^{\prime}})-\tfrac{1}{8}\sum_{\textsc{p}\textsc{p}^{\prime}\textsc{q}}\text{sym}(\nu^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{p}^{\prime}}S^{\textsc{p}^{\prime}}_{\textsc{q}})-\tfrac{1}{16}\sum_{\begin{subarray}{c}\textsc{p}\textsc{p}^{\prime}\\\
\textsc{q}\textsc{q}^{\prime}\end{subarray}}\text{sym}(v^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{q}}S^{\textsc{p}^{\prime}}_{\textsc{q}^{\prime}}S^{\textsc{p}^{\prime}}_{\textsc{q}^{\prime}}),$
(215) $\displaystyle\kappa^{(A)}_{\textsc{p}_{1}\textsc{p}_{2}}$
$\displaystyle=\sum_{\textsc{q}}\bar{\nu}^{\textsc{p}_{1}\textsc{q}}_{\textsc{q}\textsc{p}_{2}}+\tfrac{1}{2}\sum_{\textsc{q}\textsc{q}^{\prime}}\nu^{\textsc{p}_{1}\textsc{q}^{\prime}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{2}}_{\textsc{q}^{\prime}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{q}}\nu^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{p}_{2}}S^{\textsc{p}_{1}}_{\textsc{q}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{q}}\nu^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{p}}S^{\textsc{p}}_{\textsc{q}}+\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}\textsc{q}^{\prime}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}^{\prime}}_{\textsc{q}^{\prime}}S^{\textsc{p}^{\prime}}_{\textsc{q}^{\prime}}+\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}\textsc{q}^{\prime}}v^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{1}}_{\textsc{q}^{\prime}}S^{\textsc{p}_{2}}_{\textsc{q}^{\prime}},$
(216) $\displaystyle\kappa^{(B)}_{\textsc{q}_{1}\textsc{q}_{2}}$
$\displaystyle=\sum_{\textsc{p}}\bar{\nu}^{\textsc{p}\textsc{q}_{2}}_{\textsc{q}_{1}\textsc{p}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{p}^{\prime}}\nu^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{p}^{\prime}}S^{\textsc{p}^{\prime}}_{\textsc{q}_{2}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{q}}\nu^{\textsc{p}\textsc{q}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}}_{\textsc{q}_{1}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{q}}\nu^{\textsc{p}\textsc{q}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}}_{\textsc{q}}+\tfrac{1}{4}\sum_{\textsc{p}\textsc{p}^{\prime}\textsc{q}}v^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}^{\prime}}_{\textsc{q}}S^{\textsc{p}^{\prime}}_{\textsc{q}}+\tfrac{1}{4}\sum_{\textsc{p}\textsc{p}^{\prime}\textsc{q}}v^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{q}}S^{\textsc{p}^{\prime}}_{\textsc{q}_{1}}S^{\textsc{p}^{\prime}}_{\textsc{q}_{2}},$
(217)
$\displaystyle\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{p}_{3}\textsc{p}_{4}}$
$\displaystyle=\sum_{\textsc{q}}\nu^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{p}_{4}}S^{\textsc{p}_{3}}_{\textsc{q}}+\tfrac{1}{2}\sum_{\textsc{q}\textsc{q}^{\prime}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{3}}_{\textsc{q}^{\prime}}S^{\textsc{p}_{4}}_{\textsc{q}^{\prime}},$
(218)
$\displaystyle\Lambda^{\textsc{q}_{1}\textsc{q}_{2}}_{\textsc{q}_{3}\textsc{q}_{4}}$
$\displaystyle=\sum_{\textsc{p}}\nu^{\textsc{p}\textsc{q}_{4}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}}_{\textsc{q}_{3}}+\tfrac{1}{2}\sum_{\textsc{p}\textsc{p}^{\prime}}v^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}^{\prime}}_{\textsc{q}_{3}}S^{\textsc{p}^{\prime}}_{\textsc{q}_{4}},$
(219)
$\displaystyle\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}}$
$\displaystyle=\tfrac{1}{2}\sum_{\textsc{p}\textsc{q}}(S^{\textsc{p}}_{\textsc{q}}S^{\textsc{p}}_{\textsc{q}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{q}_{2}}+v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}}_{\textsc{q}_{1}}S^{\textsc{p}}_{\textsc{q}_{2}}+v^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}_{1}}_{\textsc{q}}S^{\textsc{p}_{2}}_{\textsc{q}})+\sum_{\textsc{p}}\nu^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{p}}S^{\textsc{p}}_{\textsc{q}_{2}}+\sum_{\textsc{q}}\nu^{\textsc{p}_{1}\textsc{q}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}_{2}}_{\textsc{q}},$
(220)
$\displaystyle\Lambda^{\textsc{p}_{1}\textsc{q}_{2}}_{\textsc{q}_{1}\textsc{p}_{2}}$
$\displaystyle=\bar{\nu}^{\textsc{p}_{1}\textsc{q}_{2}}_{\textsc{q}_{1}\textsc{p}_{2}}+\tfrac{1}{2}\sum_{\textsc{p}}\nu^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{p}_{2}}S^{\textsc{p}_{1}}_{\textsc{q}_{2}}+\tfrac{1}{2}\sum_{\textsc{q}}\nu^{\textsc{p}_{1}\textsc{q}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{2}}_{\textsc{q}_{1}}+\tfrac{1}{4}\sum_{\textsc{p}\textsc{q}}v^{\textsc{p}\textsc{p}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{1}}_{\textsc{q}_{2}}S^{\textsc{p}_{2}}_{\textsc{q}_{1}},$
(221)
$\displaystyle\Lambda^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{p}_{4}}$
$\displaystyle=\nu^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}_{1}\textsc{p}_{4}}+\tfrac{1}{2}\sum_{\textsc{q}}v^{\textsc{p}_{1}\textsc{p}_{2}}_{\textsc{q}\textsc{q}}S^{\textsc{p}_{4}}_{\textsc{q}_{1}},$
(222)
$\displaystyle\Lambda^{\textsc{p}_{1}\textsc{q}_{4}}_{\textsc{q}_{1}\textsc{q}_{2}}$
$\displaystyle=\nu^{\textsc{p}_{1}\textsc{q}_{4}}_{\textsc{q}_{1}\textsc{q}_{2}}+\tfrac{1}{2}\sum_{\textsc{p}}v^{\textsc{p}\textsc{p}}_{\textsc{q}_{1}\textsc{q}_{2}}S^{\textsc{p}_{1}}_{\textsc{q}_{4}}.$
(223)

####  F.1.2 Active space picture

The active space SAPT operators in the Majorana represetnation are exactly
equivalent apart from the definitions of the two-index and four-index tensors,
which are mapped as: $v\rightarrow\tilde{v}$,
$\Lambda\rightarrow\tilde{\Lambda}$, and so forth. In the spatial orbital
basis, the renormalized active space tensor coefficients for the
electrostatic-exchange operator are given by:

$\displaystyle VP_{0,\mathrm{active}}^{(m)}$
$\displaystyle=VP_{0,\mathrm{active}}-\sum_{t}\tilde{f}^{(A)}_{tt}+\sum_{tt^{\prime}}(\tilde{v}^{tt^{\prime}}_{t^{\prime}t}-\tilde{v}^{tt}_{t^{\prime}t^{\prime}})-\sum_{u}\tilde{f}^{(A)}_{uu}+\sum_{uu^{\prime}}(\tilde{v}^{uu^{\prime}}_{u^{\prime}u}-\tilde{v}^{uu}_{u^{\prime}u^{\prime}})-\sum_{tu}\tilde{\nu}^{tt}_{uu}$
(224)
$\displaystyle-\tfrac{1}{2}\sum_{tuu^{\prime}}\tilde{\nu}^{tu^{\prime}}_{uu}S^{t}_{u^{\prime}}-\tfrac{1}{2}\sum_{tt^{\prime}u}\tilde{\nu}^{tt}_{ut^{\prime}}S^{t^{\prime}}_{u}-\tfrac{1}{2}\sum_{tu}\tilde{\nu}^{tu}_{ut}-\tfrac{1}{2}\sum_{\begin{subarray}{c}tt^{\prime}\\\
uu^{\prime}\end{subarray}}\tilde{v}^{tt}_{uu}S^{t^{\prime}}_{u^{\prime}}S^{t^{\prime}}_{u^{\prime}},$
$\displaystyle\tilde{\kappa}^{(A)}_{t_{1}t_{2}}$
$\displaystyle=2\tilde{f}^{(A)}_{t_{1}t_{2}}-2\sum_{t}(\tilde{\nu}^{t_{1}t}_{tt_{2}}-\tilde{\nu}^{t_{1}t_{2}}_{tt}-\tilde{\nu}^{tt}_{t_{1}t_{2}})+2\sum_{u}\tilde{\nu}^{t_{1}t_{2}}_{uu}+\sum_{uu^{\prime}}\tilde{\nu}^{t_{1}u^{\prime}}_{uu}S^{t_{2}}_{u^{\prime}}$
(225)
$\displaystyle+\sum_{tu}(\tilde{\nu}^{tt}_{ut_{2}}S^{t_{1}}_{u}+\tilde{\nu}^{t_{1}t_{2}}_{ut}S^{t}_{u})+\sum_{u}\tilde{\nu}^{t_{1}u}_{ut_{2}}+\sum_{tuu^{\prime}}\tilde{v}^{t_{1}t_{2}}_{uu}S^{t}_{u^{\prime}}S^{t^{\prime}}_{u^{\prime}}+\sum_{tuu^{\prime}}\tilde{v}^{tt}_{uu}S^{t_{1}}_{u^{\prime}}S^{t_{2}}_{u^{\prime}}$
$\displaystyle\tilde{\kappa}^{(A)}_{u_{1}u_{2}}$
$\displaystyle=2\tilde{f}^{(A)}_{u_{1}u_{2}}\\!-\\!2\sum_{u}(\tilde{\nu}^{u_{1}u}_{uu_{2}}\\!-\\!\tilde{\nu}^{u_{1}u_{2}}_{uu}-\\!\tilde{\nu}^{uu}_{u_{1}u_{2}})\\!+2\\!\sum_{t}\tilde{\nu}^{tt}_{u_{1}u_{2}}+\\!\sum_{tt^{\prime}}\tilde{\nu}^{tt}_{u_{1}t^{\prime}}S^{t^{\prime}}_{u_{2}}$
(226)
$\displaystyle+\sum_{tu}(\tilde{\nu}^{tu_{2}}_{uu}S^{t}_{u_{2}}\\!+\tilde{\nu}^{tu}_{u_{1}u_{2}}S^{t}_{u})+\\!\sum_{t}\tilde{\nu}^{tu_{2}}_{u_{1}t}+\sum_{tt^{\prime}u}\tilde{v}^{tt}_{u_{1}u_{2}}S^{t^{\prime}}_{u}S^{t^{\prime}}_{u}+\sum_{tt^{\prime}u}\tilde{v}^{tt}_{uu}S^{t^{\prime}}_{u_{1}}S^{t^{\prime}}_{u_{2}},$
$\displaystyle\tilde{\Lambda}^{t_{1}t_{2}}_{t_{3}t_{4}}$
$\displaystyle=2\tilde{\nu}^{t_{1}t_{2}}_{t_{3}t_{4}}+\sum_{u}\tilde{\nu}^{t_{1}t_{2}}_{ut_{4}}S^{t_{3}}_{u}+\sum_{uu^{\prime}}\tilde{v}^{t_{1}t_{2}}_{uu}S^{t_{3}}_{u^{\prime}}S^{t_{4}}_{u^{\prime}},$
(227) $\displaystyle\tilde{\Lambda}^{u_{1}u_{2}}_{u_{3}u_{4}}$
$\displaystyle=2\tilde{\nu}^{u_{1}u_{2}}_{u_{3}u_{4}}+\sum_{t}\tilde{\nu}^{tu_{4}}_{u_{1}u_{2}}S^{t}_{u_{3}}+\sum_{tt^{\prime}}\tilde{v}^{tt}_{u_{1}u_{2}}S^{t^{\prime}}_{u_{3}}S^{t^{\prime}}_{u_{4}},$
(228) $\displaystyle\tilde{\Lambda}^{t_{1}t_{2}}_{u_{1}u_{2}}$
$\displaystyle=2\tilde{\nu}^{t_{1}t_{2}}_{u_{1}u_{2}}+\sum_{tu}(S^{t}_{u}S^{t}_{u}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}+\tilde{v}^{t_{1}t_{2}}_{uu}S^{t}_{u_{1}}S^{t}_{u_{2}}+\tilde{v}^{tt}_{u_{1}u_{2}}S^{t_{1}}_{u}S^{t_{2}}_{u})+\sum_{t}\tilde{\nu}^{t_{1}t_{2}}_{u_{1}t}S^{t}_{u_{2}}+\sum_{u}\tilde{\nu}^{t_{1}u}_{u_{1}u_{2}}S^{t_{2}}_{u},$
(229) $\displaystyle\tilde{\Lambda}^{t_{1}u_{2}}_{u_{1}t_{2}}$
$\displaystyle=\tilde{\nu}^{t_{1}u_{2}}_{u_{1}t_{2}}+\sum_{t}\tilde{\nu}^{tt}_{u_{1}t_{2}}S^{t_{1}}_{u_{2}}+\sum_{u}\tilde{\nu}^{t_{1}u_{2}}_{uu}S^{t_{2}}_{u_{1}}+\sum_{tu}\tilde{v}^{tt}_{uu}S^{t_{1}}_{u_{2}}S^{t_{2}}_{u_{1}},$
(230) $\displaystyle\tilde{\Lambda}^{t_{1}t_{2}}_{u_{1}t_{4}}$
$\displaystyle=\tilde{\nu}^{t_{1}t_{2}}_{u_{1}t_{4}}+\sum_{u}\tilde{\nu}^{t_{1}t_{2}}_{uu}S^{t_{4}}_{u_{1}},$
(231) $\displaystyle\tilde{\Lambda}^{t_{1}u_{4}}_{u_{1}u_{2}}$
$\displaystyle=\tilde{\nu}^{t_{1}u_{4}}_{u_{1}u_{2}}+\sum_{t}\tilde{\nu}^{tt}_{u_{1}u_{2}}S^{t_{1}}_{u_{4}}.$
(232)

