text:
\section{Code-Sample to calculate \ensuremath{[2]_{\textrm{R12}}~} with \textsc{Tequila}}\label{sec:codesample}
\lstinputlisting[language=python]{corr-w-tq-pno.py}
\lstinputlisting[language=python]{corr-w-tq-gbs.py}

\begin{figure*}[ht]
    \centering
    \begin{subfigure}{.38\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/cabsref/h2-ref-std.pgf}
        \end{adjustbox}
        \caption{H$_2$ with MP2 as surrogate model}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.38\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/cabsref/h2-ref-r12.pgf}
        \end{adjustbox}
        \caption{H$_2$ with MP2-F12 as surrogate model}
    \end{subfigure}
    \begin{subfigure}{.38\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/cabsref/lih-ref-std.pgf}
        \end{adjustbox}
        \caption{LiH with MP2 as surrogate model}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.38\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/cabsref/lih-ref-r12.pgf}
        \end{adjustbox}
        \caption{LiH with MP2-R12 as surrogate model}
    \end{subfigure}
    \caption{Explicitly correlated \ensuremath{[2]_{\textrm{R12}}~}-correction over the dimensionality of RI basis for H$_2$ and LiH given a fixed OBS of MRA-PNOs, determined according to the \emph{good} procedure.  The final point is annotated with the corrected energy. H$_2$: \bas{cc-pVDZ-F12-OPTRI} for GBS-CABS and 3~PNOs added to the CABS in the mixed case. LiH:  \bas{def2-SVP-RIFIT} as GBS-CABS, the mixed procedure adds another 6~PNOs to the CABS.}
        \label{fig:cabsconv}
\end{figure*}

\section{PNOs as CABS}\label{sec:pnocabs}
The following data comes from \cite{schleich2020regularization}.

First, we take a look at the ``convergence'' of the \ensuremath{[2]_{\textrm{R12}}~}-correction with respect to the dimensionality of the RI space to allow a judgement whether additional F12-regularization of the MRA-PNOs is beneficial, and how MRA-PNOs perform as a CABS in comparison to designated Gaussian basis sets. Note that we do not consider the RI-error here as this is rather a proof-of-principle for the convergence behaviour of MRA-PNOs vs. GBS for CABS than actual production calculations. Results therefor are depicted in Fig.~\ref{fig:cabsconv}.

In the case of the hydrogen molecule, the best energies are achieved by the GBS-CABS, while the mixed-approach does not improve the result further.
This means that the GBS-CABS, \bas{cc-pVDZ-F12-OPTRI}, is likely to be very well suited as RI in this case, which is expected because it is optimized for this use.
The regularized equivalent yields a barely improved result.
Looking at the PNO-CABS, we see that the unregularized surrogate performs almost as well as the much larger GBS-CABS, while the PNO-CABS grown by MP2-R12 give a worse result despite a correction of almost the same magnitude using even less orbitals as CABS.
The reason thereof is \rv{-- and this is a crucial point --}{} that the reference is different.
In the case of MP2 as surrogate, both the PNOs for OBS and CABS are unregularized, and accordingly, for the MP2-F12 CABS, we use OBS that have been generated within the same process coming from MP2-F12.
We chose this approach because this way, the PNOs are generated within one consistent procedure and then easily combined by CABS+ \cite{Valeev2004a}. Different combinations, mixing MP2 and MP2-F12 for OBS and CABS, can be thought of as well but lead to a less efficient procedure that creates a fair amount of unnecessary orbitals that are to be projected out.

Taking a look at the converse result for lithium hydride, we see that here, the MP2-F12 options do considerably worse than before. The higher magnitude of the correction is again related to the worse reference energy using PNOs built by MP2-F12. \rv{}{Building a higher-quality CABS in this case might account for that but comes at additional cost.} Further the choice of GBS-CABS (\bas{def2-SVP-RIFIT}) leads to generally low-performing GBS-CABS in this case, which is much less extensive than the we chose for the hydrogen molecule. The noticable improvement by the mixed approach in the case of MP2-F12 supports this. Yet still, the same basis set for LiH would amount to 110 basis functions, which is more than three times than for the PNO-CABS in this case.

To formulate an overall recommendation, we come back to the \ensuremath{[2]_{\textrm{R12}}~}-correction, which in the framework of \rv{}{Ref.~}\cite{Roskop2014} scales \rv{cubically}{cubically within the framework of approximation C~\cite{Kedzuch2005} or quadratically within approximation D~\cite{May2005,pavovsevic2016sparsemaps}} in the RI-dimensionality. \rv{}{Ref.~\cite{pavovsevic2016sparsemaps} proposes a method using PNOs to reduce the cost of MP2-F12 to quasi-linear dependence in the RI-basis except for some inexpensive quadratic terms; a similar approach could be employed to reduce this cost for \ensuremath{[2]_{\textrm{R12}}~}, too.} Given the seemingly almost equivalent performance of unregularized PNOs for OBS and CABS and the fact, that there are significantly less CABS functions in this case, this approach seems most desirable within this context.

\section{Introduction}

Quantum computing is an emergent computational paradigm with the potential to disrupt some areas of scientific and financial computing. While recent advances in the development of the hardware bring the time of error-corrected quantum computers closer~\cite{Arute2019,AIQuantum2020,Zhong2021}, current quantum computers, often called NISQ (noisy intermediate scale quantum computers), still do not offer practically useful computational advantage given their lack of noise resilience, short coherence times and a rather low, yet steadily increasing, available number of qubits~\cite{Preskill2018}.

Popular approaches to NISQ~computing are constituted by hybrid quantum-classical procedures, which rely on the interplay of a classical and a quantum computer. One important class of such hybrid quantum-classical algorithms is given by variational quantum algorithms (VQAs) that rely on the definition of a parametric cost function, which is evaluated on a quantum computer, and then optimized numerically on a classical computer.\cite{cerezo2020variational, bharti2021noisy} In the context of quantum chemistry, particularly ground-state calculations, this cost function is set to be the energy expectation value of a molecular system.

To make algorithms more suitable for NISQ~hardware, its weaknesses on the hardware end are usually addressed by error mitigation techniques, more efficient mappings of the underlying problem to the quantum computer's language or compilation strategies; see Refs.~\cite{bharti2021noisy,endo2021hybrid} for an extensive overview.
As an alternative, or in particular as an addition to these efforts, one can -- just as in classical computing -- exploit some physical knowledge to make the simulation more efficient. In this context, one aims to reduce quantum resources (in terms of qubits needed and gate count / circuit depth), usually coming at the cost of some additional computational efforts on a classical machine.\cite{McClean2017,Takeshita2020,Motta2020,Kottmann2020a,zhang2021mutual}
Efforts to do so include \emph{a priori} and \emph{a posteriori} computations, which either pre-modify the input for an existing quantum procedure or make use of the output to obtain an external correction, both with the ultimate goal to increase accuracy while maintaining the amount of quantum resources necessary. Conversely, this can be translated into a reduction of quantum resources for a certain level of accuracy.

In this work, we demonstrate a procedure that allows to reduce the \rv{}{number of} qubits required for a certain quantum chemical  quantum calculation involving electronic systems, that allow for an efficient output of the one- and two-particle reduced density matrices (1- and 2-RDM). This perturbative correction is based on methods usually summarized as ``R12'', ``F12'' or explicitly correlated methods, cf. section~\ref{sec:expcorrel}.
Recently, \emph{a priori} modifications inspired by Boys' and Handy's transcorrelated Hamiltonian~\cite{Boys1969a,Boys1969b} -- \rv{a similarity-transformed $ H' = \ensuremath{\mathrm{e}}^{-G} H \ensuremath{\mathrm{e}}^G $ with $G$ such that $H'$ is cusp-free}{where a similarity-transformation of $H$ renders $ H' = \ensuremath{\mathrm{e}}^{-G} H \ensuremath{\mathrm{e}}^G $ cusp-free given an appropriate $G$} -- have been adapted for the use in near-term quantum simulations. McArdle et. al.~\cite{McArdle2020} apply Boys' and Handy's original formulation with a non-Hermitian $H'$ within the imaginary time evolution ansatz~\cite{McArdle2019}, whereas Motta et. al.~\cite{Motta2020} implement a regularized Hamiltonian straightforwardly in a VQE simulation. The latter makes use of the canonical transcorrelation introduced in Ref.~\cite{Yanai2012}, which modifies the Hamiltonian as $H'=\ensuremath{\mathrm{e}}^{A^\dagger} H \ensuremath{\mathrm{e}}^{A}$ with some anti-Hermitian $A$ such that $H'$ is still Hermitian; the Baker-Campbell-Hausdorff-expansion of $H'$ is truncated so that only one- and two-body terms are kept.

On the contrary, we opt to use an \emph{a posteriori} correction based on a perturbative \emph{explicitly correlated} method, namely the spin-free variant of \ensuremath{[2]_{\textrm{R12}}~}~\cite{Torheyden2009,Kong2011,Roskop2014}. This approach is to be contrasted with other post-corrections in form of the family of quantum subspace expansion \rv{}{(QSE)} techniques~\cite{McClean2017,McClean2020,Takeshita2020}; an application of VQE together with the transcorrelated approach, \ensuremath{[2]_{\textrm{R12}}~} or a so called ``CABS~singles'' correction~\cite{Kong2010a} has already been suggested in Ref.~\cite{Cao2018}.
Brief comments on the distinction of our approach with this one will follow in Section~\ref{sec:trot}.

Aside of providing a framework for the application of the explicitly correlated \ensuremath{[2]_{\textrm{R12}}~}-method within the variational quantum eigensolver, the goal of this work is to raise additional awareness of explicitly correlated methods within the quantum computing community.
This work builds on a recent Master's thesis~\cite{schleich2020regularization} and contains updated data.

\paragraph*{A comment on notation}
Within this work, we will make use of the Einstein summation convention, where the indices are clarified in Tab.~\ref{tab:indices}. Frequently we label orbitals only with their indices $ \{p\}\equiv\{\phi_p\} $.  For second-quantized creation and annihilation operators as well as matrix elements, we follow the notation of the works by Kutzelnigg and Mukherjee, which is prevalent in the literature of explicitly correlated methods (defined e.g. in Refs.~\cite{Kutzelnigg1997,Kutzelnigg2003}). Excitation operators and matrix elements up to order two can be found in Tab.~\ref{tab:indices} as well.
Higher-order excitations and matrix elements can be defined in a similar manner but are not needed in this context.
If not stated otherwise, we make use of atomic units $\hslash=e=m_\mathrm{e}=1$ except for {\AA}ngstrom for the unit of length.

\section{Background}
\rv{}{For the sake of self-containment, we will first give a high-level description of explicitly correlated techniques and a historical overview before presenting and motivating the \ensuremath{[2]_{\textrm{R12}}~}-correction in more detail. After that, we will propose a workflow to embed \ensuremath{[2]_{\textrm{R12}}~} into the hybrid quantum-classical variational quantum eigensolver routine.
}
\subsection{Explicit Correlation}\label{sec:expcorrel}
\subsubsection{Basics}
It is well-known that the wavefunction of an electronic system exhibits a \emph{correlation cusp} at the points of inter-electronic coalescence~\cite{Kato1957,Lakin1965,Pack1966,Kutzelnigg1992}. A generic numerical representation of such kinks in the functions with smooth functions imposes problems and leads to a rather high number of basis functions to represent the true wavefunction up to a certain accuracy. Yet it is rather obvious that an explicit incorporation of the inter-electronic distance $r_{12}=\lvert r_1 - r_2 \rvert$ helps to overcome this slow convergence, as demonstrated for the simple case of the ground-state of Helium by Hylleraas~\cite{Hylleraas1929},   later again proposed~\cite{Kato1957,Hirschfelder1963} and supported by rigorous work~\cite{Schwartz1963,Hill1985,Kutzelnigg1992}.
 The essential take-away message can be formulated as follows: Upon occurrence of electron-electron correlation, the deployment of explicitly correlated basis function leads to more accurate energies, given that both the partial-wave expansion~\cite{Schwartz1963,Kutzelnigg1992} as well as a natural-orbital expansion~\cite{Kutzelnigg2008}  decay faster in such a basis (\rv{$\leftrightarrow$}{thus} fewer basis functions needed for the same level of accuracy).
For a comprehensive overview of explicitly correlated methods, we refer to the reviews~\cite{Kong2012,Klopper2006,Ten-no2012,Gruneis2017}.

