#  Dimensionality reduction of many-body problem using coupled-cluster sub-
system flow equations: classical and quantum computing perspective

Karol Kowalski<EMAIL_ADDRESS>Physical Sciences Division, Pacific
Northwest National Laboratory, Richland, WA 99354, USA

(August 28, 2024)

###### Abstract

We discuss reduced-scaling strategies employing recently introduced sub-system
embedding sub-algebras coupled-cluster formalism (SES-CC) to describe quantum
many-body systems. These strategies utilize properties of the SES-CC
formulations where the equations describing certain classes of sub-systems can
be integrated into a computational flows composed of coupled eigenvalue
problems of reduced dimensionality. Additionally, these flows can be
determined at the level of the CC Ansatz by the inclusion of selected classes
of cluster amplitudes, which define the wave function ”memory” of possible
partitionings of the many-body system into constituent sub-systems. One of the
possible ways of solving these coupled problems is through implementing
procedures, where the information is passed between the sub-systems in a self-
consistent manner. As a special case, we consider local flow formulations
where the local character of correlation effects can be closely related to
properties of sub-system embedding sub-algebras employing localized molecular
basis. We also generalize flow equations to the time domain and to downfolding
methods utilizing double exponential unitary CC Ansatz (DUCC), where reduced
dimensionality of constituent sub-problems offer a possibility of efficient
utilization of limited quantum resources in modeling realistic systems.

##  I  Introduction

Over the last few decades, the coupled-cluster (CC) theory Coester (1958);
Coester and Kummel (1960); Čížek (1966); Paldus, Čížek, and Shavitt (1972);
Purvis and Bartlett (1982); Koch and Jørgensen (1990); Paldus and Li (1999);
Crawford and Schaefer (2000); Bartlett and Musiał (2007) has evolved into one
of the most accurate and dominant theory to describe various quantum systems
across spatial scales hence addressing fundamental problems in many-body
physics Arponen (1983); Arponen, Bishop, and Pajanne (1987a, b); Arponen and
Bishop (1991a, 1993a, 1993a); Robinson, Bishop, and Arponen (1989); Arponen
and Bishop (1991b); Bishop and Lührmann (1978, 1982); Arponen  _et al._
(1988a); Emrich and Zabolitzky (1984) (for an excellent review of these
developments see Ref.Bishop (1991)), quantum field theory,Funke, Kaulfuss, and
Kümmel (1987); Kümmel (2001); Hasberg and Kümmel (1986); Bishop, Ligterink,
and Walet (2006); Ligterink, Walet, and Bishop (1998) quantum
hydrodynamics,Arponen  _et al._ (1988b); Bishop  _et al._ (1989) nuclear
structure theory,Dean and Hjorth-Jensen (2004); Kowalski  _et al._ (2004);
Hagen  _et al._ (2008) quantum chemistry,Scheiner  _et al._ (1987); Sinnokrot,
Valeev, and Sherrill (2002); Slipchenko and Krylov (2002); Tajti  _et al._
(2004); Crawford (2006); Parkhill, Lawler, and Head-Gordon (2009); Riplinger
and Neese (2013); Yuwono, Magoulas, and Piecuch (2020) and material
sciences.Stoll (1992); Hirata  _et al._ (2004); Katagiri (2005); Booth  _et
al._ (2013); Degroote  _et al._ (2016); McClain  _et al._ (2017); Wang and
Berkelbach (2020); Haugland  _et al._ (2020) Many strengths of the single-
reference CC formalism (SR-CC) or coupled-cluster methods(CCM) originates in
the exponential parametrization of the ground-state wave function
$|\Psi\rangle$

$|\Psi\rangle=e^{T}|\Phi\rangle\;,$ (1)