Eqs. (213) and (214) in the active space picture are given by,

$\displaystyle\hat{V}_{\mathrm{active}}^{\prime}$
$\displaystyle=-\tfrac{1}{4}\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\tau\end{subarray}}\tilde{v}^{t_{1}t_{2}}_{u_{1}u_{2}}\hat{\gamma}_{t_{1}\sigma,0}\hat{\gamma}_{t_{2}\sigma,1}\hat{\gamma}_{u_{1}\tau,0}\hat{\gamma}_{u_{2}\tau,1},$
(233) $\displaystyle\hat{P}_{\mathrm{active}}^{\prime}$
$\displaystyle=-\tfrac{i}{4}\sum_{\begin{subarray}{c}t_{1}t_{2}\\\
\sigma\end{subarray}}\tilde{p}_{t_{1}t_{2}}^{(A)}\hat{\gamma}_{t_{1}\sigma,0}\hat{\gamma}_{t_{2}\sigma,1}-\tfrac{i}{4}\sum_{\begin{subarray}{c}u_{1}u_{2}\\\
\sigma\end{subarray}}\tilde{p}_{u_{1}u_{2}}^{(B)}\hat{\gamma}_{u_{1}\sigma,0}\hat{\gamma}_{u_{2}\sigma,1}+\tfrac{1}{4}\sum_{\begin{subarray}{c}\mathbf{t}\mathbf{u}\\\
\sigma\end{subarray}}\text{sym}(S^{t_{1}}_{u_{2}}S^{t_{2}}_{u_{1}})\hat{\gamma}_{t_{1}\sigma,0}\hat{\gamma}_{t_{2}\sigma,1}\hat{\gamma}_{u_{1}\sigma,0}\hat{\gamma}_{u_{2}\sigma,1}.$
(234)

with

$\displaystyle\tilde{p}^{(A)}_{t_{1}t_{2}}$
$\displaystyle=\sum_{u}S^{t_{1}}_{u}S^{t_{2}}_{u}+2\sum_{j}S^{t_{1}}_{j}S^{t_{2}}_{j},$
(235) $\displaystyle\tilde{p}^{(B)}_{u_{1}u_{2}}$
$\displaystyle=\sum_{t}S^{t}_{u_{2}}S^{t}_{u_{1}}+2\sum_{i}S^{i}_{u_{1}}S^{i}_{u_{2}}.$
(236)

In the following two sections, we present the _sparse_ and _tensor
factorization_ encoding schemes which can apply to both the all-electron
(full-space) and active space pictures.

###  F.2 Sparse representation

In the _sparse_ encoding scheme [33] scheme, a data-loading oracle loads the
non-zero entries of the tensor coefficients reducing the overall cost compared
to an equivalent _dense_ scheme. The Jordan-Wigner mapping from the Majorana
operators for monomer A is given by,

$\displaystyle\hat{\gamma}_{p\sigma,0}=\hat{X}_{p,\sigma}\hat{Z}_{p-1,\sigma}\cdots\hat{Z}_{0,\sigma},$
(237)
$\displaystyle\hat{\gamma}_{p\sigma,1}=\hat{Y}_{p,\sigma}\hat{Z}_{p-1,\sigma}\cdots\hat{Z}_{0,\sigma}.$
(238)

Similarly, for monomer B we have:

$\displaystyle\hat{\gamma}_{q\tau,0}=\hat{X}_{q,\tau}\hat{Z}_{q-1,\tau}\cdots\hat{Z}_{0,\tau},$
(239)
$\displaystyle\hat{\gamma}_{q\tau,1}=\hat{Y}_{q,\tau}\hat{Z}_{q-1,\tau}\cdots\hat{Z}_{0,\tau}.$
(240)

The $\mathsf{PREPARE}$ and $\mathsf{SELECT}$ operators for these operators are
given explicitly in [35]. Here, we summarize the total $\ell_{1}$ norm for all
of the SAPT operators:

$\displaystyle\lambda^{(s)}_{V}$
$\displaystyle=\sum_{q_{1}q_{2}}\Big{|}\sum_{p}{v}^{pp}_{q_{1}q_{2}}\Big{|}+\sum_{p_{1}p_{2}}\Big{|}\sum_{q}v^{p_{1}p_{2}}_{qq}\Big{|}+\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
q_{1}q_{2}\end{subarray}}|v^{p_{1}p_{2}}_{q_{1}q_{2}}|,$ (241)
$\displaystyle\lambda^{(s)}_{P}$
$\displaystyle=\tfrac{1}{2}\sum_{q_{1}q_{2}}\Big{|}\sum_{p}S^{p}_{q_{2}}S^{p}_{q_{1}}\Big{|}+\tfrac{1}{2}\sum_{p_{1}p_{2}}\Big{|}\sum_{q}S^{p_{1}}_{q}S^{p_{2}}_{q}\Big{|}+\tfrac{1}{2}\Big{[}\sum_{pq}|S^{p}_{q}|\Big{]}^{2},$
(242) $\displaystyle\lambda^{(s)}_{VP}$
$\displaystyle=\lambda^{(s)}_{VP_{A}}+\lambda^{(s)}_{VP_{B}}+\lambda^{(s)}_{VP_{1\mathrm{m}}}+\lambda^{(s)}_{VP_{1\ell}}+\lambda^{(s)}_{VP_{2}}+\lambda^{(s)}_{VP_{3}}+\lambda^{(s)}_{P}\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
q_{1}q_{2}\end{subarray}}|v^{p_{1}p_{2}}_{q_{1}q_{2}}|,$ (243)

where

$\displaystyle\lambda^{(s)}_{VP_{A}}$
$\displaystyle=\tfrac{1}{2}\sum_{p_{1}p_{2}}|\kappa^{(A)}_{p_{1}p_{2}}|+\tfrac{1}{2}\sum_{\begin{subarray}{c}p_{1}>p_{2}\\\
p_{3}>p_{4}\end{subarray}}|\Lambda^{p_{1}p_{2}}_{p_{3}p_{4}}-\Lambda^{p_{1}p_{4}}_{p_{3}p_{2}}|+\tfrac{1}{4}\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
p_{3}p_{4}\end{subarray}}|\Lambda^{p_{1}p_{2}}_{p_{3}p_{4}}|,$ (244)
$\displaystyle\lambda^{(s)}_{VP_{B}}$
$\displaystyle=\tfrac{1}{2}\sum_{q_{1}q_{2}}|\kappa^{B}_{q_{1}q_{2}}|+\tfrac{1}{2}\sum_{\begin{subarray}{c}q_{1}>q_{2}\\\
q_{3}>q_{4}\end{subarray}}|\Lambda^{q_{1}q_{2}}_{q_{3}q_{4}}-\Lambda^{q_{1}q_{4}}_{q_{3}q_{2}}|+\tfrac{1}{4}\sum_{\begin{subarray}{c}q_{1}q_{2}\\\
q_{3}q_{4}\end{subarray}}|\Lambda^{q_{1}q_{2}}_{q_{3}q_{4}}|,$ (245)
$\displaystyle\lambda^{(s)}_{VP_{1\mathrm{m}}}$
$\displaystyle=\tfrac{1}{2}\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
q_{1}q_{2}\end{subarray}}|\Lambda^{p_{1}p_{2}}_{q_{1}q_{2}}|,$ (246)
$\displaystyle\lambda^{(s)}_{VP_{1\ell}}$
$\displaystyle=\tfrac{1}{2}\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
q_{1}q_{2}\end{subarray}}|\Lambda^{p_{1}q_{2}}_{q_{1}p_{2}}|,$ (247)
$\displaystyle\lambda^{(s)}_{VP_{2}}$
$\displaystyle=\tfrac{1}{2}\\!\\!\\!\\!\sum_{\begin{subarray}{c}p_{1}p_{2}p_{3}p_{4}\\\
q_{1}q_{2}\end{subarray}}\\!\\!\\!\\!\\!|\Lambda^{p_{1}p_{2}}_{q_{1}p_{4}}S^{p_{3}}_{q_{2}}|,$
(248) $\displaystyle\lambda^{(s)}_{VP_{3}}$
$\displaystyle=\tfrac{1}{2}\\!\\!\\!\\!\sum_{\begin{subarray}{c}p_{1}p_{2}\\\
q_{1}q_{2}q_{3}q_{4}\end{subarray}}\\!\\!\\!|\Lambda^{p_{1}q_{4}}_{q_{1}q_{2}}S^{p_{2}}_{q_{3}}|$
(249)

Here, the superscript ${(s)}$ is used to denote the sparse representation. The
total number $L_{F}$ of operator terms for each of the SAPT operators
$\hat{F}$ will scale as
$L_{V}^{(s)}=\mathcal{O}(N_{A}^{2}N_{B}^{2}),L_{P}^{(s)}=\mathcal{O}(N_{A}^{2}N_{B}^{2})$,
and $L_{VP}^{(s)}=\mathcal{O}(N_{A}^{4}N_{B}^{4})$ using a naive block
encoding. As we show below, however, it is possible to reduce this scaling
with respect to the total number of terms $L_{F}$ by using factorization
circuits especially designed to describe the product of two operators, e.g.
$\hat{V}$ and $\hat{P}$. This well help reduce the scaling of certain terms,
such as the second, third, and fourth terms: $\widehat{VP}_{2}$,
$\widehat{VP}_{3}$ and $\widehat{VP}_{4}$. For instance, the compilation
scaling of the fourth term in the electrostatic exchange operator, Eq. (212),
will reduce to $L_{VP}^{(s)}=\mathcal{O}(N_{A}^{4}+N_{B}^{4})$. It is also
worth pointing out that we used the anti-symmetry property of the fermionic
operators to reduce the $\ell_{1}$ norm of the intra-monomer contributions as
pointed out in Ref. [36].

###  F.3 Factorized tensor representation

While the sparse representation serves to provide a first-pass implementation
of symmetry-adapted perturbation theory, it does not scale well with the
number of terms $L$, nor does it scale well with the $\ell_{1}$ norm of the
operator. In the following, we present a tensor factorization scheme that is
analogous to the low-rank double factorization scheme [15] used in Hamiltonian
simulation. The advantage afforded by this encoding scheme is two-fold. First,
it will be shown to drastically reduce the $\ell_{1}$ norm of each of the
three operators. Fundamentally, this is explained by the fact that the
Schatten 1-norm is strictly less than or equal to the entry-wise 1-norm of
tensors. This property was pointed out by Von Burg et al. and becomes apparent
in the numerical benchmark data sets we present in the results section.
Second, it reduces the total number of terms $L$ required for the data-loading
oracle. It is important to emphasize that without an appropriate factorization
scheme, the implementation of SAPT on the quantum computer would be much less
scalable. The complete tensor factorization procedure is presented in the main
text. Using this procedure, all of the unique four-index tensors in first
order SAPT theory may be expanded as:

$\displaystyle v^{p_{1}p_{2}}_{q_{1}q_{2}}$
$\displaystyle=\sum_{tkl}s^{(v)}_{t}\alpha^{(A_{v})}_{tk}\alpha^{(B_{v})}_{tl}U^{(v)}_{tkp_{1}}U^{(v)}_{tkp_{2}}V^{(v)}_{tlq_{1}}V^{(v)}_{tlq_{2}},$
(250) $\displaystyle\text{sym}(\Lambda^{p_{1}p_{2}}_{p_{3}p_{4}})$
$\displaystyle=\sum_{tkl}s^{(A_{2})}_{t}\alpha^{(A_{2})}_{tk}\alpha^{(A_{2})}_{tl}U^{(A_{2})}_{tkp_{1}}U^{(A_{2})}_{tkp_{2}}U^{(A_{2})}_{tlp_{3}}U^{(A_{2})}_{tlp_{4}},$
(251) $\displaystyle\text{sym}(\Lambda^{q_{1}q_{2}}_{q_{3}q_{4}})$
$\displaystyle=\sum_{tkl}s^{(B_{2})}_{t}\alpha^{(B_{2})}_{tk}\alpha^{(B_{2})}_{tl}V^{(B_{2})}_{tkq_{1}}V^{(B_{2})}_{tkq_{2}}V^{(B_{2})}_{tlq_{3}}V^{(B_{2})}_{tlq_{4}},$
(252) $\displaystyle\text{sym}(\Lambda^{p_{1}q_{2}}_{q_{1}p_{2}})$
$\displaystyle=\sum_{tkl}s^{(1\ell)}_{t}\beta^{(1\ell)}_{tk}\beta^{(1\ell)}_{tl}U^{(1\ell)}_{tkp_{1}}V^{(1\ell)}_{tkq_{2}}U^{(1\ell)}_{tlp_{2}}V^{(1\ell)}_{tlq_{1}},$
(253) $\displaystyle\text{sym}(\Lambda^{p_{1}p_{2}}_{q_{1}q_{2}})$
$\displaystyle=\sum_{tkl}s^{(1\mathrm{m})}_{t}\alpha^{(A_{1\mathrm{m}})}_{tk}\alpha^{(B_{1\mathrm{m}})}_{tl}U^{(1\mathrm{m})}_{tkp_{1}}U^{(1\mathrm{m})}_{tkp_{2}}V^{(1\mathrm{m})}_{tlq_{1}}V^{(1\mathrm{m})}_{tlq_{2}},$
(254) $\displaystyle\Lambda^{p_{1}p_{2}}_{q_{1}p_{4}}$
$\displaystyle=\sum_{tkl}s^{(2)}_{t}\alpha^{(2)}_{tk}\beta^{(2)}_{tl}U^{(2)}_{tkp_{1}}U^{(2)}_{tkp_{2}}\tilde{U}^{(2)}_{tlp_{4}}V^{(2)}_{tlq_{1}},$
(255) $\displaystyle\Lambda^{p_{1}q_{4}}_{q_{1}q_{2}}$
$\displaystyle=\sum_{tkl}s^{(3)}_{t}\alpha^{(3)}_{tk}\beta^{(3)}_{tl}U^{(3)}_{tlp_{1}}\tilde{V}^{(3)}_{tkq_{4}}V^{(3)}_{tlq_{1}}V^{(3)}_{tlq_{2}},$
(256)

which is exact when none of the terms are truncated. In the following
subsections, we use this expansion to summarize the tensor-factorized SAPT
operators applicable in both the all-electron (full space) and active space
pictures. Throughout the following sections, we make use of the orbital-
transformed Majorana operators,

$\displaystyle\tilde{\gamma}_{k\sigma,i}$
$\displaystyle=\sum_{p}U_{pk}\hat{\gamma}_{\textsc{p},i}=\hat{G}^{(A)\dagger}_{\sigma}\hat{\gamma}_{k\sigma,i}\hat{G}^{(A)}_{\sigma},$
(257) $\displaystyle\tilde{\gamma}_{l\tau,i}$
$\displaystyle=\sum_{q}V_{ql}\hat{\gamma}_{l\tau,i}=\hat{G}^{(B)\dagger}_{\tau}\hat{\gamma}_{l\tau,i}\hat{G}^{(B)}_{\tau},$
(258)

where $i=\\{0,1\\}$ and the second equality in both expressions is defined
with respect to the Givens operator,
$\hat{G}^{(A)}_{\sigma}=\exp(\sum_{pk}[\log\mathbf{U}]_{pk}\hat{a}^{\dagger}_{\textsc{p}}\hat{a}_{k\sigma})$
and
$\hat{G}^{(B)}_{\tau}=\exp(\sum_{ql}[\log\mathbf{V}]_{ql}\hat{b}^{\dagger}_{q\tau}\hat{b}_{l\tau})$
respectively.