However, making use of an explicitly correlated basis in the existing methods (such as CI, leading to so called Hylleraas CI~\cite{sims1971hyllci}) does not allow an efficient implementation due to the appearance of very high-dimensional integrals~\cite{Cances2003,Kong2012}. One of the two most prominent ways to exploit explicit correlation is given by perturbative descendants of Kutzelniggs R12-theory~\cite{Kutzelnigg1985,Klopper1987,Klopper1991,Kutzelnigg1991,Kutzelnigg1992}, where MP2-R12 can be seen as the basis for all upcoming, more sophisticated, perturbation methods such as \ensuremath{[2]_{\textrm{R12}}~} or the approach in Ref.~\cite{Ten-no2007correction}. The other one consists of regularizing the system throughout a similarity-transformed Hamiltonian via Boys and Handy's transcorrelated approach~\cite{Boys1969,Boys1969a}, made more applicable throughout approximations such that the effective Hamiltonian is still Hermitian in Refs.~\cite{Luo2010,Yanai2012}. In a recent benchmark~\cite{Kersten2016}, both the perturbative \ensuremath{[2]_{\textrm{R12}}~}-correction, that we will explain in detail in section~\ref{sec:trot} and the canonical transcorrelated approach from Ref.~\cite{Yanai2012} performed comparably well.
Here, we focus on the universal perturbative \ensuremath{[2]_{\textrm{R12}}~} method and will provide a discussion to distinguish the approach from the transcorrelated Hamiltonian when applied to quantum algorithms.

Initially, the explicitly correlated basis has been built by correlating one-electron basis functions, achieved through multiplication with the inter-electronic distance $r_{12}$. Motivated by that fact, that a simple factor of $r_{12}$ does not inherit the correct asymptotics, Gaussian correlation factors $\exp{-r_{12}^2}$ (\emph{Gaussian geminals}~\cite{boys1960integral,singer1960gaussian}) and Slater-type correlation factors $\exp\{ -r_{12}\}$~\cite{Kutzelnigg1985,Ten-no2004slater} have been introduced. Within a study that investigated the performance of different correlation factors, it was found that Slater-type factors often have the most favorable behaviour~\cite{Johnson2017}.
Hence following the \ensuremath{[2]_{\textrm{R12}}~}-literature, we use correlation factors and its matrix elements given as follows:
\begin{equation} \label{eq:f12-integ-correlfactor}
    f_{12} = -\frac{1}{\gamma} \ensuremath{\mathrm{e}}^{-\gamma r_{12}}, \quad r^{\kappa \lambda}_{\mu \nu} = \braket{\mu \nu | f_{12} | \kappa \lambda}.
\end{equation}
Note that when using \texttt{Psi4}~\cite{Smith2020} in our simulations, the correlation factor $f_{12}$ is represented as a linear combination of Gaussians.

\subsubsection{\ensuremath{[2]_{\textrm{R12}}~}-Correction}\label{sec:trot}
Further on, we provide a brief outline to \ensuremath{[2]_{\textrm{R12}}~} as in Refs.~\cite{Torheyden2009,Kong2011,Roskop2014}, recalling that this is a perturbative explicitly correlated method.

Let $\mathcal{H}$ be the Hilbert space of the true wavefunction of a certain electronic system with $N_\mathrm{el}$ \rv{}{electrons} in Born-Oppenheimer approximation, described by its Hamiltonian operator
\begin{equation}
H=h^q_p a^p_q + \frac{1}{2} g^{rs}_{pq} a^{pq}_{rs},
\end{equation}
where $h^p_q$ denotes one-electron integrals and $g^{pq}_{rs}$ two-electron integrals over the Coulomb interaction:
\begin{equation}\label{eq:coulomb-int}
    g^{pq}_{rs} = \braket{rs | {(r_{12})}^{-1} | pq}.
\end{equation}

\newcommand{\ket{\Phi_\mathrm{ref}}}{\ket{\Phi_\mathrm{ref}}}
\newcommand{\Phi_\mathrm{ref}}{\Phi_\mathrm{ref}}
Further we have a \textit{reference} solution $\ket{\Phi_\mathrm{ref}}$ using an arbitrary method to find an approximate solution to the ground-state problem for $H$  using a certain basis consisting of a \emph{finite} number of one-electron orbitals $\{p\}_{p=1}^{ N_\mathrm{OBS}}$ that span $\mathcal{P}\subset \mathcal{H}$.
The perturbative correction we use here aims to correct for the incompleteness of this one-electron basis by excitations into the complementary space $\mathcal{Q} = \mathcal{H}\setminus\mathcal{P}$.
Note that formally, up to this point, the procedure \rv{would be}{is} equivalent to the Quantum Subspace Expansion~\cite{McClean2017,McClean2020,Takeshita2020}.
Yet here, the basis for the complementary space is enforced to be explicitly dependent on the inter-electronic coordinate and thus is deemed to exhibit faster convergence; it is tailored to account specifically for completeness due to the correlation cusp. Conversely, we expect that for systems where the dominant error originates from insufficient representation of the correlation cusp, the present method needs less basis functions for a qualitative representation of the complementary space than  QSE-based procedures with a primitive basis and is thus likely to be more efficient. On the other hand, the application of QSE is not limited to quantum chemistry~\cite{McClean2020}.
It is worth mentioning that all \rv{these}{} perturbative methods relying on the RDMs require the evaluation of complex equations (e.g. via the generalized Wick's theorem~\cite{Kutzelnigg1997}), as well as a set of non-trivial cumulant approximations to avoid measurement of higher-order RDMs~\cite{Kutzelnigg1999,Kutzelnigg2010,Torheyden2009,McClean2017}. While the QSE depends on the solution of a generalized eigenvalue problem, \ensuremath{[2]_{\textrm{R12}}~} amounts in the \textsc{sp}-ansatz~\cite{Ten-No2004}, which fixes amplitudes so that they fulfill the cusp condition as shown in Eq.~\eqref{eq:sp-ansatz}, to a set of tensor contractions.

\begin{table}[ht]
    \centering
    \caption{Glossary (operator and index conventions, acronyms sorted by subject matter)}
    \begin{footnotesize}
    \begin{tabular}{p{4.5cm}p{3.5cm}}
    \hline
        Indices ($\{p\} \equiv \{\phi_p\}, \ket{p}\equiv \ket{\phi_p}$) &  Explanation \\
        \hline
         $\{p,q,r,s\}$ & Orbital basis set (OBS) for orbital space $\mathcal{P}$ / reference\\
         $\{i,j\}$ & Occupied within OBS  \\
         $\{a,b\}$ & Unoccupied within OBS  \\
         $\{\kappa, \lambda\} \rightsquigarrow \{\kappa', \lambda'\}$ & Formally complete basis and finite RI basis to approximate full Hilbert space $\mathcal{H}$ with RI space \\
         $\{\alpha, \beta\}\rightsquigarrow \{\alpha',\beta'  \}$ & Complementary basis and finite complementary auxiliary basis (CABS) to resolve $\mathcal{Q} = \mathcal{H}\setminus\mathcal{P}$  \\
         \hline
         Operators and integrals & Following e.g.~\cite{Kutzelnigg2003} \\
         \hline
         $a^p = a_p^\dagger , \quad a_p$ & Fermionic creation and annihilation operators \\
         $a^p_q = a^p a_q,  {  } a^{pq}_{rs} = a^pa^q a_sa_r, \ldots $ & Particle-number conserving excitation operators \\
         $X^p_q = \braket{q(1) | X(1) | p(1)}$ &  One-electron integrals \\
         $X^{pq}_{rs} = \braket{r(1)s(2) | X(1,2) | p(1)q(2)}$ & Two-electron integrals  \\
         $g^{pq}_{rs}$ & Coulomb integrals, cf. Eq.~\eqref{eq:coulomb-int} \\
         $r^{pq}_{rs} $ & F12 integrals, cf.~Eq.~\eqref{eq:f12-integ-correlfactor} \\
         $\gamma^p_q, \gamma^{pq}_{rs}$ & One- and two-particle reduced density matrices (1- and 2-RDM), cf.~Eq.~\eqref{eq:rdms} \\
         $f^\kappa_\lambda$ & Generalized Fock operator, cf.~Eq.~\eqref{eq:genfock}\\
         \hline
         Acronyms & Explanation and Reference \\
         \hline
         NISQ & Noisy intermediate-scale quantum (computing) \cite{Preskill2018,bharti2021noisy} \\
         VQA/VQE & Variational Quantum Algorithms/Eigensolver \cite{cerezo2020variational,Peruzzo2014} \\
         PNO & Pair-natural orbitals \cite{Kottmann2020,Kottmann2020a} \\
         MRA & Multiresolution analysis \cite{Harrison2016} \\
         RDM & Reduced density matrix \\
         QSE & Quantum subspace expansion \cite{McClean2017,McClean2020,Takeshita2020} \\
         OBS & Orbital basis set \cite{Valeev2004a} \\
         CBS & Complete basis set \cite{Valeev2004a}\\
         CABS & Complementary auxiliary basis set \cite{Valeev2004a}\\
         RI & Resolution of the identity \cite{Valeev2004a}\\
         \hline
    \end{tabular}
    \end{footnotesize}
    \label{tab:indices}
\end{table}