####  F.3.1 Factorization of electrostatic operator: $\hat{V}$

Following tensor factorization procedure from the main text, we find the final
form of the electrostatic operator,

$\displaystyle\hat{V}$
$\displaystyle=\sum_{pq}v^{pp}_{qq}+\tfrac{i}{2}\sum_{k,\sigma}s_{k}^{(A)}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}+\tfrac{i}{2}\sum_{l,\sigma}s_{l}^{(B)}\tilde{\gamma}_{l\sigma,0}\tilde{\gamma}_{l\sigma,1}-\tfrac{1}{4}\sum_{\begin{subarray}{c}tkl\\\
\sigma\tau\end{subarray}}s^{(v)}_{t}\alpha^{(A_{v})}_{tk}\alpha^{(B_{v})}_{tl}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}\tilde{\gamma}_{l\tau,0}\tilde{\gamma}_{l\tau,1},$
(259)
$\displaystyle=\sum_{pq}v^{pp}_{qq}-\tfrac{1}{2}\sum_{k,\sigma}s_{k}^{(A)}\hat{G}_{\varnothing}^{(A)\dagger}\hat{Z}_{k\sigma}\hat{G}_{\varnothing}^{(A)}-\tfrac{1}{2}\sum_{l,\sigma}s_{l}^{(B)}\hat{G}_{\varnothing}^{(B)\dagger}\hat{Z}_{l\sigma}\hat{G}_{\varnothing}^{(B)}$
(260) $\displaystyle+\tfrac{1}{4}\sum_{\begin{subarray}{c}tkl\\\
\sigma\tau\end{subarray}}[s^{(v)}_{t}\alpha^{(A_{v})}_{tk}\alpha^{(B_{v})}_{tl}]\Big{(}\hat{G}_{t}^{(A)\dagger}\hat{Z}_{k\sigma}\hat{G}_{t}^{(A)}\Big{)}\otimes\Big{(}\hat{G}_{t}^{(B)\dagger}\hat{Z}_{l\tau}\hat{G}_{t}^{(B)}\Big{)}.$

In the second line, we used the following Jordan-Wigner identity for each
monomer,

$\displaystyle\hat{\gamma}_{k\sigma,0}\hat{\gamma}_{k\sigma,1}$
$\displaystyle=i\hat{Z}_{k\sigma},$ (261)
$\displaystyle\hat{\gamma}_{l\tau,0}\hat{\gamma}_{l\tau,1}$
$\displaystyle=i\hat{Z}_{l\tau},$ (262)

and we have also defined the Givens operators,
$\hat{G}^{(X)}_{\varnothing}=\hat{G}^{(X)}_{\varnothing,\alpha}\otimes\hat{G}^{(X)}_{\varnothing,\beta}$
and $\hat{G}^{(X)}_{t}=\hat{G}^{(X)}_{t,\alpha}\otimes\hat{G}^{(X)}_{t,\beta}$
for $X\in\\{A,B\\}$, where each spin-block is defined as:

$\displaystyle\hat{G}^{(A)}_{\varnothing,\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(f)}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(263) $\displaystyle\hat{G}^{(B)}_{\varnothing,\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(f)}_{t}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right),$
(264)

and

$\displaystyle\hat{G}^{(A)}_{t,\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(v)}_{t}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(265) $\displaystyle\hat{G}^{(B)}_{t,\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(v)}_{t}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right).$
(266)

The corresponding $\ell_{1}$ norm for the tensor-factorized electrostatic
operator is given by,

$\lambda_{V}^{(\mathrm{tf})}=\sum_{k}|s_{k}^{(A)}|+\sum_{l}|s_{l}^{(B)}|+\sum_{tkl}|s^{(v)}_{t}\alpha^{(A_{v})}_{tk}\alpha^{(B_{v})}_{tl}|.$
(267)

###  F.4 Factorization of Exchange Operator: $\hat{P}$

We consider the tensor factorization procedure for the active space exchange
operator since it is more general than the all-electron (full space) version.
Substituting the factorization of the intermolecular overlap matrix, we
obtain:

$\displaystyle\hat{P}_{\mathrm{active}}$
$\displaystyle=-\tfrac{1}{2}\sum_{pq}S^{p}_{q}S^{p}_{q}-\tfrac{i}{4}\sum_{k,\sigma}s_{k}^{(P_{A_{1}})}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}-\tfrac{i}{4}\sum_{l,\sigma}s_{l}^{(P_{B_{1}})}\tilde{\gamma}_{l\sigma,0}\tilde{\gamma}_{l\sigma,1}$
(268) $\displaystyle+\tfrac{1}{16}\sum_{\begin{subarray}{c}kl\\\
\sigma\end{subarray}}s_{k}s_{l}(\tilde{\gamma}_{l\sigma,0}\tilde{\gamma}_{k\sigma,1}+\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{l\sigma,1})\otimes(\tilde{\gamma}_{l\sigma,0}\tilde{\gamma}_{k\sigma,1}+\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{l\sigma,1})$
$\displaystyle=-\tfrac{1}{2}\sum_{pq}S^{p}_{q}S^{p}_{q}+\tfrac{i}{4}\sum_{k,\sigma}s_{k}^{(P_{A_{1}})}\hat{G}^{(P_{A_{1}})\dagger}_{\varnothing}\hat{Z}_{k\sigma}\hat{G}^{(P_{A_{1}})}_{\varnothing}+\tfrac{1}{4}\sum_{l,\sigma}s_{l}^{(P_{B_{1}})}\hat{G}^{(P_{B_{1}})\dagger}_{\varnothing}\hat{Z}_{l\sigma}\hat{G}^{(P_{B_{1}})}_{\varnothing}$
(269)
$\displaystyle-\tfrac{1}{4}\sum_{k,\sigma}s_{k}^{2}\Big{(}\hat{G}^{(A_{S})\dagger}\hat{Z}_{k\sigma}\hat{G}^{(A_{S})}\Big{)}\otimes\Big{(}\hat{G}^{(B_{S})\dagger}\hat{Z}_{k\sigma}\hat{G}^{(B_{S})\dagger}\Big{)}$
$\displaystyle+\tfrac{1}{16}\sum_{\begin{subarray}{c}k\neq l\\\
\sigma\end{subarray}}s_{k}s_{l}\Big{(}\hat{G}^{(A_{S})\dagger}[\hat{X}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{X}_{l\sigma}+\hat{Y}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{Y}_{l\sigma}]\hat{G}^{(A_{S})}\Big{)}\otimes\Big{(}\hat{G}^{(B_{S})\dagger}[\hat{X}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{X}_{l\sigma}+\hat{Y}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{Y}_{l\sigma}]\hat{G}^{(B_{S})}\Big{)},$

where

$\vec{Z}^{\sigma}_{kl}=\begin{cases}\hat{Z}_{k-1,\sigma}\cdots\hat{Z}_{l+1,\sigma}&\text{if}\;\;k>l\\\
\hat{Z}_{l-1,\sigma}\cdots\hat{Z}_{k+1,\sigma}&\text{if}\;\;l>k.\end{cases}$
(270)

We have also defined the Givens operators,
$\hat{G}^{(P_{X_{1}})}_{\varnothing}=\hat{G}^{(P_{X_{1}})}_{\varnothing,\alpha}\otimes\hat{G}^{(P_{X_{1}})}_{\varnothing,\beta}$
and
$\hat{G}^{(X_{S})}=\hat{G}^{(X_{S})}_{\alpha}\otimes\hat{G}^{(X_{S})}_{\beta}$
for $X\in\\{A,B\\}$, where each spin-block is defined as:

$\displaystyle\hat{G}^{(P_{A_{1}})}_{\varnothing,\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(P_{A_{1}})}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(271) $\displaystyle\hat{G}^{(P_{B_{1}})}_{\varnothing,\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(P_{B_{1}})}_{t}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right),$
(272)

and

$\displaystyle\hat{G}^{(A_{S})}_{\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(s)}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(273) $\displaystyle\hat{G}^{(B_{S})}_{\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(s)}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right).$
(274)

The corresponding $\ell_{1}$ norm is given by,

$\lambda_{P}^{(\mathrm{tf})}=\tfrac{1}{2}\sum_{k}|s_{k}^{(P_{A_{1}})}|+\tfrac{1}{2}\sum_{l}|s_{l}^{(P_{B_{1}})}|+\tfrac{1}{2}\left(\sum_{n}s_{n}\right)^{2}.$
(275)

###  F.5 Factorization of electrostatic-exchange operator: $\widehat{VP}$

The electrostatic-exchange factorization is decomposed with respect to six
different four-index contributions, which we outline below. The first two
correspond to monomer-only contributions, $\widehat{VP}_{A}$ and
$\widehat{VP}_{B}$ respectively. In the following, we provide the
factorization procedure which arrives at the same result as the standard
double factorization procedure derived by Von Burg et al.

####  $\widehat{VP}_{A}$/$\widehat{VP}_{B}$:

Substituting the factorization of the two-body terms into Eqs. (206)-(207), we
find

$\displaystyle\widehat{VP}_{A}$
$\displaystyle=\sum_{k}\tilde{s}_{k}^{(A)}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}-\sum_{tkl}s^{(A_{2})}_{t}\alpha^{(A_{2})}_{tk}\alpha^{(A_{2})}_{tl}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}\tilde{\gamma}_{l\tau,0}\tilde{\gamma}_{l\tau,1},$
(276) $\displaystyle\widehat{VP}_{B}$
$\displaystyle=\sum_{k}\tilde{s}_{k}^{(B)}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}-\sum_{tkl}s^{(B_{2})}_{t}\alpha^{(B_{2})}_{tk}\alpha^{(B_{2})}_{tl}\tilde{\gamma}_{k\sigma,0}\tilde{\gamma}_{k\sigma,1}\tilde{\gamma}_{l\tau,0}\tilde{\gamma}_{l\tau,1}.$
(277)

The transformed Majorana operators may be found in Ref. [15] with the
appropriate orbital transformation matrices found in Eqs. (251)-(252). Since
the two-body terms for each monomer have a complete-square form, it is
possible to use a quantum signal processing technique to load a second order
Chebyshev polynomial in order implement the second body term [15, 8]. This
effectively reduces the $\ell_{1}$ norm of the two-body term by a factor of
two, leading to the following expressions:

$\displaystyle\lambda_{VP_{A}}$
$\displaystyle=\tfrac{1}{2}\sum_{k}|\tilde{s}_{k}^{(A)}|+\tfrac{1}{4}\sum_{t}|s_{t}^{(A_{2})}|\Big{(}\sum_{k}|\alpha^{(A_{2})}_{tk}|\Big{)}^{2},$
(278) $\displaystyle\lambda_{VP_{B}}$
$\displaystyle=\tfrac{1}{2}\sum_{l}|\tilde{s}_{l}^{(B)}|+\tfrac{1}{4}\sum_{t}|s_{t}^{(B_{2})}|\Big{(}\sum_{l}|\alpha^{(B_{2})}_{tl}|\Big{)}^{2}.$
(279)

####  $\widehat{VP}_{1\mathrm{m}}$:

The first order mixed-spin term is equivalent to the electrostatic operator
apart from one-body contributions. The tensor factorized operator may be
written explicitly as,

$\displaystyle\widehat{VP}_{1\mathrm{m}}$
$\displaystyle=\tfrac{1}{8}\sum_{\begin{subarray}{c}tkl\\\
\sigma\tau\end{subarray}}s^{(1\mathrm{m})}_{t}\alpha_{kt}^{(A_{1\mathrm{m}})}\alpha_{lt}^{(B_{1\mathrm{m}})}\tilde{\gamma}^{(A_{1\mathrm{m}})}_{k\sigma,0}\tilde{\gamma}^{(A_{1\mathrm{m}})}_{k\sigma,1}\tilde{\gamma}^{(B_{1\mathrm{m}})}_{l\tau,0}\tilde{\gamma}^{(B_{1\mathrm{m}})}_{l\tau,1},$
(280) $\displaystyle=-\tfrac{1}{8}\sum_{\begin{subarray}{c}tkl\\\
\sigma\tau\end{subarray}}[s^{(1\mathrm{m})}_{t}\alpha_{kt}^{(A_{1\mathrm{m}})}\alpha_{lt}^{(B_{1\mathrm{m}})}]\Big{(}\hat{G}_{t}^{(A_{1\mathrm{m}})\dagger}\hat{Z}_{k\sigma}\hat{G}_{t}^{(A_{1\mathrm{m}})}\Big{)}\otimes\Big{(}\hat{G}_{t}^{(B_{1\mathrm{m}})\dagger}\hat{Z}_{l\tau}\hat{G}_{t}^{(B_{1\mathrm{m}})}\Big{)}.$
(281)

where the Givens operators for each spin-block are defined as:

$\displaystyle\hat{G}^{(A_{1\mathrm{m}})}_{t,\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(1\mathrm{m})}_{t}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(282) $\displaystyle\hat{G}^{(B_{1\mathrm{m}})}_{t,\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(1\mathrm{m})}_{t}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right).$
(283)

The corresponding $\ell_{1}$ norm is given by:

$\lambda_{VP_{1\mathrm{m}}}=\tfrac{1}{2}\sum_{tkl}|s^{(1\mathrm{m})}_{t}\alpha_{kt}^{(A_{1\mathrm{m}})}\alpha_{lt}^{(B_{1\mathrm{m}})}|.$
(284)

####  $\widehat{VP}_{1\ell}$:

The first order spin-locked contribution is similar to the exchange operator
but requires a four-index factorization procedure outlined in previous terms.
The tensor factorized operator is written as:

$\displaystyle\widehat{VP}_{1\ell}$
$\displaystyle=\tfrac{1}{4}\sum_{\begin{subarray}{c}tkl\\\
\sigma\end{subarray}}s^{(1\ell)}_{t}\beta^{(1\ell)}_{tk}\beta^{(1\ell)}_{tl}\tilde{\gamma}^{A_{1\ell}}_{k\sigma,0}\tilde{\gamma}^{A_{1\ell}}_{l\sigma,1}\tilde{\gamma}^{B_{1\ell}}_{k\sigma,0}\tilde{\gamma}^{B_{1\ell}}_{l\sigma,1},$
(285) $\displaystyle=-\tfrac{1}{4}\sum_{\begin{subarray}{c}tk\\\
\sigma\end{subarray}}s^{(1\ell)}_{t}\beta^{(1\ell)}_{tk}\beta^{(1\ell)}_{tk}\Big{(}\hat{G}_{t}^{(A_{1\ell})\dagger}\hat{Z}_{k\sigma}\hat{G}_{t}^{(A_{1\ell})}\Big{)}\otimes\Big{(}\hat{G}_{t}^{(B_{1\ell})\dagger}\hat{Z}_{k\sigma}\hat{G}_{t}^{(B_{1\mathrm{m}})}\Big{)}$
(286) $\displaystyle+\tfrac{1}{16}\sum_{\begin{subarray}{c}t\sigma\\\ k\neq
l\end{subarray}}s^{(1\ell)}_{t}\beta^{(1\ell)}_{tk}\beta^{(1\ell)}_{tl}\Big{(}\hat{G}_{t}^{(A_{1\ell})\dagger}[\hat{X}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{X}_{l\sigma}+\hat{Y}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{Y}_{l\sigma}]\hat{G}_{t}^{(A_{1\ell})}\Big{)}\otimes\Big{(}\hat{G}_{t}^{(B_{1\ell})\dagger}[\hat{X}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{X}_{l\sigma}+\hat{Y}_{k\sigma}\vec{Z}_{kl}^{\sigma}\hat{Y}_{l\sigma}]\hat{G}_{t}^{(B_{1\ell})}\Big{)}.$

where we used the permutation symmetries of the four-index tensor to go from
the first line to the second line and we have defined the Givens operators for
each spin-block as:

$\displaystyle\hat{G}^{(A_{1\ell})}_{t,\sigma}$
$\displaystyle=\exp\left(\sum_{p>k}[\log\mathbf{U}^{(1\ell)}_{t}]_{pk}(\hat{E}^{\sigma}_{pk}-\hat{E}^{\sigma}_{kp})\right),$
(287) $\displaystyle\hat{G}^{(B_{1\ell})}_{t,\tau}$
$\displaystyle=\exp\left(\sum_{q>l}[\log\mathbf{V}^{(1\ell)}_{t}]_{ql}(\hat{E}^{\tau}_{ql}-\hat{E}^{\tau}_{lq})\right).$
(288)

The corresponding $\ell_{1}$ norm given by:

$\lambda_{VP_{1\ell}}=\tfrac{1}{2}\sum_{tkl}|s^{(1\ell)}_{t}\beta^{(1\ell)}_{tk}\beta^{(1\ell)}_{tl}|.$
(289)

####  $\widehat{VP}_{2}$/$\widehat{VP}_{3}$:

The second and third terms have similar structure. Here, we provide the
explicit form for the second term taking into account all of the permutations
involved for $\text{sym}(\Lambda^{p_{1}p_{2}}_{q_{1}p_{4}}S^{p_{3}}_{q_{2}})$:

$\displaystyle\widehat{VP}_{2}$
$\displaystyle=-\tfrac{i}{8}\sum_{\begin{subarray}{c}tkl\\\
n\sigma\tau\end{subarray}}[s_{n}s^{(2)}_{t}\alpha^{(2)}_{tk}\beta^{(2)}_{tl}]\tfrac{1}{8}\Big{(}\tilde{\gamma}^{(2_{\alpha})}_{k\sigma,0}\tilde{\gamma}^{(2_{\alpha})}_{k\sigma,1}\Big{[}\tilde{\gamma}^{(A_{S})}_{n\tau,0}\tilde{\gamma}^{(A_{2\beta})}_{l\tau,1}\otimes\tilde{\gamma}^{(B_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(B_{S})}_{n\tau,1}+\tilde{\gamma}^{(A_{S})}_{n\tau,0}\tilde{\gamma}^{(A_{2\beta})}_{l\tau,1}\otimes\tilde{\gamma}^{(B_{S})}_{n\tau,0}\tilde{\gamma}^{(B_{2\beta})}_{l\tau,1}$
(290)
$\displaystyle+\tilde{\gamma}^{(A_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(A_{S})}_{n\tau,1}\otimes\tilde{\gamma}^{(B_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(B_{S})}_{n\tau,1}+\tilde{\gamma}^{(A_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(A_{S})}_{n\tau,1}\otimes\tilde{\gamma}^{(B_{S})}_{n\tau,0}\tilde{\gamma}^{(B_{2\beta})}_{l\tau,1}\Big{]}$
$\displaystyle+\Big{[}\tilde{\gamma}^{(A_{S})}_{n\tau,0}\tilde{\gamma}^{(A_{2\beta})}_{l\tau,1}\otimes\tilde{\gamma}^{(B_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(B_{S})}_{n\tau,1}+\tilde{\gamma}^{(A_{S})}_{n\tau,0}\tilde{\gamma}^{(A_{2\beta})}_{l\tau,1}\otimes\tilde{\gamma}^{(B_{S})}_{n\tau,0}\tilde{\gamma}^{(B_{2\beta})}_{l\tau,1}$
$\displaystyle+\tilde{\gamma}^{(A_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(A_{S})}_{n\tau,1}\otimes\tilde{\gamma}^{(B_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(B_{S})}_{n\tau,1}+\tilde{\gamma}^{(A_{2\beta})}_{l\tau,0}\tilde{\gamma}^{(A_{S})}_{n\tau,1}\otimes\tilde{\gamma}^{(B_{S})}_{n\tau,0}\tilde{\gamma}^{(B_{2\beta})}_{l\tau,1}\Big{]}\tilde{\gamma}^{(2_{\alpha})}_{k\sigma,0}\tilde{\gamma}^{(2_{\alpha})}_{k\sigma,1}\Big{)},$

where all of the orbital-transformed Majorana operators are explicitly defined
as:

$\displaystyle\tilde{\gamma}_{k\sigma,i}^{(2_{\alpha})}$
$\displaystyle=\sum_{p}U^{(2)}_{tkp}\hat{\gamma}_{p\sigma,i},$ (291)
$\displaystyle\tilde{\gamma}_{l\sigma,i}^{(A_{2\beta})}$
$\displaystyle=\sum_{p}\tilde{U}^{(2)}_{tlp}\hat{\gamma}_{p\sigma,i},$ (292)
$\displaystyle\tilde{\gamma}_{l\sigma,i}^{(B_{2\beta})}$
$\displaystyle=\sum_{q}V^{(2)}_{tlq}\hat{\gamma}_{q\sigma,i},$ (293)
$\displaystyle\tilde{\gamma}_{n\sigma,i}^{(A_{S})}$
$\displaystyle=\sum_{p}U^{(s)}_{np}\hat{\gamma}_{p\sigma,i},$ (294)
$\displaystyle\tilde{\gamma}_{n\sigma,i}^{(B_{S})}$
$\displaystyle=\sum_{q}V^{(s)}_{nq}\hat{\gamma}_{q\sigma,i}.$ (295)

The corresponding $\ell_{1}$ norm is given by:

$\displaystyle\lambda_{VP_{2}}$
$\displaystyle=\tfrac{1}{2}\lambda_{s}\sum_{tkl}|s^{(2)}_{t}\alpha^{(2)}_{tk}\beta^{(2)}_{tl}|.$
(296)

The $\ell_{1}$ norm of the third term is also found to be given by,

$\displaystyle\lambda_{VP_{3}}$
$\displaystyle=\tfrac{1}{2}\lambda_{s}\sum_{tkl}|s^{(3)}_{t}\alpha^{(3)}_{tk}\beta^{(3)}_{tl}|.$
(297)

####  $\widehat{VP}_{4}$:

The fourth term consists of the symmetric product of two operators,
$\hat{V}^{\prime}$ and $\hat{P}^{\prime}$ defined above. The corresponding
$\ell_{1}$ norm is simply given by,

$\lambda_{VP_{4}}=\lambda_{P}^{(\mathrm{tf})}\sum_{tkl}|s^{(v)}_{t}\alpha^{(A_{v})}_{tk}\alpha^{(B_{v})}_{tl}|.$
(298)

To implement this operator on the quantum computer, we have two options which
we discuss in more detail in Appendix G.3. We can block encode the entire
operator $\widehat{VP}_{4}$ which will consist of $L_{V}L_{P}$ total terms
without truncation. Using the tensor factorization procedures from previous
sections, we have found empirically that the number of terms of the
electrostatic operator, $L_{V}$, will scale between
$\mathcal{O}(N^{2})-\mathcal{O}(N^{3})$ depending on whether we increase the
number of basis orbitals in the continuum limit (increasing the basis set
size) or increase the number of active space orbitals with fixed filling
fraction. Since $L_{p}$ scales as $\mathcal{O}(N^{2})$ then the quantum
circuit for $\widehat{VP}_{4}$ will have an overall complexity of
$\mathcal{O}(N^{5})$ using a naive implementation. For large system sizes,
this implies that the compilation overhead for this operator will be
substantial. While we have shown in the main manuscript that the eigenstate
reflection circuit is the primary bottleneck in the SAPT-EVE algorithm, it is
still worth understanding how we can improve this scaling to ensure that the
block encoding of $\widehat{VP}_{4}$ is not the rate-limiting step once
improved eigenstate preparation techniques are developed in the future. As a
result, we propose the self-inverse product of block encoded operators circuit
outlined in more detail in Appendix G.3. While the total normalization
constant (which affects the total run-time of the expectation value estimation
algorithm) will remain equal to the product, $\lambda_{a}\lambda_{b}$, the
compilation cost is additive with respect to the two operators, $H_{a}$ and
$H_{b}$, scaling asymptotically as $\mathcal{O}(L_{a}+L_{b})$ rather than
$\mathcal{O}(L_{a}L_{b})$.

##  Appendix G SAPT block encoding

block encodings provide a powerful framework for performing non-unitary
operations on a quantum computer. In the following, we describe the block
encoding of the electrostatic, exchange and dominant terms of the
electrostatic-exchange operator. We provide the block encoding of all of the
SAPT operators in the active space picture because they contain more terms
than the full space picture. The full space operator block encoding represents
a special case of the active space operator with some terms excluded. This
Appendix is divided into four sections: Sec. G.1 presents an overview of the
block encoding methodology, Sec. G.2 presents the data-loading oracle, Sec.
G.3 presents the product of block encoded operator circuit proposed in this
work, and Sec. G.4 outlines the full circuits for different SAPT operators.

###  G.1 Overview of block encoding framework

We consider the general block encoding of an operator $\hat{F}$ expressed as a
linear combination of unitaries (LCU),

$\hat{F}=\sum_{n}\alpha_{n}H_{n}$ (299)

where $\alpha_{n}$ are real coefficients and $H_{n}$ are unitary and Hermitian
operators that are assumed to be self-inverse, $H_{n}^{2}=\mathbb{1}$. This
operator can be prepared on a quantum computer using standard block encoding
techniques where the Hermitian operator $\hat{F}$ is embedded within a larger
unitary operator, $\mathcal{B}[\hat{F}]$. To perform the appropriate block
encoding, we require:

$\displaystyle\mathsf{PREPARE}_{F}\ket{0}\ket{\psi}$
$\displaystyle=\sum_{n}\sqrt{\frac{\alpha_{n}}{\lambda}}\ket{n}\\!\ket{\psi}$
(300) $\displaystyle\mathsf{SELECT}_{F}$
$\displaystyle=\sum_{n}\ket{n}\\!\\!\bra{n}\otimes H_{n}$ (301)

where $\mathsf{PREPARE}_{F}$ is a unitary circuit that prepares the
coefficients in the LCU representation of the operator $\hat{F}$ in Eq. (299),
and $\mathsf{SELECT}_{F}$ is a reflection which coherently loads each unitary
in the LCU representation of $\hat{F}$ [57]. These subroutines satisfy the
block encoding equation,

$\braket{0}{\mathsf{PREPARE}_{F}^{\dagger}\,\mathsf{SELECT}_{F}\,\mathsf{PREPARE}_{F}}{0}=\hat{F}/\lambda$
(302)

Here, $\lambda=\sum_{n}|\alpha_{n}|$ denotes the $\ell_{1}$ norm for the
vector of coefficients $\\{\alpha_{n}\\}$ and is needed to ensure that the
operator remains unitary. It is well established in qubitization as well as
other energy estimation algorithms with block encodings that the $\ell_{1}$
norm significantly affects the resource cost of the algorithm. This will also
be true for observable estimation algorithms such as the SAPT-EVE algorithm.