The \ensuremath{[2]_{\textrm{R12}}~} correction then can formally be derived\footnote{This is one way to derive it. One could also blindly start with some perturbation and the second-order Hylleraas functional, interpreting the RI-approximation in the usual sense to break down higher-order integrals. But since this approach well showcases where the correction for \emph{basis-set incompleteness} comes from, we choose to sketch it here. A more thorough outline can be found in Ref.~\cite{schleich2020regularization}.} using L\"owdin partitioning~\cite{Lowdin1962,Valeev2008,Torheyden2008,Torheyden2009}. Then, one can approximate the full complementary space $\mathcal{Q}=\mathrm{span}\{\alpha\}$ using a complementary auxiliary basis set (CABS)  $\{\alpha'\}_{\alpha'=1}^{N_\mathrm{CABS}}$ with finite $N_\mathrm{CABS}$ -- this CABS basis can be an arbitrary basis of functions and does not need to come from a Gaussian basis set, even if the OBS is built from Gaussians.
Note that here, this approximation is formally equivalent to the resolution of the identity approximation in the evaluation of molecular integrals of usual R12/F12 methods (usually abbreviated by RI, insertion of $I =  \ket{\kappa}\bra{\kappa} \approx  \ket{\kappa'}\bra{\kappa'} $ to avoid higher-order integrals).
We construct the RI basis as $\{\kappa'\}=\{p\}\cup\{\alpha'\}$, i.e., as the union of the orbitals $\{p\}$ resulting from the regular orbital basis and $\{\alpha'\}$ coming from the complementary auxiliary basis. The partitioning into a reference $\ket{\Phi_\mathrm{ref}}\in\mathcal{P}$ and a perturbation (to account for incompleteness due to $\mathcal{Q}$) then leads to a correction given by the minimization of a second-order Hylleraas functional
\begin{equation}
  J_H^{(2)}=\braket{\psi^{(1)} | H^{(0)} | \psi^{(1)}   } + 2 \braket{\psi^{(0)} | H^{(1)}| \psi^{(1)}  },
\end{equation}
named after Hylleraas' general approach to perturbation theory~\cite{Hylleraas1929}.
The reference Hamiltonian corresponds to the model Hamiltonian $H^{(0)}=H$, and as zeroth-order state, the solution obtained by the reference method $\ket{\psi^{(0)}}=\ket{\Phi_\text{ref}}$ is picked.
Further, the first-order Hamiltonian is chosen to be a normalized, generalized Fock operator
\begin{equation}\label{eq:genfock}
    H^{(1)}=F = f^\kappa_\lambda a^\lambda_\kappa - E_0 = (h^\kappa_\lambda + \bar{g}^{\lambda r}_{\kappa s} \gamma^s_r) a^\lambda_\kappa - E_0,
\end{equation}
while the appearing 1-RDM corresponds to the reference method (not HF as in the standard Fock operator)~\cite{Kutzelnigg2003}. The bar over the Coulomb-tensor denotes antisymmetrization. This induces maximal resemblance to the MP2-R12 method~\cite{Torheyden2009} .
Explicit correlation is brought into play with the first-order perturbation
\begin{equation}
  \ket{\psi^{(1)}}= QR \ket{\Phi_\text{ref}},
\end{equation}
where $Q$ denotes a projector ensuring that contributions of OBS are projected out \rv{as well}{} as well as only semi-internal first-order excitations are kept (which was found to be important in contrast to Ref.~\cite{Ten-no2007correction}).\cite{Torheyden2009,Kong2011,Kong2012} Further excitations from occupied to CABS orbitals are excluded, being the origin of one-particle incompleteness, which are to be treated by a ``CABS~Singles'' correction as in Ref.~\cite{Kong2010a}. The correlation operator $R$ is defined here as
\begin{equation}
  R=\frac{1}{2} d^{rs}_{pq} r^{pq}_{\kappa\lambda} a^{\kappa\lambda}_{rs}
\end{equation}
and generates two-particle correlation, performing excitations into the complementary space $\mathcal{Q}$, which is represented by a set of explicitly correlated functions.
Using a set of approximations\footnote{Cumulant approximations~\cite{Kutzelnigg1999} as well as the standard, extended and generalized Brillouin conditions~\cite{Kutzelnigg1991} and a set of so called screening approximations~\cite{Valeev2008,Torheyden2009,Kong2011}.}, one could then seek to minimize the Hylleraas functional; this would typically result in an ill-conditioned linear system~\cite{Kong2012}, which usually is avoided by making use of the \textsc{sp}-Ansatz by Ten-No~\cite{Ten-No2004}.
In this formulation, the amplitudes $d^{rs}_{ps}$ are fixed such that the cusp conditions both for singlet~($\sim$\textsc{s}) and triplet~($\sim$\textsc{p}) states are fulfilled.
Using spin-free orbitals, the \textsc{sp}-ansatz reads
\begin{equation}\label{eq:sp-ansatz}
    d^{rs}_{pq}=\frac{3}{8}\delta^r_{p}\delta^s_{q}+\frac{1}{8}\delta^r_{q}\delta^s_{p}.
\end{equation}
This way, \ensuremath{[2]_{\textrm{R12}}~} amounts to a set of tensor contractions, whose evaluation scales as $\mathcal{O}{(N_{\mathrm{el}}^6)}$ assuming $N_\mathrm{OBS}, N_\mathrm{CABS} \in \order{N_{\mathrm{el}}}$ within the framework of the so called  approximation C, introduced in Ref.~\cite{Kedzuch2005} and applied to \ensuremath{[2]_{\textrm{R12}}~} in Ref.~\cite{Roskop2014}.
Instead of restating the rather complex equations, we refer to the original works~\cite{Torheyden2009,Kong2011,Roskop2014}. We further point out that a reduction of the cost can be achieved by correlating only a subset of the reference orbitals that may be chosen according to their occupation number~\cite{Kong2011,Roskop2014}\rv{}{, as well as by following ideas from Ref.~\cite{pavovsevic2016sparsemaps} to lower the order of dependency on the RI dimension}. Diagonalization of the 1-RDM to obtain these comes at almost no cost, since they need to be available anyway.
The framework of approximation C of \ensuremath{[2]_{\textrm{R12}}~} requires evaluation of additional molecular integrals, as well as only  the 1- and 2-RDM with respect to the given reference
\begin{align}\label{eq:rdms}
\gamma^p_q = \braket{\Phi_\text{ref}|a^{p}_{q}|\Phi_\text{ref}}, \quad \gamma^{pq}_{rs}=\braket{\Phi_\text{ref} |   a^{pq}_{rs}| \Phi_\text{ref}}.
\end{align}
Insofar, this correction can be applied  to universal references, as long as the 1- and 2-RDM can be made available.
Here, we exclusively use the spin-free formulation, called SF-\ensuremath{[2]_{\textrm{R12}}~}. Consequently, excitations, RDMs etc. are to be understood as spin-free obtained by summation over spin -- for ease of notation, we treat this implicitly.
A more detailed interpretation of \ensuremath{[2]_{\textrm{R12}}~} can be found in Ref.~\cite{schleich2020regularization}.

We note that the application of the correction is not necessarily restricted to ground state calculations but can be applied to excited states in a similar manner.

\subsection{VQE and VQE + \ensuremath{[2]_{\textrm{R12}}~}}
In this work, we rely on the Variational Quantum Eigensolver~\cite{Peruzzo2014,McClean2016} that can be assigned to the class of variational quantum algorithms, which is one of the most promising classes of algorithms in the NISQ~era of quantum computing~\cite{Cao2018,McArdle2018,cerezo2020variational,bharti2021noisy}. Here, one defines an objective function in form of the expected value of the Hamiltonian operator with respect to a parametrized quantum state. The quantum state is composed by a sequence of parametrized unitary operations in form of quantum gates (often called {ansatz}). Minimization of the objective function yields an approximation to the ground-state energy of the system; the accuracy of this approach greatly depends on the capability of the ansatz to express the true ground state~\cite{Romero2019, anand2021quantum, Lee2019, Grimsley2019, Grimsley2020, Izmaylov2020,ryabinkin2018qubit}.

This way, we obtain a reference for the perturbative approach in form of $\ket{\Phi_\text{ref}} = U(\theta^\star) \ket{0}$ with
\begin{equation}
    \theta^\star = \arg\min_\theta \braket{0|U^\dagger(\theta) {H} U(\theta)|0}
\end{equation}
 and $U(\theta)$ a sequence of parametrized quantum gates. Throughout this work, we will restrict ourselves on Unitary Coupled Cluster-type ans\"atze (UCC)~\cite{anand2021quantum,Peruzzo2014,Romero2019,Kottmann2020a,Kottmann2020b,kottmann2021feasible}.
Given a basis set $\{p\}$, a generic UCC-type wavefunction can be obtained by
\begin{equation}
  \ket{\psi_\mathrm{UCC}(\theta)} = \ensuremath{\mathrm{e}}^{\hat{T} - \hat{T}^\dagger} \ket{\Phi_\mathrm{HF}}  ,
\end{equation}
where $\hat{T}= t^i_a a^a_i + \frac{1}{4}  t^{ij}_{ab} a^{ab}_{ij}$ is the usual cluster operator.
The amplitudes $t^i_a, t^{ij}_{ab}$ serve as variational parameters in this approach.
In practice, a wide range of approximations is raised in order to implement or approximate such a wavefunction~\cite{anand2021quantum}.

In Fig.~\ref{fig:workflow}, we summarized the proposed workflow of a VQE+\ensuremath{[2]_{\textrm{R12}}~} computation. Upon having chosen the molecule of desire along with a choice of orbital basis and CABS basis, one can build the fermionic Hamiltonian $H_\mathrm{ferm}$ in the orbital basis as well as molecular integrals needed for the F12-correction (see Ref.~\cite{Kong2011,Roskop2014}) in the RI-basis.
Additionally, one needs to provide a F12-exponent $\gamma$, cf. Eq.~\eqref{eq:f12-integ-correlfactor}.
Within this work, we fixed $\gamma=1.4$, which overall led to good results. When aiming for the best energy possible, one might perform computations with a set of values and then pick the lowest one if the CBS-limit is not available. In general we found that \ensuremath{[2]_{\textrm{R12}}~} is not very likely to yield energies below the CBS-limit despite it being a non-variational method. Next, one needs to map the fermionic Hamiltonian to a set of qubit operators using an operator mapping. Here, we used Jordan-Wigner, but other mappings can be employed in a similar manner; see e.g. Refs.~\cite{Cao2018,McArdle2019, bharti2021noisy}. The qubit Hamiltonian $H_\mathrm{qc}$ then serves as input for the VQE computation. After VQE has reached convergence, the 1- and 2-RDM need to be measured, which then are processed together with the molecular integrals in the RI basis to a scalar correction in the energy.

\begin{figure*}
    \centering
    \begin{adjustbox}{width=.6\linewidth}
        \begin{footnotesize}
            \inkfig{workflow_paper}
        \end{footnotesize}
    \end{adjustbox}
\caption{Workflow of VQE+\ensuremath{[2]_{\textrm{R12}}~}. Modules marked by green colored can be performed on a quantum processing unit or a classical simulator, while orange modules denote classical pre- and post-operations. Used software packages are marked in the above illustration, and reiterated in section~\ref{sec:software}.}
\label{fig:workflow}
\end{figure*}

To obtain  the necessary 1-~and~2-RDM, the VQE measurement procedure needs to be adapted to resolving the full reduced density matrices, coming at a cost $\order{N_\mathrm{OBS}^4}$ for the 2-RDM in the worst case (symmetry conditions and spin-free formulation only allow for a reduction by a constant factor).
A wide set of measurement reduction techniques has been developed for the evaluation of Hamiltonian expectation values. These dominated by two classes of such techniques: One relies on the identification of sets of commuting operators that can be measured simultaneously, e.g. Refs.~\cite{izmaylov2019unitary, Verteletskyi2020, crawford2021measurement}, which allow  for an approximately cubic cost. The other one determines basis rotations that result in a formulation with less non-mutually commutative terms, leading to less than quadratic number of terms that need to be measured separately~\cite{Huggins2019a,Yen2020}.
A comprehensive overview of advances in this area can be found in Ref.~\cite{bonet2020measurement}.
However, as pointed out in Ref.~\cite{bonet2020measurement}, most of these methods are not directly applicable for the measurement of general fermionic $k$-RDMs.
Using a basis of Majorana operators, Ref.~\cite{bonet2020measurement} shows that smart operator partitioning can yield a reduction of measurement cost up to $\Omega(N_\mathrm{OBS}^2)$.
An advisable strategy for a VQE+\ensuremath{[2]_{\textrm{R12}}~} routine then would be to implement one of the measurement reduction techniques that reliably can suppress the cost to lower than quadratic order, and make use of the strategy outlined in \rv{}{Ref.}~\cite{bonet2020measurement} for the final RDM-evaluation.
An additional set of interesting approaches improving the measurement of RDMs  based on the $n$-representability conditions is given in Ref.~\cite{Rubin2018}.
The efficacy of these approaches has been demonstrated in Ref.~\cite{gonthier2020identifying}.
Such techniques are in particular important in light of stochastic errors when sampling the RDMs. As it was investigated in Ref.~\cite{booth2012explicitly} who applied \ensuremath{[2]_{\textrm{R12}}~} in FCIQMC calculations, sampling noise in the RDMs spreads as well to the correction; such behaviour can be expected to appear as well when extracting RDMs from VQE and thus high fidelity in doing so at moderate cost is desirable.