###  G.2 Data-loading oracle

The efficient implementation of the $\mathsf{PREPARE}$ and $\mathsf{SELECT}$
oracles has been the subject of ongoing work over the past few years. In the
following, we briefly review current state-of-the-art implementations which
make use of the data-lookup oracle $\mathsf{QROM}$ (quantum read-only memory)
that performs the task [35, 58],

$\mathsf{QROM}\ket{\mathbf{\mathbf{x}}}\ket{0}=\ket{\mathbf{\mathbf{x}}}\ket{a_{x}}.$
(303)

As pointed out in [38], by utilizing additional qubits (beyond the standard
$\log L$ qubits), it is possible to reduce the overall Toffoli cost. The
asymptotic Toffoli count for this type of $\mathsf{QROM}$ is given by,

$C_{T}=\mathcal{O}\Big{(}\Big{\lceil}\frac{L}{k}\Big{\rceil}+b(k-1)\Big{)},$
(304)

where $k$ represents a tunable power-of-two number copies of the first
register. Minimizing this quantity with respect to $k$, we obtain:

$k=\sqrt{\frac{L}{b}}$ (305)

which requires $\mathcal{O}(\sqrt{L})$ additional qubits. This optimization
will be exploited throughout in order to minimize the overall runtime of the
algorithm. We also take advantage of the efficient quantum adders, coherent
alias sampling, and unprepare subroutines discussed in Babbush  _et al._ [35].

###  G.3 Product of block encoded operators

Before proceeding with the exposition of the block encoding of the SAPT
observables, we first discuss the block encoding of products of operators
which is required for the electrostatic-exchange $\widehat{VP}$ operator. For
illustration purposes, we consider two independent operators $\hat{F}_{1}$ and
$\hat{F}_{2}$ given by,

$\displaystyle\hat{F}_{1}$
$\displaystyle=\sum_{n}^{L_{1}}\alpha^{(1)}_{n}H^{(1)}_{n}\;\;\text{and}\;\;\lambda_{F_{1}}=\sum_{n}|\alpha^{(1)}_{n}|,$
(306) $\displaystyle\hat{F}_{2}$
$\displaystyle=\sum_{n}^{L_{2}}\alpha^{(2)}_{n}H^{(2)}_{n}\;\;\text{and}\;\;\lambda_{F_{2}}=\sum_{n}|\alpha^{(2)}_{n}|.$
(307)

The block encoding of the product of two operators, $\hat{F}_{2}\hat{F}_{1}$,
can proceed in different ways. For instance, consider the following product of
block encoding representation,

$\displaystyle\mathsf{PREPARE}^{\dagger}_{F_{2}F_{1}}\mathsf{SELECT}_{F_{2}F_{1}}\mathsf{PREPARE}_{F_{2}F_{1}}.$
(308)

This block encoding uses two sets of non-contiguous auxiliary qubit registers
with sizes $\log L_{1}$ and $\log L_{2}$ respectively. For instance, the
select oracle is explicitly written as:

$\mathsf{SELECT}_{F_{2}F_{1}}=\sum_{\begin{subarray}{c}n=0\\\
m=0\end{subarray}}^{\begin{subarray}{c}n=L_{1}-1\\\
m=L_{2}-1\end{subarray}}\ket{m}\\!\bra{m}\otimes\ket{n}\\!\bra{n}\otimes
H_{m}^{(2)}H_{n}^{(1)}.$ (309)

On the other hand, the prepare oracle may be expressed as the Kronecker
product of each individual prepare oracle,
$\mathsf{PREPARE}_{F_{2}F_{1}}=\mathsf{PREPARE}_{F_{2}}\otimes\mathsf{PREPARE}_{F_{1}}$.
Based on these definitions, one can verify that the projection with respect to
the zero-state of both auxiliary registers recovers the appropriate product of
operators, $\hat{F}_{2}\hat{F}_{1}$,

$\displaystyle\bra{0}_{2}\bra{0}_{1}\mathsf{PREPARE}_{F_{2}}^{\dagger}\mathsf{PREPARE}_{F_{1}}^{\dagger}\mathsf{SELECT}_{F_{2}F_{1}}\mathsf{PREPARE}_{F_{1}}\mathsf{PREPARE}_{F_{2}}\ket{0}_{1}\ket{0}_{2}=\hat{F}_{2}\hat{F}_{1}/(\lambda_{F_{1}}\lambda_{F_{2}}).$
(310)

This implementation, however, will have a QROM cost scaling as
$\mathcal{O}(L_{1}L_{2})$ which becomes prohibitive for large values of
$L_{1}$ and $L_{2}$. Instead, we consider the product of block encoded
operator (PBEO) circuit in Fig. 6.

Figure 6: Not self-inverse block encoding of a product of operators. Product
of normalized hermitian operators $\hat{F}_{2}$ and$\hat{F}_{1}$ (with
$\ell_{1}$ norms $\lambda_{F_{2}}$, $\lambda_{F_{1}}$, respectively), that
each act on a register $\mathsf{sim}$. The block encoding of the product
features the block encoding of each operator, such that a register
$\mathsf{enc}$ of auxiliary qubits is shared. When the number of auxiliary
qubits to encode $\hat{F}_{j}$ is $\log L_{j}$, then $\mathsf{enc}$ only needs
to hold enough qubits to encode the operators separately, i.e. a total of
$\max\\{\log L_{1},\log L_{2}\\}$ qubits, rather than their combined cost
$\log L_{1}+\log L_{2}$. This comes at the cost of only one additional qubit
labeled $\mathsf{aux}$, as well as a Toffoli gate over the entire
$\mathsf{enc}$ register.

This circuit is a simplification of the one first presented in the work by Von
Burg _et al._ [15]. To ensure that this operator remains self-inverse, we make
a modification by adding a single auxiliary qubit initialized in the Hadamard
basis as shown in Fig. 7,

Figure 7: Self-inverse circuit for block encoding of an operator product.
Using the non-self-inverse block encodings of a product of normalized
hermitian operators $\hat{F}_{2}$ and $\hat{F}_{1}$ (with $\ell_{1}$ norms
$\lambda_{F_{2}}$, $\lambda_{F_{1}}$, respectively), we can turn said block
encoding into a self-inverse encoding of
$(\hat{F}_{1}\hat{F}_{2}+\hat{F}_{2}\hat{F}_{1})/2$ using one query to the
controlled versions of each – the original block encoding and its inverse. The
cost for this is an additional qubit, which has to be initialized on the
Hadamard basis.

which is manifestly self-inverse. The advantage of this approach comes from
the asymptotic cost of block encoding which now scales as
$\mathcal{O}(L_{1}+L_{2})$, providing a substantial reduction Toffoli cost in
the large $L$ limit.

###  G.4 SAPT operators

Due to the factorized representations of the various SAPT operators detailed
in Appendix F.3, the circuit implementations of $\mathsf{PREPARE}$ and
$\mathsf{SELECT}$ must be greatly modified from their most naive, canonical
forms. Below we present the block encoding circuits for each SAPT term with
qualitative descriptions of the reproduced circuit diagrams to provide
intuition for the constructions. We highlight the following notational
conventions used throughout the circuit diagrams:

  * •

We use hexagonal controls to indicate “multiplexing”, or “uniformly-
controlling” a routine [59].

  * •

We distinguish temporary registers and persistent qubit registers whose state
must remain coherent throughout the entirety of an algorithm by where the
drawn wires begin and end; persistent registers (such as index registers and
system qubits) extend the entire length of a diagram, while temporary qubits
begin at particular subroutines that allocate them and terminate at
subroutines that deallocate them.

    * –

For example, data-loaders allocate a number of so-called clean auxiliary
qubits, and the uncomputation of data-loaders deallocate (measure out) those
same qubits; in these diagrams, the target registers where data is loaded will
never flow into a compute portion of a data-loader or flow out of the
uncompute portion of a data-loader, but rather, will emerge and terminate
(respectively) from these routines. As such, these are “temporary" qubit
registers.

  * •

As an extension of the Gidney elbow notation introduced in Ref. [59] for a
Toffoli gate targeting a newly-allocated clean ancilla qubit, we also depict
certain subroutines with emerging elbows to denote when a routine ANDs the
result of a computation (via a Toffoli gate) onto a newly-allocated, single
clean ancillary qubit.

    * –

For example, a comparator circuit flips the state of a zero qubit depending on
the result of a particular inequality check. This “result” qubit can be
allocated by this comparator, and so we depict this resulting qubit and the
Toffoli that targets it as an elbow emerging from a comparator box.

### block encoding circuit for the factorized electrostatics operator
$\hat{V}$

Figure 8: block encoding of the electrostatic operator. Here, the gates
labeled ‘$|\vec{\lambda}\rangle$ ’, ‘$|f_{k}\rangle$’ and ‘$|f_{l}\rangle$’
implement linear superpositions, while hexagonal gates are data loaders
multiplexing over indices such as $t$, $k$ and $l$. Controlled gates labeled
‘$+$’ (‘$-$’) denote in place additions (subtractions) of the control number
to the target number. The labels $\Theta_{A}$ and $\Theta_{B}$ are sets of
angles for the Givens rotations $G_{A}$ and $G_{B}$. The $S$ gate acts on the
computational basis as $S=\mathrm{diag}(1,i)$. We also highlight the
$\mathsf{PREPARE}$ and $\mathsf{SELECT}$ sections of this circuit near the top
of the figure. $\mathsf{H}$ is the Hadamard gate.

Just as the factorized form of the electrostatics operator (after a Jordan-
Wigner transformation) shown in Eq. (260) is analogous to the double
factorization procedure used for electronic structure Hamiltonians, the block
encoding circuit in Figure 8 is analogous to the double factorization circuit
introduced in von Burg  _et al._ [15] and modified in Lee  _et al._ [8]. In
fact, the circuit largely resembles Fig. 16 in Appendix C of Lee  _et al._
[8].

Only a handful of the qubit registers in the circuit are persistent qubit
registers; the rest are temporary auxiliary qubits used for loading
information, and conditioning subsequent operations on those registers. The
persistent registers are: a register that indexes over the first rank
(logarithmic in $N_{1}\leq\text{max}(N_{A}^{2},N_{B}^{2})$), a register that
indexes over the second internal rank for monomer A (logarithmic in
$N_{2}^{(A)}\leq N_{A}$), a register that indexes over the second internal
rank for monomer B (logarithmic in $N_{2}^{B}\leq N_{B}$), a single ancilla
used to index over spin for each monomer, and a pair of registers representing
alpha (up) and beta (down) spin-orbitals of monomers A and B.

In spirit, the circuit proceeds just as the canonical double factorization
circuit [8]: the $\mathsf{PREPARE}$ portion begins by preparing a
superposition over the coefficients indexed by the outer rank on one register,
then uses this register to load data necessary for state preparation on a
separate register for the inner rank of each leaf in the double factorization,
and performs this multiplexed state preparation over the inner rank. The
$\mathsf{SELECT}$ portion proceeds by coherently loading angles for basis-
transforming Givens rotations indexed by the inner, second rank, and then
applies these Givens rotations onto the system qubits before applying single-
qubit Pauli Zs.

The apparent differences between the original double factorization circuit [8]
and the one depicted in this work arise because we are dealing with two
monomers rather than a single set of system qubits. For this reason, we must
coherently load two sets of data for the inner rank state preparations over
each monomer individually, and for rotations that target each monomer’s system
qubits register. Additionally, rather than only applying single-qubit Pauli Z
gates on the system qubits after the Givens rotations, we must select between
$Z_{A}\otimes Z_{B}$, $Z_{A}\otimes\mathbb{1}$, and $\mathbb{1}\otimes Z_{B}$.
To do this, we introduce two additional auxiliary qubits to act as one-body,
single monomer flags to either turn on or turn off the application of a Pauli
Z on the opposite monomer (top wires of the $\text{data}_{A}$ and
$\text{data}_{B}$ subroutines).

The bulk of the cost for this block encoding is in loading the Givens rotation
angles for monomers A and B, and then executing these rotations. For more
details on subroutine-wise costs, see Ref. [8].

### block encoding circuit for the factorized exchange operator $\hat{P}$

Figure 9: block encoding of exchange operator. Here, the gates labeled
‘$|s_{k}s_{l}\rangle$’ implement linear superpositions of data entries $k$ and
$l$, while hexagonal gates are data loaders multiplexing over said indices.
The gates labeled ‘$k\stackrel{{\scriptstyle?}}{{=}}l$’ compare the numbers in
the $k$ and $l$ registers with an adder. Wires born from and ending with
rectangular gates denote Gidney elbows, see Ref. [35] Figure 4. The gates
labeled ‘$G_{i,A}$’ and ‘$G_{j,B}$’ are Givens rotations of monomers A and B,
where $i,j\in\\{1,2\\}$ label whether it corresponds to the intra-monomer or
inter-monomer Givens rotations respectively. The $S$ gate acts on the
computational basis as $S=\mathrm{diag}(1,i)$. We have also highlighted the
$\mathsf{PREPARE}$ and $\mathsf{SELECT}$ sections of this circuit near the top
of the figure.

The factorization circuit used for the exchange circuit in Fig. 9 is similar
in spirit to the single factorization circuit introduced in Berry  _et al._
[34] and modified in Lee  _et al._ [8]. Just as in that case, the
$\mathsf{PREPARE}$ circuit prepares a superposition over the coefficients
resulting from the factorization (in this case, the two indices obtained from
the single-rank decomposition of both monomers), and then the
$\mathsf{SELECT}$ portion must choose between a number of different Pauli
strings. The differences here again owe to the fact that we are dealing with
two monomers rather than a single system register.

It is worth noting that in the full space formulation, the Givens operators
for the inter-monomer and intra-monomer terms are equivalent, i.e.
$G_{2,X}=G_{1,X}$ where $X\in\\{A,B\\}$ (see Eq. (269)), while in the active
space case, they are different, i.e. $G_{2,X}\neq G_{1,X}$.