\section{Results}

\subsection{Computational Setup}

\subsubsection{Construction of Complementary Basis}
Within this work, we follow the CABS+ procedure introduced in Ref.~\cite{Valeev2004a} to generate a complementary basis. This means, given an orbital (reference) basis and complementary basis, the complementary space is specfied by orthogonalization of the joint RI basis $\{\kappa'\}$, obtained by the union of orbital and complementary basis, and projecting out the orbital space. Formally, one constructs a coefficient matrix $C_{\alpha'\kappa'}$, which maps from the RI to the complementary set, such that $\sum_{\kappa'} C_{\alpha'\kappa'}\braket{\kappa'|p} = 0~\forall p$.

In contrast to that, we comment on the possibility of a ``quick \& dirty''-approach via the choice of an active space.
Here, one starts with a given basis set, that is employed in an active space fashion.
That is, there is a set of frozen virtual orbitals which do not contribute to electron correlation effects.
Consequently, we find ourselves in the same setting as the work by Takeshita et. al.~\cite{Takeshita2020} who make use of QSE to account for the ``active space incompleteness''.
However, we do not present results for this situation since we found, in accordance to Valeev's arguments~\cite{Valeev2004a}, that corrections obtained by CABS+ are more accurate and are not sensitive to the choice of active space. Still this does depend on the specific intention behind the computation -- if a valid active space is chosen, this procedure allows in the case of the virtual QSE to retrieve the expressibility of the full orbital basis~\cite{urbanek2020chemistryqse}, and a similar behaviour can be expected for \ensuremath{[2]_{\textrm{R12}}~}.

Accordingly it is advised to follow the CABS+ procedure, as done in Ref.~\cite{Motta2020}.
But, if one were to perform an active space calculation (and an appropriate set of orbitals is chosen), the application of \ensuremath{[2]_{\textrm{R12}}~} is expected to be beneficial.
While we have no numerical evidence to support this, we expect \ensuremath{[2]_{\textrm{R12}}~} to perform better than similar methods that excite into conventional basis functions, such as the subspace expansion-based approach in Ref.~\cite{Takeshita2020}, whenever electron-electron cusp effects are dominant, re-emphasizing that the F12 method constructs an explicitly correlated basis that is expected to ``converge faster'' and hence should require fewer basis functions for a given energy correction.

As basis functions, we will mostly resort to the recently introduced MRA-PNOs (globally orthonormalized, directly determined pair natural orbitals obtained by multiresolution analysis)~\cite{Kottmann2020,Kottmann2020a}. This enables us to, on one hand, build up on the existing data thereof, and on the other hand, investigate the potential of a combination of PNOs together with a F12-method.
Even the most simple MRA-PNO generation at this time generates at least $4N_{\mathrm{diag}}+8N_{\mathrm{offdiag}}$ orbitals given a number of diagonal and off-diagonal pairs, which means that in most cases there are a few unused PNOs. This raises the question whether PNOs are suited for a deployment as CABS-basis.
Building on the theory, it is to be expected that explicitly correlated methods also deliver faster convergence in terms of natural orbitals~\cite{Kutzelnigg2008}, which means we would expect that PNOs built from a MP2-R12 surrogate, as already done in Ref.~\cite{Kottmann2020}, inherit this behaviour. \rv{}{Such behaviour was also found in Refs.~\cite{tew2011local,schmitz2014explicitly}.} Further, a basis of natural orbitals can be seen as a least-squares approximation~\cite{Lowdin1955,Kong2012} and we again expect the PNOs to be somewhat near-optimal despite the pair-approximation. This means that PNOs in general and in particular MP2-F12 PNOs might be interesting candidates. To that end, we provide some data from Ref.~\cite{schleich2020regularization} and additional discussion in appendix~\ref{sec:pnocabs}. Based on these results and the discussion, hereinafter, we will pair VQE-results based on MRA-PNOs for the reference with a PNO-CABS (all with plain MP2 for the surrogate computation), given that they have turned out to outperform PNOs obtained by MP2-F12 if obtained in a self-contained procedure and are also far more compact as an RI-approximation than Gaussian basis functions (\rv{}{The reason herefore is that the reference computation when using PNOs generated by R12 performs worse. While this is expected, we found that in our case, this is so significant that the correction does not fully account for that, although the correction is higher; also } see appendix~\ref{sec:pnocabs}). We emphasize here however that our RI by PNOs is not necessarily perfectly accurate but rather more compact and accurate than a GBS of a comparable size. Additionally the point we want to make is that any PNOs generated that are not used in OBS can increase accuracy when used for the complementary basis.

Of course any set of basis functions -- be it the most prominent choice of orbitals from Gaussian basis sets (GBS), PNOs from GBS or plane-waves -- can be used to apply \ensuremath{[2]_{\textrm{R12}}~}.
As already mentioned, we employ MRA-PNOs here to be able to compare with previous results in Ref.~\cite{Kottmann2020a}. Further, as specified therein and in Ref.~\cite{kottmann2021optimized}, the MRA-PNOs yield surprisingly good results in bond-breaking regimes despite the MP2 dependency. Additionally, in contrast to Gaussian basis sets, they exhibit faster convergence so that even for small basis set sizes, the dominant error can already be expected due to the cusp.
In section~\ref{sec:software} we will point to open-source, freely available software that we employed and wherein we implemented the computation of the correction; both MRA-PNOs and GBS are ready to be employed.

\subsubsection{Choice of {Ansatz} in VQE-Calculations}\label{sec:compu-ansatz}
In the subsequent results, we use a classical FCI calculation for systems with Gaussian basis sets (GBS); for the PNO-basis, we focus on UCC-type ans\"atze. However, the use of a specialized F12-basis set in the case of GBS is recommended, as this enables the reference computation to focus on the representation of the cusp-free part of the wavefunction and the correction takes care of the cusp.
This work can be understood as an extension to Ref.~\cite{Kottmann2020a} and thus we aim to show the potential of MRA-PNOs together with \ensuremath{[2]_{\textrm{R12}}~}. Consequently the results we show mostly deploy MRA-PNOs as a basis but we showcase a few exemplary simulations using FCI/GBS as a verification that general bases can be made use of.

For all MRA-PNO-based computations, we exploit the pair-structure in form of the SPA-ansatz from Ref.~\cite{kottmann2021optimized}, which is \rv{a generalization of the PNO-UpCCG(S)D ansatz in Ref.~\cite{Kottmann2020a} (inspired by Ref.~\cite{Lee2019}) and is}{} repeated here in brevity; also see Ref.~\cite{anand2021quantum} for a description thereof. Given a set of HF reference orbitals $\{i\}$, for each $(ij)$-pair we generate orthonormalized PNOs  $\tilde{\mathcal{S}}_{ij} = \bigcup_{\tilde{a}_{ij}=1}^{r_{ij}} \{ \ket{\tilde{a}_{ij}} \}$. As an orthonormalization procedure, we use a Cholesky decomposition, which has turned out deliver superior results as opposed to e.g. symmetric orthonormalization because it changes the ``most important'' PNOs, ordered by occupation number, the least. Then, we restrict to pair-excitations from the references to the associated PNOs ($U_{\tilde{\mathrm{ D}}}$) and generalized excitations within each PNO ($U_{\mathrm{G}\tilde{\mathrm{D}}}$). One might also think of single excitations, which we do not consider in this work. We can express the \rv{PNO-UpCCGD or}{} SPA-UpCCGD wavefunction as
\begin{equation}
    \ket{\text{SPA-UpCCGD}} = U_{\mathrm G \tilde{\mathrm D}} U_{\tilde{\mathrm D}} U_{\mathrm{HF}} \ket{0}
\end{equation}
where $\ket{\Phi_0} = \ket{\Phi_{\mathrm{HF}}} = U_{\mathrm{HF}} \ket{0}$  and
\begin{align}
    U_{\tilde{\mathrm D}}  &= \prod_{i=1}^{N_{\mathrm{el}/2}} \prod_{a\in \tilde{\mathcal{S}}_{ii}} \exp \left\{ \frac{\theta}{2} \tilde{G}^{aa}_{ii} \right\} \\
    U_{\mathrm G \tilde{\mathrm D}}  &= \prod_{i=1}^{N_{\mathrm{el}/2}} \prod_{a,b\in \tilde{\mathcal{S}}_{ii}} \exp\left\{ \frac{\theta}{2} \tilde{G}^{aa}_{bb} \right\}.
\end{align}
The pair-excitation generator $\tilde G$ is defined as
\begin{equation}
    \tilde{G}^{aa}_{ii} = a^{a_\uparrow a_\downarrow}_{i_\uparrow i_\downarrow} - \mathrm{h.c.}

\end{equation}
In what follows, whenever we use PNOs as a basis, we will employ the SPA-UpCCD ansatz and use ``SPA'' to denote this.
Ref.~\cite{Kottmann2020a} demonstrated that whenever the PNO approximation is a good representation of the molecular system behind, this PNO-restricted parametrization provides potential for a drastic reduction in number of parameters and CNOT-counts as opposed to a UpCCGSD-wavefunction, which itself is compact in comparison to a full UCCSD-parametrization~\cite{Lee2019}. \rv{}{More detailed demonstrations of the performance of SPA can be found in Ref.~\cite{kottmann2021optimized}, aiding to classify it in comparison to popular classical methods.}

Further we provide results for a \emph{cheap} and \emph{good} set of PNOs obtained by MRA-PNO computations.
The \emph{cheap} set signifies a minimal version of PNOs, with the least rich class of excitations from Ref.~\cite{Kottmann2020} (``dipole+'') that uses only the spare orbitals from the generation which one does not opt to use for the orbital space.
In this sense, this procedure would be equivalent to using an active space of some GBS and exploiting the spare orbitals in a correction scheme.
Apart from that, the \emph{good} set of PNOs  allows ten macro-iterations in their generation -- convergence in the PNO-generation usually is achieved beforehand when a certain maximum number of orbitals is specified -- and uses a richer excitation ansatz (``multipole'') with a prescribed goal of the number of PNOs.
We set this number accordingly such that \ensuremath{[2]_{\textrm{R12}}~} converges, which for our purposes can be seen as a measure that the identity is resolved well enough. This method then would be somewhat analogous to adding a RI-optimized CABS to some orbital basis set.

\subsubsection{Software}\label{sec:software}
We have outlined the software packages that have been used in Fig.~\ref{fig:workflow}.
The leading framework organizing the workflow for the work presented here is \textsc{Tequila}~\cite{Kottmann2020b}, while we rely on the methodology in Ref.~\cite{kottmann2021feasible} to obtain analytical gradients at the cost of only two energy expectation values, independent of excitation rank.

As a quantum chemistry backend for classical CI calculations and the construction of a CABS made up by Gaussian orbitals, we use \textsc{Psi4}~\cite{Smith2020}.
To compute MRA-PNOs, we employed the software package \textsc{Madness}~\cite{Harrison2016}.
For the CABS-functionality, our implementation uses modified forks of \textsc{Psi4} and \textsc{Madness}~\footnote{
\textsc{Psi4}: Currently, CABS-functionality is not available in the main repository; a hacky but ready-to-use implementation is accessible at \url{https://github.com/philipp-q/psi4/tree/ri_space}.
For \textsc{Madness}, we use the forked branch \url{https://github.com/kottmanj/madness/tree/pno_integrals_cabs}, which is described in more detail in Ref.~\cite{Kottmann2020a}. Installation instructions for \textsc{Madness} together with \textsc{Tequila} can be found at \url{https://github.com/kottmanj/madness/tree/tequila}. Integration into the main repository is planned here.
}. F12-integrals in Eq.~\eqref{eq:f12-integ-correlfactor} from \textsc{Madness} are computed directly as Slater-type ($\ensuremath{\mathrm{e}}^{-\gamma r_{12}}$), when obtained from \textsc{Psi4} via GBS they are approximated by a linear combination of Gaussians.