Furthermore, the Pauli strings are implemented using a similar control logic
to that detailed in Ref. [34]. We have the operations $\vec{Z}Y_{k}$ and
$\vec{Z}X_{l}$ targeting the same sets of qubits (depicted in Fig. 9 of Ref.
[35]), which implement $X\vec{Z}X$, $Y\vec{Z}Y$, or $Z$ (up to signs and
phases) depending on whether $k<l$, $k>l$, or $k=l$. The signs and phases are
cleaned up by the Pauli $Z$ and $S$ gates in the diagram. To distinguish the
one-body single-monomer cases ($Z_{k}\otimes\mathbb{1}$ and $\mathbb{1}\otimes
Z_{l}$), we also introduce additional one-body single-monomer temporary qubits
emerging (or terminating) at the joint state preparation (or its conjugate).
These auxiliary qubits conditionally control the application of a $k$-indexed
$Z_{k}$ operation or an $l$-indexed $Z_{l}$ operation, also controlled on
$k\neq l$ (zero-controls at the top of the diagram).

Here the only persistent qubit registers are the two index registers which sum
over the first rank indices for both monomers, a single ancilla in the plus
state to symmetrize over both index registers after joint state preparation, a
single ancilla plus state to index over spin, another single ancilla on which
we apply a Pauli Z to correct a phase on the resulting Pauli strings, and the
up-and-down spin-orbital registers for both monomers.

### block encoding circuit for the factorized electrostatic-exchange operator
$\widehat{VP}$

As detailed in Appendix F.5, the factorization for the complete electrostatic-
exchange operator is quite verbose. For illustration purposes, here we present
the exact block encoding of only the $\widehat{VP}_{1\ell}$ and
$\widehat{VP}_{4}$ terms. The latter is the asymptotically most expensive part
of the block encoding. We then describe how to combine all the resulting terms
to realize a block encoding of the entire electrostatic-exchange operator
$\widehat{VP}$.

Figure 10: block encoding of the electrostatic-exchange operator,
$\widehat{VP}_{1\ell}$. Here, the gates labeled ‘$|\vec{\lambda}\rangle$’
implement linear superpositions, while hexagonal gates are data loaders
multiplexing over indices such as $t$, $k$, and $l$, in the corresponding
registers. The gates labeled ‘$k\stackrel{{\scriptstyle?}}{{=}}l$’ compare the
numbers in the $k$ and $l$ registers with an adder. The labels
‘$\widetilde{f}_{kl}^{(t)}$’ denote coefficients, and $\Theta_{A}$,
$\Theta_{B}$ are sets of angles for the Givens rotations $G_{A}$ and $G_{B}$.
The $S$ gate acts on the computational basis as $S=\mathrm{diag}(1,i)$. The
controlled ‘$+$’ gate is an in-place addition of numbers from the control
register into the target register. We have highlighted the $\mathsf{PREPARE}$
and $\mathsf{SELECT}$ sections of this circuit near the top of the figure.

####  G.4.1 $\widehat{VP_{1\ell}}$

The complete block encoding circuit for the electrostatic-exchange operator
$\widehat{VP}_{1\ell}$ from Eq. (286) is shown in Fig. 10. This operator is
conceptually a hybrid between the electrostatic and exchange operators
discussed in the previous sections. Compared to the previous operators,
$\widehat{VP}_{1\ell}$ does not contain any one-body terms. However, unlike
the exchange operator, it requires a double factorization procedure where the
first factorization is denoted by the index $t$ and the second
factorization(s) are denoted by $k$ and $l$ respectively, where $k$ and $l$
are analogous to the $k$ and $l$ indices in the exchange operator.

The only persistent registers are the ones holding the indices over the
various factorization indices ($t$, $k$, and $l$), a plus state to symmetrize
over the $k$ and $l$ registers, qubits labeled $q$ and $\theta$ to help
implement different Pauli strings in the case distinction (analogous to the
case distinction discussed for the exchange operator), a plus state to
symmetrize over spin, and the pairs of up and down spin-orbital system qubits
for each monomer.

Just as in the exchange operator block encoding, we need control logic to
distinguish various Pauli string cases. Here the case distinction is exactly
like the one described in Ref. [34], except we have two monomers. For each
monomer, we must apply $X\vec{Z}X+Y\vec{Z}Y$ and $\mathbb{1}-Z$, which we can
do via inequality checks for $k$ and $l$ and using the $q$ wire in a plus
state to choose between $Z$ and $\mathbb{1}$ for the case of $k=l$. The
controlled Pauli Z and controlled S are used to correct for the sign and
imaginary unit that arise from the product of the appropriate Pauli operators.

####  G.4.2 $\widehat{VP_{4}}$

Finally, we consider the implementation of,
$\widehat{VP}_{4}=\tfrac{1}{2}(\hat{V}\hat{P}+\hat{P}\hat{V})$, written as the
symmetric product of two operators. Using the established block encoding
circuit primitives for $\mathcal{B}[\hat{V}]$ and $\mathcal{B}[\hat{P}]$ found
above, it is then possible to use the circuits in Figs. 6 and 7 to implement
the self-inverse product of block encoding of $\widehat{VP}_{4}$. Explicitly,
the complete circuit for the $\widehat{VP}_{4}$ term is given in Fig. 11.

Figure 11: block encoding of the electrostatic-exchange operator,
$\widehat{VP}_{4}$. The symmetric and self-inverse product of $\hat{V}$ and
$\hat{P}$ is block encoded on the simulator registers of monomers A and B,
with a number $\log L_{\mathrm{max}}+2$ of auxiliary qubits, where
$L_{\mathrm{max}}=\max\\{L_{V},L_{P}\\}$, meaning that the two types of block
encodings reuse each other’s auxiliary qubits. The price of that is just one
extra qubit, plus the qubit to implement the superposition. The circuit calls
three instances of the block encoding circuits, where we can make the choice
for the more expensive one, here: $\mathcal{B}[\hat{V}/\lambda_{V}]$, to be
called only once.

####  G.4.3 Combining all terms

In order to implement the complete electrostatic-exchange operator, we recall
that the following terms are required:

$\widehat{VP}_{\mathrm{s}}=\widehat{VP}_{A}+\widehat{VP}_{B}+\widehat{VP}_{1\mathrm{m}}+\widehat{VP}_{1\ell}+\widehat{VP}_{2}+\widehat{VP}_{3}+\widehat{VP}_{4}.$
(311)

The first two terms correspond to monomer-only terms that are exactly
analogous to the standard monomer Hamiltonians which use the standard double
factorization procedure for its implementation. The
$\widehat{VP}_{1\mathrm{m}}$ term corresponds to a unique term that only
arises in the active space picture and is exactly equal to the electrostatic
operator without any 1-body contributions. We have also accounted for
$\widehat{VP}_{1\ell}$ and $\widehat{VP}_{4}$ in the previous paragraphs. As
discussed in Appendix F.5, in the limit of a large number of orbitals, this
operator will always correspond to the most dominant in terms of Toffoli cost.
To implement the complete electrostatic-exchange operator, we use the
following quantum circuit primitive from Ref. [15]:

Figure 12: Linear combination of block encodings. The block encoding of a
hermitian operator $\sum_{j}F_{j}/||\vec{\alpha}||_{1}$ (where $\alpha_{j}$ is
the one-norm of $F_{j}$, and $F_{j}$ is hermitian) acting on a register
$\mathsf{sim}$, is block encoded by preparing the state
$\sum_{j}a_{j}|j\rangle$ in one auxiliary qubit register $\mathsf{aux\,1}$,
such that $a_{j}=|\alpha_{j}|/||\vec{\alpha}||_{1}$, and then multiplexing the
block encoding of the $F_{j}/\alpha_{j}$ over the computational basis state
$|j\rangle$ in $\mathsf{aux\,1}$ using the auxiliary register
$\mathsf{aux\,2}$ to encode the operator.

Which only requires three additional ancillae in order to implement all seven
terms appropriately.

##  Appendix H Benchmark set for small molecules

The benchmark molecules have been chosen according to previous benchmark
studies found in the classical SAPT literature [60]. The $\mathtt{XYZ}$ files
for all of the small benchmark cases may be found in in the benchmark energy
and geometry database for noncovalent complexes: http://www.begdb.org.
Resource estimates for the SAPT-EVE algorithm for the small molecule benchmark
system as well as the heme-artemisinin system are also presented here for
completeness.

##  Appendix I Benchmark for drug design: Heme-artemisinin

###  I.1 Computational Details

The initial pre-complex structure of heme and artemisinin was prepared using
the MOE suit employing the Amber10:EHT force field [61]. Unrestricted density
functional theory calculations (DFT) [62, 63] were performed to obtain the
transition state (TS) structure using Gaussian 16, Revision C.01 (G16) [64].
The $\omega$B97X-D functional [65] and a mixed basis set (Fe: def2-TZVP [66],
N: def2-SVPD and def2-SVP for C, O, and H) were used with standard G16
settings for SCF convergence, exchange-correlation grid and geometry
optimization convergence parameters. The level shift was enabled
($vshift=1000$) to accelerate the SCF convergence. Relaxed potential-energy
surface scans along the Fe-O bond prepared the initial structure of the
transition state. The transition state was subsequently located using the G16
standard TS search algorithm. We used ROHF in combination with the cc-pVDZ
[67, 68, 69] basis to generate orbitals and integrals for the active space
selection and subsequent DMRG [70, 71] calculations using the Pyscf [72] and
block2 [73] packages. All important data describing the molecular systems used
in this paper can be found on the open access repository [74].

Figure 13: (Top) Proposed mechanism of the decomposition of artemisinin with
heme. The key initial reaction step involves the coordination of artemisinin
to the Fe center and a homolytic cleavage of the O-O bond. (Bottom) Summary of
the key orbitals for both artemisinin and heme in order to describe the
complete decomposition mechanism, which must be included in the active space
(see main text for a detailed justification).

###  I.2 Active space selection

The active space was generated for each monomer separately. For heme, we
followed the approach from Ref. [53] and included Fe 3d and 4d orbitals, $\pi$
and $\pi^{*}$ orbitals of nitrogen and carbon in the heme ring, and bonding
and antibonding Fe-N orbitals. This resulted in a (42o,42e) active space. We
picked the orbitals individually from the Pipek-Mezej localized orbitals based
on a high-spin ROHF reference state. The natural orbital occupation from DMRG
calculation are plotted in Fig. 14. For the dimer-centered basis, we started
with the converged monomer-centered ROHF density matrix to converge the ROHF
calculation for which the same procedure was repeated.

For artemisinin, we have used the atomic valence active space (AVAS) method
[75]. We have also included all $2s$ and $2p$ orbitals of the peroxo moiety
(C-O-O-C) in the selection prompt. In addition, we included four more atoms
close to the moiety, which play an important role during the decomposition
(see Fig. 13, colored in red). This yields an active of (40o, 48e) for the
monomer-centered basis, the natural orbital occupation numbers are plotted in
Fig. 14. The same procedure was repeated for the dimer-centered basis.

Figure 14: Natural occupation numbers vs. the index of the active space
orbitals from DMRG calculations; left: the heme monomer (bond dimension
$M=2000$); right: artemisinin monomer ($M=1000$). Deviation from integer
values ($0,1,2$) indicates a strongly correlated nature that is not well
described by single-reference wavefunctions. Both the heme and the artemisinin
molecule have orbitals around the Fermi-level that substantially deviate from
integer occupation, while the heme natural orbital occupations also show long
tails of smaller corrections away from the Fermi-level that would require
large active space calculations to describe correctly.

The active space for the dimer system has been generated by first selecting
the artemisinin orbitals through AVAS from a high-spin ROHF reference. The
remaining orbitals were localized through the Pipek-Mezej routine and the same
set of orbitals from the heme monomer were then selected by hand. This yields
a total active space of (82o, 90e) for the supersystem, which is intractable
for most quantum chemistry methods. The active space choice is not unique, and
FTQC simulations would be required actually to confirm this choice. However,
we believe by carefully analyzing each monomer DMRG result, using chemical
intuition and established mechanistic insights, that this is a representable
active space for studying the initial reaction step for the artemisinin
decomposition.

For the resource estimates, we also require the energy gap between the ground
state of the Hamiltonian and the first excited state and the overlap with the
initial state. We do not have access to the full configuration interaction
energies or states but we use accurate DMRG calculations as a proxy. In Table
2, we list the results of our active space calculations for the heme and
artemisinin monomers in the monomer-centered and dimer-centered bases. We
chose bond dimension $M=2000$ for the heme and $M=1000$ for artemisinin. The
ground states and excited states are found in the same spin sector: $S=1$ for
the heme and $S=0$ for artemisinin.

Table 2: Molecular data from DMRG calculations for benchmark heme artemisinin system. $E_{0}$/$E_{1}$ are ground and first-excited state energies measured in Hartrees (Ha). $\Delta=E_{1}-E_{0}$ is the energy gap. The last column represents the overlap between the restricted open-shell Hartree Fock wavefunction and the DMRG wavefunction. Data is provided on both monomer-centered basis (MCB) and dimer-centered basis (DCB). System | Basis |  $E_{0}$[Ha] |  $E_{1}$[Ha] |  $\Delta$[Ha] | $\left|\braket{\Psi_{\rm ROHF}}{\Psi_{\rm DMRG}}\right|^{2}$
---|---|---|---|---|---
Heme | MCB | -2469.858136 | -2469.851239 | 0.006897 | 0.068174
| DCB | -2469.858690 | -2469.851043 | 0.007646 | 0.067537
Artemisinin | MCB | -916.399835 | -916.278659 | 0.121176 | 0.800254
| DCB | -916.405542 | -916.284444 | 0.121098 | 0.800654

##  Appendix J Supermolecular resource data

Supermolecular resource data is provided in Tab. 3 for the small molecule
benchmark system in the all-electron (full space) picture as well as the
active-space-optimized heme-artemisinin dimer system. In all cases, the
supermolecular (SM) resource estimation consists of three standard quantum
phase estimation runs with the block-encoded double-factorized Hamiltonians
for each system (for more details, see Refs. [15, 8]). Note that we have
performed an appropriate error budgeting procedure to reduce the overall
Toffoli gate cost of the total supermolecular calculation.

|  | System Parameters | Gates | Qubits
---|---|---|---|---
|  | $N_{AB}$ | $N_{A}$ | $N_{B}$ | $\lambda_{H_{AB}}$ | $\lambda_{H_{A}}$ |  $\lambda_{H_{B}}$ | SM | $\textrm{E}_{AB}$ | $\textrm{E}_{A}$ |  $\textrm{E}_{A}$ | SM | $\textrm{E}_{AB}$ | $\textrm{E}_{A}$ | $\textrm{E}_{A}$
Water | STO-3G | 14 | 7 | 7 | 142.8 | 53.9 | 53.9 | $2.11\\!\times\\!10^{9}$ | $1.30\\!\times\\!10^{9}$ | $4.09\\!\times\\!10^{8}$ |  $4.09\\!\times\\!10^{8}$ | 709 | 709 | 440 | 440
6-31G | 26 | 13 | 13 | 226.5 | 73.1 | 73.1 | $5.70\\!\times\\!10^{9}$ | $3.60\\!\times\\!10^{9}$ | $1.05\\!\times\\!10^{9}$ |  $1.05\\!\times\\!10^{9}$ | 1154 | 1154 | 675 | 675
cc-pVDZ | 48 | 24 | 24 | 1027.7 | 328.1 | 328.1 | $4.98\\!\times\\!10^{10}$ | $3.16\\!\times\\!10^{10}$ | $9.07\\!\times\\!10^{9}$ |  $9.06\\!\times\\!10^{9}$ | 2096 | 2096 | 1144 | 1144
Ammonia | STO-3G | 16 | 8 | 8 | 127.3 | 44.6 | 44.6 | $2.06\\!\times\\!10^{9}$ | $1.28\\!\times\\!10^{9}$ | $3.87\\!\times\\!10^{8}$ |  $3.87\\!\times\\!10^{8}$ | 777 | 777 | 474 | 474
6-31G | 30 | 15 | 15 | 278.3 | 84.0 | 84.0 | $7.93\\!\times\\!10^{9}$ | $5.13\\!\times\\!10^{9}$ | $1.40\\!\times\\!10^{9}$ |  $1.40\\!\times\\!10^{9}$ | 1330 | 1330 | 743 | 743
cc-pVDZ | 58 | 29 | 29 | 1405.1 | 433.8 | 433.8 | $8.20\\!\times\\!10^{10}$ | $5.30\\!\times\\!10^{10}$ | $1.45\\!\times\\!10^{10}$ |  $1.45\\!\times\\!10^{10}$ | 2535 | 2535 | 1331 | 1331
Benz.-Water | STO-3G | 43 | 36 | 7 | 592.6 | 453.6 | 53.9 | $2.97\\!\times\\!10^{10}$ | $1.67\\!\times\\!10^{10}$ | $1.21\\!\times\\!10^{10}$ |  $8.38\\!\times\\!10^{8}$ | 1897 | 1897 | 1594 | 455
6-31G | 79 | 66 | 13 | 1495.6 | 1158.5 | 73.1 | $1.36\\!\times\\!10^{11}$ | $7.72\\!\times\\!10^{10}$ | $5.58\\!\times\\!10^{10}$ |  $2.74\\!\times\\!10^{9}$ | 3440 | 3440 | 2850 | 696
cc-pVDZ | 138 | 114 | 24 | 6104.4 | 4563.9 | 328.1 | $1.01\\!\times\\!10^{12}$ | $5.82\\!\times\\!10^{11}$ | $4.08\\!\times\\!10^{11}$ |  $2.23\\!\times\\!10^{10}$ | 6237 | 6237 | 5084 | 1176
Heme-Art. | cc-pVDZ[opt] | 83 | 43 | 40 | 942.0 | 232.2 | 361.8 | $7.91\\!\times\\!10^{10}$ | $5.14\\!\times\\!10^{10}$ | $1.28\\!\times\\!10^{10}$ |  $1.49\\!\times\\!10^{10}$ | 3515 | 3515 | 1852 | 1744

Table 3: Resource estimates of the supermolecular approach. $N_{X}$ denotes
the number of spatial orbitals, $\lambda_{X}$ denotes the $\ell_{1}$ norm for
the double factorized system Hamiltonian ($\hat{H}^{(\mathrm{df})}_{X}$) in
units of Hartree, where $X\in\\{A,B,AB\\}$. The total cost of the
supermolecular (SM) algorithm is measured in terms of Toffoli gates and is
decomposed according to each of the three eigenphase estimates ($E_{A}$,
$E_{B}$, and $E_{AB}$) required to compute the interaction energy in units of
Hartree. The same decomposition is performed with respect to the total number
of qubits. It is important to note that we have performed an appropriate error
budgeting procedure in order to minimize the overall Toffoli gate cost of the
total supermolecular calculation via the Lagrange method, setting the targeted
errors of each QPE as $\varepsilon_{X}\propto\sqrt{\lambda_{X}}$.

##  Appendix K Comprehensive resource data

(a)  (b)

Figure 15: Quantum resource estimation in the active space picture. (a)
Toffoli gate cost for target heme-artemisinin system as a function of active
space spatial orbitals. The active space orbitals are not optimized and are
symmetrically chosen around the Fermi level. (b) Toffoli and logical qubit
dependence of the SAPT-EVE algorithm for the three SAPT observables,
$\hat{F}=\\{\hat{V},\hat{P},\widehat{VP}\\}$, for the non-optimal active space
system on the left. The size of the data points corresponds to the size of the
$\ell_{1}$ norm for the corresponding observable. The numbers on the right
figure correspond to the number of active space orbitals.

Figure 16: SAPT-EVE algorithm call graph for the electrostatic operator,
$\hat{V}$. The total call graph depicts the estimated resources required for
computing all of the SAPT operators for heme-artemisinin benchmark system
using QSP-EVE. The graph displays the distribution of the costs among various
subroutines, and where B[H] is the block encoding of the Hamiltonian and
$\mathcal{R}_{\tau}$ is the controlled block encoding of the observable. All
costs are given in terms of Toffoli gate counts, with each subroutine node
depicting the per-call cost, the total number of calls and cost when taken
over the full algorithm (i.e. overall parent calls). Note that some routines
are deliberately omitted in the count, as they contribute very little. Edge
numbers define the number of calls of the target routine within a single call
of its parent routine. Darker shading indicates a greater total gate cost for
the subroutine. The ASP comprises several low-precision phase estimation
routines in the repeat-until-success circuit.

Figure 17: SAPT-EVE algorithm call graph for the exchange operator, $\hat{P}$.
The total call graph depicts the estimated resources required for computing
all of the SAPT operators for heme-artemisinin benchmark system using QSP-EVE.
The graph displays the distribution of the costs among various subroutines,
and where B[H] is the block encoding of the Hamiltonian and
$\mathcal{R}_{\tau}$ is the controlled block encoding of the observable. All
costs are given in terms of Toffoli gate counts, with each subroutine node
depicting the per-call cost, the total number of calls and cost when taken
over the full algorithm (i.e. over all parent calls). Note that some routines
are deliberately omitted in the count, as they contribute very little. Edge
numbers define the number of calls of the target routine within a single call
of its parent routine. Darker shading indicates a greater total gate cost for
the subroutine. The ASP comprises several low-precision phase estimation
routines in the repeat-until-success circuit.

Figure 18: SAPT-EVE algorithm call graph for the electrostatic-exchange
operator $\widehat{VP}$. The total call graph depicts the estimated resources
required for computing all of the SAPT operators for heme-artemisinin
benchmark system using QSP-EVE. The graph displays the distribution of the
costs among various subroutines, and where B[H] is the block encoding of the
Hamiltonian and $\mathcal{R}_{\tau}$ is the controlled block encoding of the
observable. All costs are given in terms of Toffoli gate counts, with each
subroutine node depicting the per-call cost, the total number of calls and
cost when taken over the full algorithm (i.e. over all parent calls). Note
that some routines are deliberately omitted in the count, as they contribute
very little. Edge numbers define the number of calls of the target routine
within a single call of its parent routine. Darker shading indicates a greater
total gate cost for the subroutine. The ASP comprises several low-precision
phase estimation routines in the repeat-until-success circuit.

Table 4: Molecular properties (sub-table $a$) and resource estimates for the
electrostatic operator $\hat{V}$ (sub-tables $b$ and $c$), electrostatic-
exchange operator $\widehat{VP}_{\mathrm{s}}$ (sub-tables $d$ and $e$), and
exchange operator $\hat{P}$ (sub-tables $f$ and $g$). $N_{A}/N_{B}$ denote
spatial orbitals of monomers A/B. $\lambda_{A}/\lambda_{B}$ denote $\ell_{1}$
norms for Hamiltonians $H_{A}/H_{B}$ in units of Hartree.
$\Delta_{A}/\Delta_{B}$ denote the spectral gaps for monomers $A/B$ in units
of Hartree. $\lambda_{F}$ denotes the $\ell_{1}$ norm for observable,
$\hat{F}$. $\lambda_{V}/\lambda_{VP}$ are in units of Hartree, while
$\lambda_{P}$ is unitless. $\varepsilon_{F}$ is the allocated precision for
observable $\hat{F}\in\\{\hat{V},\hat{P},\widehat{VP}_{\mathrm{s}}\\}$.
Spectral gaps, $\ell_{1}$ norms and precisions are displayed in units of
Hartree. The final cost of the algorithm measured in total number of Toffoli
gates is decomposed in terms of total cost (EF), initial state preparation
cost (ASP), outer QPE cost (oQPE), eigenstate reflection cost
($\mathcal{R}_{\pi}$), inner QPE cost for each monomer (iQPEA and iQPEB),
block encoding cost for each monomer Hamiltonian (B[HA] and B[HB]), second
reflection cost ($\mathcal{R}_{\tau}$). We have also performed the same
decomposition for the total number of qubits. Note that the block encoding
cost for the electrostatic-exchange operator, $\widehat{VP}_{\mathrm{s}}$,
include the $\widehat{VP}_{A}$, $\widehat{VP}_{B}$,
$\widehat{VP}_{1\mathrm{m}}$, $\widehat{VP}_{1\ell}$, and $\widehat{VP}_{4}$
contributions as discussed in the main text.

$(a)$ Properties
System Basis $N_{A}$ $N_{B}$ $\lambda_{A}$ $\lambda_{B}$ $\Delta_{A}$
$\Delta_{B}$ $\lambda_{V}$ $\lambda_{VP}$ $\lambda_{P}$ $\varepsilon_{V}$
$\varepsilon_{VP}$ $\varepsilon_{P}$ Water STO-3G 7 7 53.92 53.92 0.455 0.455
0.43 0.11 0.04 $9.04\\!\times\\!10^{-4}$ $4.72\\!\times\\!10^{-4}$
$4.33\\!\times\\!10^{-4}$ 6-31G 13 13 73.12 73.08 0.311 0.311 3.22 7.09 0.96
$2.96\\!\times\\!10^{-4}$ $6.14\\!\times\\!10^{-4}$ $1.26\\!\times\\!10^{-4}$
cc-pVDZ 24 24 328.11 328.11 0.302 0.302 13.05 39.5 1.53
$2.20\\!\times\\!10^{-4}$ $6.09\\!\times\\!10^{-4}$ $3.32\\!\times\\!10^{-5}$
Ammonia STO-3G 8 8 44.59 44.59 0.532 0.532 0.55 0.07 0.03
$1.02\\!\times\\!10^{-3}$ $3.79\\!\times\\!10^{-4}$ $3.09\\!\times\\!10^{-4}$
6-31G 15 15 84.03 84.03 0.289 0.289 3.98 8.44 0.97 $2.96\\!\times\\!10^{-4}$
$6.06\\!\times\\!10^{-4}$ $1.03\\!\times\\!10^{-4}$ cc-pVDZ 29 29 433.76
433.76 0.28 0.28 16.92 54.9 1.72 $2.04\\!\times\\!10^{-4}$
$6.05\\!\times\\!10^{-4}$ $2.61\\!\times\\!10^{-5}$ Benz.-Water STO-3G 36 7
453.61 53.92 0.211 0.455 1.94 0.14 0.02 $1.12\\!\times\\!10^{-3}$
$3.05\\!\times\\!10^{-4}$ $8.12\\!\times\\!10^{-5}$ 6-31G 66 13 1158.48 73.1
0.198 0.311 12.6 22.23 0.93 $3.14\\!\times\\!10^{-4}$
$5.78\\!\times\\!10^{-4}$ $3.32\\!\times\\!10^{-5}$ cc-pVDZ 114 24 4563.95
328.15 0.197 0.302 47.44 146.51 1.82 $1.99\\!\times\\!10^{-4}$
$5.88\\!\times\\!10^{-4}$ $9.51\\!\times\\!10^{-6}$ Heme-Art. cc-pVDZ[4] 4 4
1.38 2.03 0.007 0.121 0.37 0.05 0.05 $9.58\\!\times\\!10^{-4}$
$3.59\\!\times\\!10^{-4}$ $6.22\\!\times\\!10^{-4}$ cc-pVDZ[6] 6 6 2.81 3.97
0.007 0.121 1.9 1.59 0.82 $3.74\\!\times\\!10^{-4}$ $4.62\\!\times\\!10^{-4}$
$2.40\\!\times\\!10^{-4}$ cc-pVDZ[8] 8 8 4.74 6.93 0.007 0.121 2.71 2.34 0.88
$3.60\\!\times\\!10^{-4}$ $4.59\\!\times\\!10^{-4}$ $1.71\\!\times\\!10^{-4}$
cc-pVDZ[12] 12 12 10.51 13.61 0.007 0.121 5.39 8.2 1.48
$2.53\\!\times\\!10^{-4}$ $4.91\\!\times\\!10^{-4}$ $8.97\\!\times\\!10^{-5}$
cc-pVDZ[16] 16 16 18.98 23.44 0.007 0.121 9.91 23.97 2.31
$1.80\\!\times\\!10^{-4}$ $5.09\\!\times\\!10^{-4}$ $5.01\\!\times\\!10^{-5}$
cc-pVDZ[22] 22 22 36.67 44.71 0.007 0.121 19.56 68.73 3.28
$1.34\\!\times\\!10^{-4}$ $5.21\\!\times\\!10^{-4}$ $2.58\\!\times\\!10^{-5}$
cc-pVDZ[32] 32 32 84.56 98.26 0.007 0.121 41.28 225.01 5.02
$9.28\\!\times\\!10^{-5}$ $5.31\\!\times\\!10^{-4}$ $1.24\\!\times\\!10^{-5}$
cc-pVDZ[46] 46 46 191.62 215.95 0.007 0.121 94.15 901.4 9.01
$5.46\\!\times\\!10^{-5}$ $5.35\\!\times\\!10^{-4}$ $5.51\\!\times\\!10^{-6}$
cc-pVDZ[64] 64 64 386.59 455.65 0.007 0.121 184.92 2775.08 14.47
$3.50\\!\times\\!10^{-5}$ $5.34\\!\times\\!10^{-4}$ $2.83\\!\times\\!10^{-6}$
cc-pVDZ[90] 90 90 808.5 1068.83 0.007 0.121 347.93 8059.98 22.22
$2.31\\!\times\\!10^{-5}$ $5.37\\!\times\\!10^{-4}$ $1.51\\!\times\\!10^{-6}$
cc-pVDZ[opt] 43 40 232.2 361.78 0.007 0.121 65.54 537.3 6.35
$7.29\\!\times\\!10^{-5}$ $5.66\\!\times\\!10^{-4}$ $7.60\\!\times\\!10^{-6}$