Further dependencies include \textsc{OpenFermion}~\cite{McClean2017a} for the handling of fermionic and qubit operations as well as \textsc{Qulacs}~\cite{suzuki2020qulacs} as quantum computing backend. Further, automatic differentiation is taken care of by \textsc{Jax}~\cite{jax2018github}. \textsc{SciPy}~\cite{virtanen2020scipy} is used for the classical optimization procedures (BFGS).

In section~\ref{sec:codesample}, there is a code sample for the computation of the correction, both in the case of Gaussian orbitals and MRA-PNOs. Further, we made available a tutorial notebook on the matter on the github repository for \textsc{Tequila}~\cite{Kottmann2020b}.

\input{results}

\section{Conclusion and Outlook}
First and foremost, we have proposed a workflow for the combination of the explicitly correlated \ensuremath{[2]_{\textrm{R12}}~}-correction together with VQE, and provided the software infrastructure to carry out such computations.
Based on a few small test systems, we were able to show a significant increase in accuracy when using this correction. \rv{}{Although \ensuremath{[2]_{\textrm{R12}}~} is a non-variational perturbation method, within out experiments, we only noticed few and low overshoots beyond FCI as long as the reference method produced a somewhat sensible approximation. For non-sensical or very bad reference inputs, the behaviour of the correction also seemed unpredictable.} In particular in combination with orthogonalized MRA-PNOs~\cite{Kottmann2020,Kottmann2020a} employed in the reference method as well as complementary basis shows considerable promise, and allows to perform the quantum routine at a cost of a minimal basis  yielding accuracies  comparable to large correlation-consistent basis sets currently unfeasible for quantum algorithms. Of course, PNOs generated by a set of Gaussian orbitals can be considered equivalently. To ease the combination of PNOs with a complementary basis composed of Gaussian basis functions, an interface of the MRA part of \textsc{Madness}~\cite{Harrison2016} with e.g. \textsc{MPQC}~\cite{ChongPengCannadaLewisXiaoWangMarjoryClement} can be envisioned.

Additionally, exploiting the PNO-structure for the parametrized quantum circuit turned out to be a powerful tool to reduce parameters within the classical optimization subroutine of VQE. For larger systems such as BeH$_2$ however, this parametrization turned out to loose validity -- ways to overcome this are proposed in Ref.~\cite{Kottmann2020a}.
Here we used the separable pair ansatz~\cite{kottmann2021optimized} as an affordable way to construct quantum circuits, illustrating also that \ensuremath{[2]_{\textrm{R12}}~} can not overcome fundamental limitations in the ansatz.

We divided our MRA-PNO computations into  a set of \emph{cheap} and \emph{good} PNOs, based on the quality of the generated PNOs. We note that for small systems, the \emph{cheap} set often does not yield a large enough number of extra PNOs to induce a significant impact, larger systems with more than 1-2 pairs quickly generate a quite rich set of PNOs (see correction for BH in Fig.~\ref{fig:bh3}, where the corrections for both sets of PNOs do not differ much).
Further, known physical behaviour has to be kept in mind and monitored when growing PNOs, such as the pair structure of LiH (off-diagonal PNOs barely contribute) or critical geometries along the potential energy surface that tend to degeneracies (compare to the \emph{cheap} sets of H$_2$ and LiH in Figs.~\ref{fig:h21},\ref{fig:lih1}.). Yet in any case, whenever the reference PNOs are sufficiently good, adding remaining left-over orbitals throughout \ensuremath{[2]_{\textrm{R12}}~} in the spirit of Ref.~\cite{Takeshita2020} is beneficial.  \rv{}{Beyond that, it would be interesting to consider the constructing geminal-spanning orbitals from Ref.~\cite{pavovsevic2014geminal} instead of MP2-PNOs using MRA.}

While our findings imply a reduction in the number of qubits at the same level of accuracy, due to its perturbative nature we cannot give projections regarding circuit depth, number of entangling gates, or other cost measures. Yet in comparison to the transcorrelated approach~\cite{Motta2020}, the Hamiltonians using a perturbation method remain unchanged and do not exhibit an increased number of terms. On the other hand, the perturbation method does not allow for the same level of flexibility as a regularized Hamiltonian. In the end, the choice between these two approaches depends on a case-by-case basis and might even boil down to a matter of taste.
It remains to investigate which of the methods yield more accurate results. Within a classical benchmark~\cite{Kersten2016}, they performed equally well.

Additionally it would be interesting to examine the influence of noise on \ensuremath{[2]_{\textrm{R12}}~}, which has not been considered in this work. To this end, we refer to Ref.~\cite{urbanek2020chemistryqse} who applied the virtual quantum subspace expansion on actual quantum hardware. Beyond that, Refs.~\cite{urbanek2020chemistryqse,booth2012explicitly} proposed a technique to account for measurement noise -- similar approaches can be thought of in this case.

Finally, one might think of combinations of the quantum subspace expansion together with \ensuremath{[2]_{\textrm{R12}}~}, given that the necessary reduced density matrices are already available. However this does turn out to be not trivial because when applying two different perturbative schemes, one must be careful not to account for the same behaviour twice and ``overcompensate''. One way to approach this might be to follow the idea of a ``CABS~singles'' correction~\cite{Kong2011} to additionally account for one-electron incompleteness.

\section{Acknowledgments}

This work was supported by the U.S. Department of Energy under Award No. DE-SC0019374.

A.A.-G. acknowledges the generous support from Google, Inc.  in the form of a Google Focused Award. A.A.-G. also acknowledges support from the Canada Industrial Research Chairs  Program and the Canada 150 Research Chairs Program. P.S. acknowledges support by a fellowship within the IFI program of the German Academic Exchange Service (DAAD). Simulations were performed with computing resources granted by RWTH Aachen University under project thes0850. This research was enabled in part by support provided by Compute Canada. Computations were performed on the niagara supercomputer at the SciNet HPC Consortium\cite{niagara1, niagara2}. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. J.S.K thanks Florian Bischoff for introducing him to explicit correlation and the depths of madness back in the days. We thank the generous support of Anders G. Fr\o{}seth.

\subsection{Atomic systems}
We start by looking at the atomic systems Helium ($N_{\mathrm{el}}  = 2$) and Beryllium ($N_{\mathrm{el}}  = 4$) and consider a choice of different bases to obtain an intuition as how well MRA-PNOs+\ensuremath{[2]_{\textrm{R12}}~} perform against a naive procedure with a GBS. This should give us an indication regarding possible savings of quantum resources.

Except for the corrected VQE/MRA-energies, Fig.~\ref{fig:atomicsys} is identical to Fig.~2 in \cite{Kottmann2020a}. Compared to the uncorrected energy, \ensuremath{[2]_{\textrm{R12}}~} provides significantly lower energies, which slightly ``overshoot'' but still within a chemical accuracy of $1.6$~mEh, for the simulations with ten and twelve qubits (we denote number of qubits as $N_q$). This showcases the non-variational nature of the perturbation method.
In the case of Beryllium -- this is a behaviour that carries through the results in particular for $N_\mathrm{el}>2$, although in general no assumptions can be made that this needs to be true -- the calculations with 10 qubits are equivalently accurate as the one with 24 qubits.
Note that as anticipated in Ref.~\cite{Kottmann2020a}, the result for Beryllium ``saturates'' at a certain level.
This way, the combination of MRA-PNOs as a system-adapted basis and the \ensuremath{[2]_{\textrm{R12}}~}-correction enables a significant reduction in the necessary number of qubits for both Helium and Beryllium.

\begin{figure*}[t]
    \centering
    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/he_energies.pgf}
        \end{adjustbox}
        \caption{He, $E_{\mathrm{CBS}} = -2.9037$ \cite{dehesa1992study,nakatsuji2012discovery,Bischoff2014a}. }
    \end{subfigure}
    \hfill
    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/be_energies.pgf}
        \end{adjustbox}
        \caption{Be, $E_{\mathrm{CBS}} = -14.667$ \cite{nakatsuji2012discovery}}
    \end{subfigure}
    \caption{Atomic systems.
    Comparison of classical FCI with corrected basis-set-free VQE+\ensuremath{[2]_{\textrm{R12}}~} energies using \emph{good} PNOs. We use a SPA-UpCCD ansatz for VQE~\cite{kottmann2021optimized}. Data  for classical computations is from \cite{Kottmann2020a}.  \rv{}{Note that the corrected results of He are in parts lower than FCI and thus show the non-variational nature of the perturbative correction.} }
        \label{fig:atomicsys}
    \end{figure*}

\subsection{Molecules}\label{subsec:res-mol}
Again the shown results here widely follow Ref.~\cite{Kottmann2020a} to demonstrate the potential of a combination of \ensuremath{[2]_{\textrm{R12}}~} with MRA-PNOs.

In the following, we present potential energy surfaces for the dissociation of H$_2$, LiH, BH and BeH$_2$ (symmetric) in Fig.~\ref{fig:pes-mra}. Recall from section~\ref{sec:compu-ansatz}  that we mostly present outcomes using a MRA/PNO-basis with a distinction in \emph{cheap} and \emph{good} PNOs, i.e., using the left-over, unused orbitals vs. creating a CABS on purpose.
In addition, Fig.~\ref{fig:h2lihgbs} shows potential energy surfaces for H$_2$ and LiH in Gaussian basis sets with a specific CABS basis.
Moving on, we will discuss the results shown in the figures.

\textbf{H$_\mathbf{2}$} First, look at the hydrogen molecule with $N_\mathrm{el}=2$ in Figs.~\ref{fig:h21},\ref{fig:h22},\ref{fig:h23}. For the \emph{cheap} PNOs, only $5$ PNOs are available, which leaves only one orbital for the CABS in the 4-qubit computation.
Hence, the 8~qubit computation (with 4 spatial orbitals) is left with only one PNO to represent the complementary space. As expected, the correction is very small in this case, however it still makes sense to apply it given it only improves the result.
Given that there is only one electron pair in this case, the PNO generation can be seen as quite efficient, and the correction for the \emph{cheap} set is rather low. On the other hand, the 8~qubit computation using \emph{good} PNOs almost reaches the accuracy of the 120~qubit FCI-calculation in the regime of small bond distances. We note that for longer distances, the correction tends towards zero because in case of two basically separated hydrogen atoms, there is no more electron-electron correlation present. Further we point out an additional anomaly in Fig.~\ref{fig:h21}: At a distance of 3~\AA{}ngstrom, the 8~qubit result should lie lower (roughly -1 Eh) -- this is due to the MP2 surrogate model for the PNO generation, which has not been able to produce orbitals of the correct symmetry in this case (for more details, see Ref.~\cite{schleich2020regularization}).

\textbf{LiH}
The previous findings mostly can be confirmed by looking at LiH in Figs.~\ref{fig:lih1},\ref{fig:lih2},\ref{fig:lih3}. Note that since there are two electron pairs now, there would be 8~PNOs left to represent the CABS in the larger OBS for the \emph{cheap} PNOs. However, the associated correction almost vanished since all these orbitals stem from an off-diagonal pair -- since the LiH wavefunction very well separates in a product structure of pair one and two, these orbitals barely contribute, also visible by their low occupation number.  The \emph{good} PNOs allow for a considerable correction, with the corrected energy of both the 12~and 20~qubit computation at a similar level.
Within the dissociation curve for \emph{cheap} PNOs, the energies for both 12 and 20 qubits at $-0.6$~\AA{} are far too high -- this anomaly has the same origin as the $3.0$~\AA{}-point for H$_2$, and can again be resolved by allowing for another macro-iteration in the orbital determination.