$(b)$ $\hat{V}$: gate counts
System Basis $E_{V}$ ASP $\textrm{aQPE}_{A}$ $\textrm{aQPE}_{B}$ oQPE
$\mathcal{R}_{\pi}$ $\textrm{iQPE}_{A}$ $\textrm{B}[H_{A}]$
$\textrm{iQPE}_{B}$ $\textrm{B}[H_{B}]$ $\mathcal{R}_{\tau}$ B[V] Water STO-3G
$3.67\\!\times\\!10^{13}$ $1.15\\!\times\\!10^{10}$ $1.88\\!\times\\!10^{7}$
$1.88\\!\times\\!10^{7}$ $3.67\\!\times\\!10^{13}$ $8.95\\!\times\\!10^{9}$
$2.24\\!\times\\!10^{9}$ $2.26\\!\times\\!10^{3}$ $2.24\\!\times\\!10^{9}$
$2.26\\!\times\\!10^{3}$ $1.73\\!\times\\!10^{3}$ $1.61\\!\times\\!10^{3}$
6-31G $4.80\\!\times\\!10^{15}$ $5.69\\!\times\\!10^{10}$
$9.34\\!\times\\!10^{7}$ $9.32\\!\times\\!10^{7}$ $4.80\\!\times\\!10^{15}$
$3.66\\!\times\\!10^{10}$ $9.16\\!\times\\!10^{9}$ $4.14\\!\times\\!10^{3}$
$9.15\\!\times\\!10^{9}$ $4.14\\!\times\\!10^{3}$ $3.62\\!\times\\!10^{3}$
$3.48\\!\times\\!10^{3}$ cc-pVDZ $1.72\\!\times\\!10^{17}$
$2.48\\!\times\\!10^{12}$ $4.07\\!\times\\!10^{9}$ $4.07\\!\times\\!10^{9}$
$1.72\\!\times\\!10^{17}$ $3.28\\!\times\\!10^{11}$ $8.20\\!\times\\!10^{10}$
$8.28\\!\times\\!10^{3}$ $8.19\\!\times\\!10^{10}$ $8.27\\!\times\\!10^{3}$
$6.88\\!\times\\!10^{3}$ $6.72\\!\times\\!10^{3}$ Ammonia STO-3G
$1.97\\!\times\\!10^{13}$ $7.29\\!\times\\!10^{9}$ $1.19\\!\times\\!10^{7}$
$1.19\\!\times\\!10^{7}$ $1.97\\!\times\\!10^{13}$ $4.81\\!\times\\!10^{9}$
$1.20\\!\times\\!10^{9}$ $2.47\\!\times\\!10^{3}$ $1.20\\!\times\\!10^{9}$
$2.47\\!\times\\!10^{3}$ $2.00\\!\times\\!10^{3}$ $1.88\\!\times\\!10^{3}$
6-31G $5.60\\!\times\\!10^{15}$ $9.50\\!\times\\!10^{10}$
$1.56\\!\times\\!10^{8}$ $1.56\\!\times\\!10^{8}$ $5.60\\!\times\\!10^{15}$
$4.27\\!\times\\!10^{10}$ $1.07\\!\times\\!10^{10}$ $4.83\\!\times\\!10^{3}$
$1.07\\!\times\\!10^{10}$ $4.83\\!\times\\!10^{3}$ $4.10\\!\times\\!10^{3}$
$3.96\\!\times\\!10^{3}$ cc-pVDZ $4.28\\!\times\\!10^{17}$
$5.77\\!\times\\!10^{12}$ $9.46\\!\times\\!10^{9}$ $9.46\\!\times\\!10^{9}$
$4.28\\!\times\\!10^{17}$ $4.08\\!\times\\!10^{11}$ $1.02\\!\times\\!10^{11}$
$1.02\\!\times\\!10^{4}$ $1.02\\!\times\\!10^{11}$ $1.02\\!\times\\!10^{4}$
$8.43\\!\times\\!10^{3}$ $8.27\\!\times\\!10^{3}$ Benz.-Water STO-3G
$7.66\\!\times\\!10^{15}$ $5.24\\!\times\\!10^{12}$ $1.72\\!\times\\!10^{10}$
$1.88\\!\times\\!10^{7}$ $7.66\\!\times\\!10^{15}$ $4.67\\!\times\\!10^{11}$
$2.32\\!\times\\!10^{11}$ $1.27\\!\times\\!10^{4}$ $2.20\\!\times\\!10^{9}$
$2.26\\!\times\\!10^{3}$ $4.99\\!\times\\!10^{3}$ $4.84\\!\times\\!10^{3}$
6-31G $1.10\\!\times\\!10^{18}$ $7.03\\!\times\\!10^{13}$
$2.30\\!\times\\!10^{11}$ $9.33\\!\times\\!10^{7}$ $1.10\\!\times\\!10^{18}$
$2.09\\!\times\\!10^{12}$ $1.04\\!\times\\!10^{12}$ $2.49\\!\times\\!10^{4}$
$9.03\\!\times\\!10^{9}$ $4.14\\!\times\\!10^{3}$ $1.07\\!\times\\!10^{4}$
$1.05\\!\times\\!10^{4}$ cc-pVDZ $3.57\\!\times\\!10^{19}$
$2.09\\!\times\\!10^{15}$ $6.86\\!\times\\!10^{12}$ $4.07\\!\times\\!10^{9}$
$3.57\\!\times\\!10^{19}$ $1.70\\!\times\\!10^{13}$ $8.43\\!\times\\!10^{12}$
$4.59\\!\times\\!10^{4}$ $8.20\\!\times\\!10^{10}$ $8.28\\!\times\\!10^{3}$
$2.00\\!\times\\!10^{4}$ $1.98\\!\times\\!10^{4}$ Heme-Art. cc-pVDZ[4]
$7.71\\!\times\\!10^{12}$ $1.43\\!\times\\!10^{8}$ $4.22\\!\times\\!10^{5}$
$4.61\\!\times\\!10^{4}$ $7.71\\!\times\\!10^{12}$ $1.88\\!\times\\!10^{9}$
$8.43\\!\times\\!10^{8}$ $1.18\\!\times\\!10^{3}$ $9.90\\!\times\\!10^{7}$
$1.06\\!\times\\!10^{3}$ $1.06\\!\times\\!10^{3}$ $9.70\\!\times\\!10^{2}$ cc-
pVDZ[6] $4.48\\!\times\\!10^{14}$ $8.95\\!\times\\!10^{8}$
$2.67\\!\times\\!10^{6}$ $2.68\\!\times\\!10^{5}$ $4.48\\!\times\\!10^{14}$
$6.84\\!\times\\!10^{9}$ $3.07\\!\times\\!10^{9}$ $1.82\\!\times\\!10^{3}$
$3.55\\!\times\\!10^{8}$ $1.65\\!\times\\!10^{3}$ $1.70\\!\times\\!10^{3}$
$1.60\\!\times\\!10^{3}$ cc-pVDZ[8] $1.27\\!\times\\!10^{15}$
$3.53\\!\times\\!10^{9}$ $1.05\\!\times\\!10^{7}$ $1.13\\!\times\\!10^{6}$
$1.27\\!\times\\!10^{15}$ $1.94\\!\times\\!10^{10}$ $8.66\\!\times\\!10^{9}$
$2.47\\!\times\\!10^{3}$ $1.03\\!\times\\!10^{9}$ $2.26\\!\times\\!10^{3}$
$2.24\\!\times\\!10^{3}$ $2.12\\!\times\\!10^{3}$ cc-pVDZ[12]
$1.73\\!\times\\!10^{16}$ $2.70\\!\times\\!10^{10}$ $8.15\\!\times\\!10^{7}$
$7.02\\!\times\\!10^{6}$ $1.73\\!\times\\!10^{16}$ $6.59\\!\times\\!10^{10}$
$2.94\\!\times\\!10^{10}$ $3.77\\!\times\\!10^{3}$ $3.52\\!\times\\!10^{9}$
$3.55\\!\times\\!10^{3}$ $3.42\\!\times\\!10^{3}$ $3.29\\!\times\\!10^{3}$ cc-
pVDZ[16] $9.77\\!\times\\!10^{16}$ $1.21\\!\times\\!10^{11}$
$3.67\\!\times\\!10^{8}$ $2.89\\!\times\\!10^{7}$ $9.77\\!\times\\!10^{16}$
$1.86\\!\times\\!10^{11}$ $8.81\\!\times\\!10^{10}$ $5.30\\!\times\\!10^{3}$
$5.02\\!\times\\!10^{9}$ $4.75\\!\times\\!10^{3}$ $4.69\\!\times\\!10^{3}$
$4.55\\!\times\\!10^{3}$ cc-pVDZ[22] $5.89\\!\times\\!10^{17}$
$6.59\\!\times\\!10^{11}$ $2.01\\!\times\\!10^{9}$ $1.51\\!\times\\!10^{8}$
$5.89\\!\times\\!10^{17}$ $5.61\\!\times\\!10^{11}$ $2.65\\!\times\\!10^{11}$
$7.50\\!\times\\!10^{3}$ $1.52\\!\times\\!10^{10}$ $6.76\\!\times\\!10^{3}$
$6.67\\!\times\\!10^{3}$ $6.52\\!\times\\!10^{3}$ cc-pVDZ[32]
$7.75\\!\times\\!10^{18}$ $5.26\\!\times\\!10^{12}$ $1.61\\!\times\\!10^{10}$
$1.14\\!\times\\!10^{9}$ $7.75\\!\times\\!10^{18}$ $1.85\\!\times\\!10^{12}$
$8.73\\!\times\\!10^{11}$ $1.15\\!\times\\!10^{4}$ $5.03\\!\times\\!10^{10}$
$1.04\\!\times\\!10^{4}$ $9.98\\!\times\\!10^{3}$ $9.82\\!\times\\!10^{3}$ cc-
pVDZ[46] $1.07\\!\times\\!10^{20}$ $4.12\\!\times\\!10^{13}$
$1.27\\!\times\\!10^{11}$ $8.44\\!\times\\!10^{9}$ $1.07\\!\times\\!10^{20}$
$6.41\\!\times\\!10^{12}$ $2.87\\!\times\\!10^{12}$ $1.74\\!\times\\!10^{4}$
$3.39\\!\times\\!10^{11}$ $1.62\\!\times\\!10^{4}$ $1.51\\!\times\\!10^{4}$
$1.50\\!\times\\!10^{4}$ cc-pVDZ[64] $2.56\\!\times\\!10^{21}$
$2.48\\!\times\\!10^{14}$ $7.58\\!\times\\!10^{11}$ $5.41\\!\times\\!10^{10}$
$2.56\\!\times\\!10^{21}$ $3.81\\!\times\\!10^{13}$ $1.80\\!\times\\!10^{13}$
$2.54\\!\times\\!10^{4}$ $1.03\\!\times\\!10^{12}$ $2.33\\!\times\\!10^{4}$
$2.16\\!\times\\!10^{4}$ $2.14\\!\times\\!10^{4}$ cc-pVDZ[90]
$1.60\\!\times\\!10^{22}$ $1.60\\!\times\\!10^{15}$ $4.81\\!\times\\!10^{12}$
$4.44\\!\times\\!10^{11}$ $1.60\\!\times\\!10^{22}$ $1.19\\!\times\\!10^{14}$
$5.62\\!\times\\!10^{13}$ $3.74\\!\times\\!10^{4}$ $3.26\\!\times\\!10^{12}$
$3.44\\!\times\\!10^{4}$ $3.13\\!\times\\!10^{4}$ $3.11\\!\times\\!10^{4}$ cc-
pVDZ[opt] $9.74\\!\times\\!10^{19}$ $6.12\\!\times\\!10^{13}$
$1.79\\!\times\\!10^{11}$ $2.14\\!\times\\!10^{10}$ $9.74\\!\times\\!10^{19}$
$1.16\\!\times\\!10^{13}$ $5.51\\!\times\\!10^{12}$ $1.67\\!\times\\!10^{4}$
$3.01\\!\times\\!10^{11}$ $1.43\\!\times\\!10^{4}$ $1.34\\!\times\\!10^{4}$
$1.32\\!\times\\!10^{4}$

$(c)$ $\hat{V}$: qubit counts
System Basis $E_{V}$ ASP $\textrm{aQPE}_{A}$ $\textrm{aQPE}_{B}$ oQPE
$\mathcal{R}_{\pi}$ $\textrm{iQPE}_{A}$ $\textrm{B}[H_{A}]$
$\textrm{iQPE}_{B}$ $\textrm{B}[H_{B}]$ $\mathcal{R}_{\tau}$ B[V] Water STO-3G
706 457 380 380 674 597 449 75 449 75 635 513 6-31G 1221 686 591 591 1187 887
700 94 700 94 1141 1005 cc-pVDZ 2081 1246 1112 1112 2042 1479 1249 126 1249