In contrast to diatomic hydrogen, the corrections for systems containing atoms with more than two electrons do not vanish for long bond distances because there is still electron-electron correlation embodied in the individual atoms.

Furthermore, we observe that for both sets of PNOs, the lower-qubit computation reaches the same level as the computation with more qubits. This behaviour shows again in the case of BH.

\textbf{BH}
Considering Figs.~\ref{fig:bh1},\ref{fig:bh2},\ref{fig:bh3}, we additionally observe that while the uncorrected energies with both 12 and 22 qubits are very close (they differ by roughly $0.01~Eh$, the corrected energies are almost equal, and except for around the equilibrium differ only in the order of milli-Hartrees, where the fewer qubits computation actually performs better because of the higher-dimensional CABS.
Additionally, the results for \emph{cheap} and \emph{good} PNOs are much closer than before.
This rationale behind this is that with $N_{\mathrm{el}}=6$, with three diagonal and three off-diagonal pairs, there are $4N_{\mathrm{diag}} + 8N_{\mathrm{offdiag}} = 36$ PNOs readily generated within one macro-iteration, and given they do not differ too much in ``quality'', this already yields a decent RI-representation. For higher bond distance, this almost levels out, while for lower distances with stronger electron-electron correlation, the larger CABS still pays off.

\textbf{BeH$_\mathbf{2}$}
Results for BeH$_2$ are shown in Figs.~\ref{fig:beh1},\ref{fig:beh2},\ref{fig:beh3}; these have been obtained in a frozen-core approximation with four active reference orbitals. We point out that for larger bond distances, the PNO-approximation in the construction of the quantum circuit does not hold anymore as good.
This was already noticed in \cite{Kottmann2020a} and overcome using an adaptively enriched parametrization in the style of ADAPT-VQE~\cite{Grimsley2020}.
Here, we abstain from doing so and investigate the impact of the correction in such a regime. We see that -- as expected -- the explicitly correlated correction is not able to account for weak properties of the underlying ansatz.

\newcommand{\vspace{4pt}\hrule\vspace{4pt}}{\vspace{4pt}\hrule\vspace{4pt}}
\begin{figure*}[h]
    \centering
    \vspace{4pt}\hrule\vspace{4pt}
    {H$_2$}\\
    \vspace{4pt}\hrule\vspace{4pt}
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/h2-pes-cheap.pgf}
        \end{adjustbox}
        \caption{\emph{Cheap} PNOs \& CABS, $\dim(\text{RI})=5$}\label{fig:h21}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/h2-pes-good.pgf}
        \end{adjustbox}
        \caption{\emph{Good} PNOs \& CABS, $\dim(\text{RI})=18$}\label{fig:h22}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/h2-mra-corr.pgf}
        \end{adjustbox}
        \caption{\ensuremath{[2]_{\textrm{R12}}~}-Correction}\label{fig:h23}
    \end{subfigure}

\vspace{4pt}\hrule\vspace{4pt}
    {LiH}\\
    \vspace{4pt}\hrule\vspace{4pt}
    \centering
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/lih-pes-cheap.pgf}
        \end{adjustbox}
        \caption{\emph{Cheap} PNOs \& CABS, $\dim(\text{RI})=18$}\label{fig:lih1}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/lih-pes-good.pgf}
        \end{adjustbox}
        \caption{\emph{Good} PNOs \& CABS, $\dim(\text{RI})=41$}\label{fig:lih2}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/lih-mra-corr.pgf}
        \end{adjustbox}
        \caption{\ensuremath{[2]_{\textrm{R12}}~}-Correction}\label{fig:lih3}
    \end{subfigure}

\vspace{4pt}\hrule\vspace{4pt}
    {BH}\\
    \vspace{4pt}\hrule\vspace{4pt}
    \centering
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/bh-pes-cheap.pgf}
        \end{adjustbox}
        \caption{\emph{Cheap} PNOs \& CABS, $\dim(\text{RI})=39$}\label{fig:bh1}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/bh-pes-good.pgf}
        \end{adjustbox}
        \caption{\emph{Good} PNOs \& CABS, $\dim(\text{RI})=51$}\label{fig:bh2}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/bh-mra-corr.pgf}
        \end{adjustbox}
        \caption{\ensuremath{[2]_{\textrm{R12}}~}-Correction}\label{fig:bh3}
    \end{subfigure}

\vspace{4pt}\hrule\vspace{4pt}
    {BeH$_2$}\\
    \vspace{4pt}\hrule\vspace{4pt}
    \centering
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/beh2-pes-cheap.pgf}
        \end{adjustbox}
        \caption{\emph{Cheap} PNOs \& CABS, $\dim(\text{RI})=19$}\label{fig:beh1}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/beh2-pes-good.pgf}
        \end{adjustbox}
        \caption{\emph{Good} PNOs \& CABS, $\dim(\text{RI})=48$}\label{fig:beh2}
    \end{subfigure}
    \hfill
    \begin{subfigure}{.33\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/beh2-mra-corr.pgf}
        \end{adjustbox}
        \caption{\ensuremath{[2]_{\textrm{R12}}~}-Correction}\label{fig:beh3}
    \end{subfigure}
    \caption{Potential energy surfaces with \ensuremath{[2]_{\textrm{R12}}~}-correction and PNOs as CABS for H$_2$, LiH, BH, BeH$_2$, following Ref.~\cite{Kottmann2020a}. SPA ansatz with doubles excitations only (SPA-UpCCD~\cite{kottmann2021optimized}). \rv{}{For validation of the SPA ansatz compared to classical techniques such as HF, CC, we refere to Ref.~\cite{kottmann2021optimized}}}.
    \label{fig:pes-mra}
\end{figure*}

\begin{figure*}[t]
    \centering
    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/h2-enco.pgf}
        \end{adjustbox}
        \caption{{H$_2$}, $\dim(\text{CABS})=48$.}
    \end{subfigure}
    \hfill

    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/lih-enco.pgf}
        \end{adjustbox}
        \caption{{LiH}, $\dim(\text{CABS})=61$.}
    \end{subfigure}
    \caption{Corrected potential energy surfaces using classical FCI and Gaussian basis sets.}
    \label{fig:h2lihgbs}
\end{figure*}

The last set of potential energy surfaces we look at are comprised by the PES for the hydrogen molecule and lithium hydride in Gaussian basis sets in Fig.~\ref{fig:h2lihgbs}. They show potential energy surfaces for H$_2$ and LiH in \bas{STO-3G} and \bas{6-31G} with a specific CABS basis generated by CABS+ \cite{Valeev2004a} to demonstrate that Gaussian basis sets with designated CABS can be used in the same manner.
Here, we do not observe any anomalies with respect to the results before but see only consistent behaviour, e.g. the vanishing correction for large bond distances in the case of H$_2$.

\subsection{NPE + MAX}

\begin{figure*}[t]
   \centering
    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/npe_pnos.pgf}
        \end{adjustbox}
        \caption{Non-Parallelity Error }
    \end{subfigure}
    \hfill
    \begin{subfigure}{.45\linewidth}
        \begin{adjustbox}{width=\linewidth}
            \input{figs/max_pnos.pgf}
        \end{adjustbox}
        \caption{Maximum Error }
    \end{subfigure}
    \caption{Performance of MRA-PNOs with and without \ensuremath{[2]_{\textrm{R12}}~} in comparison to FCI in a DZ-basis. CABS for MRA are generated using the \emph{good} setup ($\dim$(CABS) as in Fig.~\ref{fig:pes-mra}; for the GBS we used cc-pVDZ-F12-OPTRI, amounting to $\dim(\text{CABS})$ of 48 for H$_2$ and 110 for both LiH, BH). }
    \label{fig:errors}
\end{figure*}

As a final result, we consider the non-parallelity error (NPE) and maximum error (MAX) defined over the computed PES for the cases of H$_2$, LiH and BH, following~\cite{Kottmann2020a}.
We compare MRA-PNOs with and without corrections with a corrected DZ-basis (with \bas{cc-pVDZ-F12-OPTRI} as CABS).
We do not use a F12-optimized \bas{cc-pVDZ-F12} for the orbital basis to keep the number of qubits closer to the MRA-PNO computations.
With $\Delta E(R) = \abs{E(R)-E^{\mathrm{FCI}}_\mathrm{ref}(R)}$, these error metrics are defined as
\begin{align}
    \Delta E_{\mathrm{NPE}} &= \max_{R\in\mathrm{PES}} \Delta E(R)  - \min_{R\in\mathrm{PES}} \Delta E(R)   \\
    \Delta E_{\mathrm{MAX}} &= \max_{R\in\mathrm{PES}} \Delta E(R).
\end{align}
Looking at Fig.~\ref{fig:errors}, we can say that in particular the MAX error is consistently and significantly reduced. Since the explicitly correlated correction is not a constant shift but typically higher for lower bond distances and thus higher electron-electron correlation, the NPE error is not necessarily reduced but also does not increase by much in our computations. This holds in particular for the smaller systems because this effect carries more weight here. Occasional increase in NPE can be traced to higher correction for short bond distances, which holds in particular for smaller systems (compare to discussion of H$_2$ in section~\ref{subsec:res-mol}). This effect is is less severe for larger systems, as witnessed by BH in Fig.~\ref{fig:errors}. For C$_2$, Ref.~\cite{roskop2016application} observed a reduction in NPE as well.

redpajama_set_name: RedPajamaArXiv


text:
Grozny
Mosque "Heart of Chechnya" named after Akhmat Kadyrov
Fountain "Running Wave"
Spiritual Administration of Muslims of the Chechen Republic
The complex of high-rise buildings "Grozny-City"
State Picture Gallery. Akhmata-Khadzhi Kadyrova
House P. Kosenko
State Gallery named after A. A. Kadyrov
THE HOUSE IN WHICH HAS BEEN THE headquarters of the 1-st arrow regiment of the XII RED ARMY
Chechen State Drama Theater named after Khanpashi Nuradilov
Church of St. Michael the Archangel
Grozny @wikipedia
Grozny (Russian: Грозный, IPA: [ˈgroznɨj]; Chechen: Соьлжа-ГӀала / Sölƶa-Ġala) is the capital city of Chechnya, Russia. The city lies on the Sunzha River. According to the 2010 Census, it had a population of 271,573; up from 210,720 recorded in the 2002 Census, but still only about two-thirds of 399,688 recorded in the 1989 Census. It was previously known as Groznaya (until 1870).
Russia @wikipedia
Russia (Russian: Росси́я, tr. Rossiya, IPA: [rɐˈsʲijə]), or the Russian Federation (Russian: Росси́йская Федера́ция, tr. Rossiyskaya Federatsiya, IPA: [rɐˈsʲijskəjə fʲɪdʲɪˈratsɨjə]), is a transcontinental country in Eastern Europe and North Asia. At 17,125,200 square kilometres (6,612,100 sq mi), Russia is by a considerable margin the largest country in the world by area, covering more than one-eighth of the Earth's inhabited land area, and the ninth most populous, with about 146.80 million people as of 2019, including Crimea. About 77% of the population live in the western, European part of the country. Russia's capital, Moscow, is one of the largest cities in the world and the second largest city in Europe; other major cities include Saint Petersburg, Novosibirsk, Yekaterinburg and Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a wide range of environments and landforms. From northwest to southeast, Russia shares land borders with Norway, Finland, Estonia, Latvia, Lithuania and Poland (both with Kaliningrad Oblast), Belarus, Ukraine, Georgia, Azerbaijan, Kazakhstan, China, Mongolia and North Korea. It shares maritime borders with Japan by the Sea of Okhotsk and the U.S. state of Alaska across the Bering Strait. However, Russia recognises two more countries that border it, Abkhazia and South Ossetia, both of which are internationally recognized as parts of Georgia. The East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, the medieval state of Rus arose in the 9th century. In 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus ultimately disintegrated into a number of smaller states; most of the Rus' lands were overrun by the Mongol invasion and became tributaries of the nomadic Golden Horde in the 13th century. The Grand Duchy of Moscow gradually reunified the surrounding Russian principalities and achieved independence from the Golden Horde. By the 18th century, the nation had greatly expanded through conquest, annexation, and exploration to become the Russian Empire, which was the third largest empire in history, stretching from Poland on the west to Alaska on the east.Following the Russian Revolution, the Russian Soviet Federative Socialist Republic became the largest and leading constituent of the Union of Soviet Socialist Republics, the world's first constitutionally socialist state. The Soviet Union played a decisive role in the Allied victory in World War II, and emerged as a recognized superpower and rival to the United States during the Cold War. The Soviet era saw some of the most significant technological achievements of the 20th century, including the world's first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the world's second largest economy, largest standing military in the world and the largest stockpile of weapons of mass destruction. Following the dissolution of the Soviet Union in 1991, twelve independent republics emerged from the USSR: Russia, Ukraine, Belarus, Kazakhstan, Uzbekistan, Armenia, Azerbaijan, Georgia, Kyrgyzstan, Moldova, Tajikistan, Turkmenistan and the Baltic states regained independence: Estonia, Latvia, Lithuania; the Russian SFSR reconstituted itself as the Russian Federation and is recognized as the continuing legal personality and a successor of the Soviet Union. It is governed as a federal semi-presidential republic. Russia's economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2018. Russia's extensive mineral and energy resources are the largest such reserves in the world, making it one of the leading producers of oil and natural gas globally. The country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction. Russia is a great power as well as a regional power and has been characterised as a potential superpower. It is a permanent member of the United Nations Security Council and an active global partner of ASEAN, as well as a member of the Shanghai Cooperation Organisation (SCO), the G20, the Council of Europe, the Asia-Pacific Economic Cooperation (APEC), the Organization for Security and Co-operation in Europe (OSCE), and the World Trade Organization (WTO), as well as being the leading member of the Commonwealth of Independent States (CIS), the Collective Security Treaty Organization (CSTO) and one of the five members of the Eurasian Economic Union (EEU), along with Armenia, Belarus, Kazakhstan and Kyrgyzstan.
Nokhchiyn kkherch
Hearth ancestors
Teahouse "Bukhara"
Sira Din Hotel
Daimohk
Hotel Nohcho-Star
ZIART SALAM-HADJI
ZIARTI IBRAHIM-HAJI
MONUMENT TO CIVIL WAR HEROES N.F. GIKALO, A.D. SHERIPOVA AND G.S. AHRIEVU
Grozny Dendrological Garden
Chernorechensky reservoir
Maternal Glory Park

redpajama_set_name: RedPajamaCommonCrawl


text:
Caribbean Citizenship by Investment Programmes join in unity for Caribbean Investment Summit 2018.
As the economic citizenship industry continues to compel a captive audience internationally, Prime Minister of St Kitts and Nevis, Dr the Honourable Timothy Harris, has today announced an investment summit to be hosted by their pioneering Citizenship by Investment Programme.
In: Newsfeed
By: CS Global Partners
Share: Facebook / Twitter
The Summit will bring together all five Caribbean nations that currently provide citizenship by investment programmes in a combined effort to lead constructive debate on the industry's most important issues.
The theme central to this event is 'Unity in the Age of Division, Emerging Trends of CIPs in an Uncertain World', focusing on industry best practice, as well as the current geopolitical climate and its associated impact. The Summit will stimulate an exchange of ideas to better understand how the economic citizenship industry will, and should, move forward.
The Summit will be based at the St Kitts Marriott Resort from 16-19 May, and will include two days of speakers and panels, sessions with local developers, and island tours.
The CEO of the St Kitts and Nevis Citizenship by Investment Unit, Les Khan, believes the event will be important locally, as well as internationally, for the industry:
"The Caribbean Investment Summit is not only relevant to our regional partners here in the Caribbean. From its inception the event has been designed to attract an international dialogue."
Khan stressed the need for the industry to unite and proactively discuss the biggest challenges facing the future of the industry:
"The conversations and challenges that the Caribbean region is facing, and has faced in the past, are not dissimilar to those seen on a global scale. This is a perfect opportunity for the industry to unite, and hold constructive debates on the issues of today, and prospects of tomorrow."
Speaking topics such as transparency versus confidentiality, and how international policies such as Common Reporting Standard may influence or alter practices in the industry are on the agenda. Climate resilience will also be a major talking point, with sustainable development a key consideration in the area. The Summit will also delve into the area of technology and how new practices such as blockchain will be regulated in the industry.
With Heads of States and Government officials from all five islands already committed to attending, the Summit is expected to attract key industry stakeholders. Further information can be found on the Caribbean Investment Summit website along all the information you may need to get involved. Those interested in sponsorship are encouraged to download the official brochure.
All enquiries are encouraged through [email protected].
Austrian Citizenship by Investment Program: Benefits, requirements, and investment options
Austria is a beautiful, developed country located in the centre of Europe. It is renowned for its natural beauty, impressive...
Antigua and Barbuda citizenship by investment programme
Antigua and Barbuda: As 2023 begins, the world still faces several unresolved conflicts and economic uncertainties. Russia's invasion of Ukraine...


The Maldives has won the award for World's Best Spa Destination at the World Spa Awards held on Sunday (4 September) at Forte Village Resort in Sardinia, Italy.
The Maldives was also named the Best Spa Destination in the Indian Ocean category.
Launched by the World Travel Awards in 2015, the World Spa Awards aims to drive up standards within spa tourism by rewarding organisations that are the leaders in the field.
The Spa by Thalgo at The Sun Siyam Iru Fushi was named Indian Ocean's Best Resort Spa as well as Maldives' Best Wellness Retreat.
COMO Cocoa Island won the award for Indian Ocean's Best Wellness Retreat. COMO was also awarded Best Resort Spa in the country.
The gala event took place alongside the World Travel Awards Europe Gala Ceremony 2016, and welcomed hundreds of hospitality leaders from across Europe and around the world.
Speaking at the Gala Ceremony, World Spa Awards President Graham Cooke said: "Following a successful launch in Vietnam last year, World Spa Awards is now firmly established on the international spa industry calendar.
Talise by Jumeirah won the prestigious title of World's Best Hotel Spa Brand while Etihad Six Senses Spa was awarded World's Best Airline Lounge Spa.
The title of World's Best Spa Brand was awarded to QMS MediCosmetics with The Armani/SPA at Armani Hotel Milano winning World's Best Spa Design award.
Anantara Spa at Banana Island Resort Doha meanwhile took home the award for World's Best Private Island Spa.
The award for World's Best Wellness Retreat went to Forte Village Resort, Italy, the host of the event.
During the ceremony, Olhuveli Beach & Spa Resort was announced as the host for the Grand Finale of the World Travel Awards 2016, to be held in December.
The World Spa Awards programme has categories for the Best Hotel Spa, Best Resort Spa, Best Wellness Retreat, and Best Spa Destination, each awarded on a national level.
The individual country level winner with the most votes not only takes the national award but also the world level award in each of the respective categories.
A number of categories exist on a world-level only, such as Best Airline Lounge Spa, Best Cruise Line Spa, Best Spa Brand, Best Medical Spa and Best Desert Spa.
Votes are cast by professionals working within the spa industry and submitted online at World Spa Awards website.


Another 15 percent of near-retirees have $500,000 saved. If you are among them, you might be gasping now: Clearly $500,000 looks puny when you assume that you will have to use $260,000 of it to pay for insurance for doctors and hospitals, and extras like dentists, glasses and medicine.
If you think you can shave something off those costs, it's not likely. Fidelity bases its estimate on what people must pay each year to get Medicare and the additional medical costs that aren't covered by Medicare.
Some people assume Medicare is free, but it's not. The average person will pay $107.60 a month next year for the monthly premiums the government charges to get Medicare doctor and hospital coverage, or what is known as Medicare Part B. Then for drug coverage, the average monthly cost will be $40.66, according to a recent Medicare trustees report.
But even after paying all of those charges, you will be far from done. Medicare pays only part of medical costs, so besides paying for their Medicare, retirees need to buy supplemental insurance policies. They are called Medigap and average about $180 a month. And you aren't done with costs after that, either. There are additional out-of-pocket expenses for medicine, dentists, eye doctors, glasses, hearing aids at $5,000 a pop and a lot more.
The average retiree gets about $1,350 a month from Social Security.
Since 2002, each year Fidelity has looked at medical costs and how they have been growing. With that, the investment firm has estimated what retirees are likely to have to pay in the years ahead. Fidelity's estimates are generally considered reasonable among financial planners.
With health care costs and life expectancy both rising, this year's estimate is about 6 percent higher than last year's and about 14 percent higher than 2014's. Fidelity Senior Vice President Sunit Patel said the estimates assume health care costs will rise 4 to 5 percent a year. That's higher than the 2 percent average from 2010 to 2015, but lower than the 6 to 8 percent that preceded the recent slowdown in costs, he said.
"The truth is no one knows with certainty what will happen over 22 years," Patel said, although the estimates help people plan.
Fidelity assumes a person will retire at 65 and live to the average retirement age calculated by actuaries. That's 85 for a man and 87 for a woman. Because women tend to live longer than men, women should assume higher total health care costs over their retirement years. Consequently, a single woman might expect to pay $135,000 for health care during retirement and a man $125,000, according to Fidelity.
Longevity is one reason why women must be especially attentive to savings and calculating retirement needs. They can face shortages of savings late in life after devoting large amounts to care for ill husbands.
One major cost that is not woven into Fidelity's health cost estimate is long-term care insurance, which helps elderly people pay for nursing assistants when they can't walk, are recovering from surgery or going through treatment for a long illness such as cancer. Currently, a daily cost of $250 is common for such care, possibly totaling about $90,000 for a year. Medicare doesn't pay any of it.
Fidelity estimated that a couple would need an additional $130,000 to pay for a long-term care insurance policy that would provide $8,000 a month during three years of care. The estimate assumes inflation adjustments in monthly benefits.
"Not everyone needs long-term care insurance," Patel said. It depends on variables such as the amount of money you could devote to the care if needed and whether family members could care for you. People who want the insurance, he said, should get it around age 50 because by 65 health conditions such as osteoporosis, diabetes or heart trouble could make people ineligible.
Gail MarksJarvis is a personal finance columnist for the Chicago Tribune and author of "Saving for Retirement Without Living Like a Pauper or Winning the Lottery." Readers may send her email at gmarksjarvis@tribune.com.
Idaho low-income earners have the one of the lowest tax burdens in the United States. See how Idaho compares to other states when it comes to paying taxes.


text:
World of Matouk
Escape To...San Miguel de Allende with Jeffry Weisman
Designer Jeffry Weisman, San Miguel local. Portrait by Meg Messina.
The charm of this Mexican town captured Weisman's heart. Photo by Marcus Jolly.
The accomplished interior designer gives us an insider guide to this colonial-era city in Mexico's central highlands.
Partners in design and life Jeffry Weisman and Andrew Fisher create spaces that The New York Times Sunday Magazine describe as "drop dead, big city chic." Known for their exuberant imagination, elegant aesthetic, and attention to detail, they've created memorable interiors for clients and have been celebrated in the pages of Architectural Digest, Garden & Gun, Veranda, Elle Decor, and House Beautiful.
Both accomplished artists and creative forces in their own right, they joined forces in 2000, founding their eponymous design firm, now with offices in San Francisco and San Miguel de Allende. In addition to their wide range of memorable interiors, they also have lines that include furniture and lighting.
Weisman and Fisher chose to live in San Miguel seven years ago after a trip to the colonial-era city known for its baroque Spanish architecture, thriving arts scene, and cultural festivals. "Andrew and I went on vacation. We had long aspired to have a home in Mexico but had ruled out all the places we had visited previously for one reason or another," says Weisman. "We spent six hours our first day on a walking tour of the 16th century Spanish colonial center of town and by the time we finished, we had committed to start looking at houses that day." (Their gorgeous 18th-century home is even available to stay...more on that below.)
Because San Miguel is a Unesco World Heritage site, most of the historic charm of the place is intact, including the scale of the historic center. Modern conveniences, such as fiber optic cable, have been discreetly hidden below the cobblestone streets. Perched at 6,500 foot elevation, the air is clear and the skies largely blue. The town is small enough to be cozy but large enough to be interesting.
Weisman and Fisher love the gentle pace and the warmth of the community. "The long history of craft was also a major selling point for us, and the reason we are able to produce our lighting and furniture collection there," he says. As a place to visit, San Miguel also ticks the requisite boxes. Safe, clean, and welcoming, Weisman attests that "we don't have a single traffic signal or stop light in the entire town, yet every driver stops at every intersection to let the person to their right go first. You rarely hear a horn honk."
We're ready to start packing. But before we head south of the border, we asked Weisman to tell us the places not to miss for a long weekend or longer. There's even a few doable side-trips.
Read on for the best of San Miguel de Allende.
A Ki'Bok barista at work. Photo Courtesy of Ki'Bok Coffee.
The stunning interior of the 1826 Restaurant in the Rosewood Hotel. Photo courtesy of Rosewood Hotels & Resorts.
Ki'Bok (Diez de Sollano y Dávalos 25, Centro, Zona Centro) "A charming corner café with outrageously good espresso and a legendary avocado and egg breakfast." The Rosewood Hotel (Nemesio Diez 11, Colonia Centro | +52-415-152-9700). "Visit the lovely courtyard and restaurant for breakfast any day of the week and don't miss San Miguel's undisputed champion weekend brunch there."
The Restaurant (Sollano 16, Centro | 213-471-2833) "Lunch in the courtyard of chef Donnie Masterton's first restaurant in San Miguel, simply named The Restaurant, is perfectly delightful. Nothing beats his shredded brussels sprouts and kale salad followed by seared tuna tacos." "The courtyard café in the Hotel Casa Blanca 7 (Juarez No. 7 Centro | 415-688-1438) which we designed has a wide range of insanely delicious Mediterranean-inspired small plates you can enjoy beneath towering palms and bowers of flowers." "El Manantial (Barranca 78, Centro, Zona Centro | +52-415-110-0007) is reportedly the oldest bar in centro and offers a fresh take on Mexican fare in a casual setting and is always a treat."
Above: The stunning space and presentation at Jacinto 1930. Photos Courtesy of Jacinto 1930.
Jacinto 1930 (Relox 18, Centro | +52-415-150-0075) "Located in the core of the Doce 18 Concept store, Jacinto is an architectural gem with exquisite, modern Mexican cuisine. Don't miss the mushroom tacos or the esquites (corn stew)." Bovine (Canal 16, Centro, Zona Centro | +52-415-121-6787) "A stylish room with perfect martinis, great steaks and seafood, and top notch service." Trazo 1810 (Hidalgo 8, Centro, Zona Centro | +52-415-121-3506) "This rooftop restaurant tucked behind a gallery of tourist art is well worth investigating. The food is consistently inventive and delicious, with a great view."
The beautiful view at night at the Rosewood Hotel's Luna Rooftop Tapas Bar. Photo courtesy of Rosewood Hotels & Resorts.
Atrio (Cuna de Allende 3, Centro, Zona Centro | +52-415-688-1405) "This rooftop bar and restaurant is located just off the jardín (our main square), perched just below the dazzling tower of the parroquia (the main church of San Miguel). You can literally watch the bell ringers pull the heavy sisal ropes that tip the bells every 15 minutes." Luna Bar (Nemesio Diez 11, Colonia Centro | +52-415-152-9700) "The rooftop bar at the Rosewood Hotel is gracious and overlooks the whole of the historic center." Hotel Casa Blanca 7 (Juarez No. 7 Centro | 415-688-1438) "The courtyard I recommended for lunch is also one of my favorite spots for drinks."
High ceilings and a sleek interior makes for an exceptional space at Recreo San Miguel. Photo by Marcus Jolly.
Another shot of the rustic meets refined interior of Recreo San Miguel. Photo by Marcus Jolly.
Recreo San Miguel (Recreo 26, Centro, Zona Centro | (+52-415-154-4820) "Incredibly chic serapes and other casual outerwear for men and women that translate perfectly to almost anywhere." Wabi (Aldama 17, Centro, Zona Centro | +52-415-152-1940) "A minimalist's paradise, Ted Wolter has filled his stunning home store with incredibly chic offerings worth shipping home." La Calaca (Mesones 93, Centro | +52-415-152-3954 ) "Evita Avery's gem of a shop is the best place to find antique textiles and ceramics in San Miguel." Doce 18 Concept Store (Relox 18, Centro, Zona Centro | +52-415-154-9201) "Sally and Roy Azar converted an abandoned foundry into the most chic shopping and dining spot." Camino Silvestre (Zacateros 46, Centro, Zona Centro | +52-415-121-3359) "From their roots as hummingbird feeder makers, Jim McKeever and Alfredo Garcia-Lucio have deftly expanded into fabulous gifts and home furnishings." Casa Midy (Pila Seca 3 | +52-415-152-0403) "The showroom of the iconic San Miguel brand created by Anne Marie Midy and Jorge Almada." Mixta (Pila Seca 3, Zona Centro | +52-415-152-7343) "Aussie expat Anna Sinclair has a fresh take on jewelry, gifts, and clothing." CORE (Pila Seca 8 Colonia Centro | +52 415 1525 017) "This uber-stylish home décor store from Mexico City has everything from massive dining tables of petrified wood to the best faucets available."
Above: The vast selection of product available to shop at Sollano 16, designed by Weisman and Fisher.
Sollano 16 (16 Calle, Diez de Sollano y Dávalos, Zona Centro | +52-415-154-8872) "We designed this shop and sell our collection here alongside well curated jewelry, fashion accessories and housewares from around the world. I can't say I'm unbiased, but I love this store."
Ignacio Allende House (Cuna de Allende 1, Centro, Zona Centro | +52-415-152-2499) "A house-museum on a corner of the jardín. This is the house where Ignacio Allende—the nobleman from San Miguel who led the revolution to free Mexico from Spanish rule—was born. The contents are not the focus; the house is. A short visit gives you a view of how the rich landowners lived in the 18th century." Casa Canal (Canal 3, Centro, Zona Centro | +52-415-152-0479) "The grandest house on the jardín, Casa Canal is now the headquarters for Banamex (part of Citibank). The courtyard and stables beyond are open to the public and are exquisite. The small state-owned shop on the front of the house offers first-class crafts and great art books." The Bellas Artes (Calle del Doctor Ignacio Hernandez Macias 75, Zona Centro | +52-415-152-0289) "This former convent has been reborn as an arts center, with galleries and spaces for art and dance classes. The galleries and the room at the back with partially-completed murals by Siqueiros are well worth a visit."
Jeffry's Mini Escapes
El Santuario de Atotonilco "A Unesco world heritage site, this jewel of a church is well worth the 20 minute cab ride. Just ask the cab to wait while you visit. This is a DON'T MISS. Try to get into the chapel on the left as you enter; it's not always open, and a ticket is required." Guanajuato and Jaral de Berrios "Guanajuato is the state capital and a grand university town. The opera house is spectacular. On the way home, take a detour to see the Jaral de Berrios, a spectacular hacienda in a state of magical decay."
The stunning exterior of Weisman and Fisher's home, Casa Acanto. Photo by Grey Crawford.
Sunset views at Casa Acanto. Photo by Grey Crawford.
Lush gardens and lovely space are aplenty at Casa Acanto. Photo by Jeffry Weisman.
L'Otel (De Los Chiquitos 1A, Centro, Zona Centro | +52-415-154-9850) "The super chic hotel above Doce 18. Access to the rooftop swimming pool lined in black and white marble is part of the package." Hotel Matilda (Aldama 53, Centro, Zona Centro | +52-415-152-1015) "An art-filled, modern hotel with a fun restaurant and pool. Dos Casas (Quebrada 101, Centro, Zona Centro | +52-415-154-4073) "A super chic small hotel. Premier San Miguel (Aldama 10-B Zona Centro | 646 536 7634) "The best house rental service operated by Katharine Hibberts. If you're staying for more than a few days, renting a house is the way to go. You can even rent our 18th-century home, Casa Acanto, through Katharine." The Rosewood "For first world comfort in large suites that are a short walk from the historic center. I love the gym too. Lastly, Hotel Casa Blanca 7 how can I not offer a plug for the one hotel we designed?"
"Relax at either of the two killer spas in town... Sense at the The Rosewood and the spa at Hotel Matilda. Both deserve the better part of an afternoon."
Spectacular views at El Charco del Ingenio. Photos provided by El Charco del Ingenio.
MY PERFECT DAY IN SAN MIGUEL…
"It would start with an espresso at Ki'Bok followed by a walk to the top of the hill the city is perched on. From there, I head to Barbacoa Rodriguez for sheep tacos, a local breakfast tradition. The Rodriguez family roasts sheep in pits carved into the earth and lined with agave leaves, and the braised meat is delicate and divine in freshly pressed corn tortillas. From there I would walk to the Charco del Ingenio, our botanical garden. After enjoying a walk through the Charco I would head home to shower and read The New York Times in our garden. Later I would stroll over to The Restaurant or Hotel Casa Blanca 7 for a light lunch in the courtyard. Then some food shopping and cooking before having a bunch of friends and visitors over for cocktails and dinner at home."
In the Bedroom...Shazalynn Cavin-Winfrey
Step inside a traditional boy's bedroom with a bit of modern flair, and learn how designer Shazalynn Cavin-Winfrey achieves personalized spaces for...
In the Bedroom...James Farmer
Discover the techniques designer James Farmer used to create this cozy, detail-oriented master bedroom that channels Southern style.
Art of Bedding
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Don't overlook the fact that this is less than a 3,000 word article. The content couldn't have been any lighter. I got no new or usable information for a research paper I'm writing. I was hoping to fill in the period of time when Hoover was the Secretary of Commerce. That section was glossed over with a sentence. The whole text would not have filled up five full size pages. Save your three dollars... it isn't worth a dime.
Herbert Hoover was a great humanitarian; he fed more than a quarter of the Europe's population after World War I. Even FDR thought he would make a great president. Unfortunately Hoover's timing couldn't have been worse; the Great Depression hit during his administration. President Hoover, like the proverbial Dutch boy, plugged the whole in the economic dam with his finger. The result was foretold, but Hoover was not the cause of the Depression and, given the economic and financial systems then in place, could not have averted it. It is highly doubtful anyone, even FDR, would have known or understood its magnitude and taken action at the onset of the Depression to avert it. Author Daniel Alef does an excellent and unbiased job of telling Hoover's story. Kudos to Alef.
Formatted for Kindle. Linked Contents.