where $T$ and $|\Phi\rangle$ correspond to cluster operator and reference
function. For example, one can define a hierarchy of CC approximations by
increasing the rank of excitations included the cluster operator. Another
important feature of CC stems from the linked cluster theorem Brandow (1967a);
Lindgren and Morrison (2012) which allows one to build efficient non-iterative
algorithms for higher-order excitations. When both approximation techniques
are combined, one can define efficient and accurate methodologies that can
deliver a high-level of accuracy in chemical simulations. Raghavachari  _et
al._ (1989); Kucharski and Bartlett (1998a, b); Crawford and Stanton (1998);
Gwaltney and Head-Gordon (2000); Gwaltney  _et al._ (2000); Gwaltney and Head-
Gordon (2001); Hirata  _et al._ (2001); Bomble  _et al._ (2005); Robinson and
Knowles (2013); Bozkaya and Schaefer III (2012); Kallay and Gauss (2008);
Meissner and Bartlett (2001); Kowalski and Piecuch (2000); Piecuch, Gour, and
Włoch (2009); Bauman, Shen, and Piecuch (2017) More recently, CC methodologies
have been integrated with stochastic Monte Carlo methods probing
configurational space and leading to near full-configuration-interaction
accuracy of calculated energies. Thom (2010); Deustua, Shen, and Piecuch
(2017); Yuwono, Magoulas, and Piecuch (2020) However, the applicability of
canonical formulations of these theories (especially to large molecular
systems) may be limited by their steep (polynomial) numerical scaling.
Unfortunately, even with rapid progress in computational technologies,
problems with data locality, data movement, and polynomial scaling of high-
rank canonical CC methods lead to insurmountable numerical problems in
modeling large systems. Although impressive progress has been achieved in the
development of local approaches for CC pair theories, Hampel and Werner
(1996); Schütz (2000); Schütz and Werner (2000, 2001); Neese, Wennmohs, and
Hansen (2009); Neese, Hansen, and Liakos (2009); Riplinger  _et al._ (2013,
2016); Pavosevic  _et al._ (2016, 2017); Saitow  _et al._ (2017); Guo  _et
al._ (2018) extension of these methods to include higher-rank excitations may
still require a significant theoretical effort. Some of these problems may be
addressed by using the mature form of quantum computers, however, due to the
limited size of existing quantum registers this can only be achieved by
developing flexible algorithms to reduce the dimensionality of quantum
problem. These problems have been scrutinized only recently, including local
and reduced-dimensionality quantum computing formulations. Bauman  _et al._
(2019); Kowalski and Bauman (2020); Bauman, Low, and Kowalski (2019); Metcalf
_et al._ (2020); Bauman  _et al._ (2020); Takeshita  _et al._ (2020); Motta
_et al._ (2020); McArdle and Tew (2020); Şahinoğlu and Somma (2020)

In the light of the above discussion, new high-accuracy CC-based techniques
for re-representing quantum many-body problem in reduced-dimensionality spaces
are in high demand. Especially interesting are approaches where the original
high-dimensionality problem can be recast in the form of coupled low-
dimensionality problems. Also, for quantum computing algorithms, the
dimensions of sub-problems coupled into a flow should be tunable to the
available quantum computing (QC) resources to provide, by controlling the
number of parameters processed at a given time, optimal utilization of
computational tools such as Variational Quantum Eigensolvers (VQE). Peruzzo
_et al._ (2014); McClean  _et al._ (2016); Romero  _et al._ (2018); Shen  _et
al._ (2017); Kandala  _et al._ (2017, 2019); Colless  _et al._ (2018); Huggins
_et al._ (2020); Cao  _et al._ (2019); Ryabinkin  _et al._ (2020)
Additionally, recent strides made in the development of unitary CC
formulations Hoffmann and Simons (1988); Bartlett, Kucharski, and Noga (1989);
Taube and Bartlett (2006); Harsha, Shiozaki, and Scuseria (2018); Lee  _et
al._ (2018); Izmaylov, Díaz-Tinoco, and Lang (2020); Grimsley  _et al._
(2019a) such as their disentangled,Evangelista, Chan, and Scuseria (2019)
adaptive VQE variants,Grimsley  _et al._ (2019b) and qubit representations of
unitary CC methods and exact quadratic truncation of the
Baker–Campbell–Hausdorff expansion Ryabinkin  _et al._ (2018); Lang,
Ryabinkin, and Izmaylov (2020) provide tools not only for next-generation VQE-
type solvers but also for unlocking properties of unitary CC formulations
needed in the analysis of reduced-dimensionality methods Bauman  _et al._
(2019); Kowalski and Bauman (2020).

In this paper, we will focus on the further extension of recently introduced
CC sub-system embedding sub-algebras CC (SES-CC) Kowalski (2018) and double
unitary CC downfolding methods (DUCC).Bauman  _et al._ (2019); Kowalski and
Bauman (2020) In a natural way, these methods allow to calculate ground-state
energies as eigenvalues of effective Hamiltonians in pre-defined active spaces
describing sub-systems of the whole quantum system. Since in the construction
of effective Hamiltonians all out-of-active-space correlation effects are
integrated out, the CC downfolding procedures can be viewed as a natural
renormalization techniques. The flow equations for single-reference SES-CC
case,Kowalski (2018) utilize this property for each active space involved in
the flow and for this reason can be considered as formulations that have
”memory” of each sub-system involved in the flow. Specifically, each sub-
system can be characterized by the corresponding effective Hamiltonian that
includes interactions with other sub-systems. We will illustrate the ability
of these approaches to capture complicated correlation effect and dynamics of
the system, through traversing large sub-spaces of the entire Hilbert space
without an unnecessary increase of the size of the numerical problem to be
solved at a given time in the flow algorithm. We will also show that it is
possible to define flows that decouple the representation of the Schrödinger
equation in large sub-spaces of the Hilbert (often defined by net dimensions
beyond classical/quantum computing capabilities) into smaller problems that
are numerically tractable.

This paper discusses extensions of the CC flow formalism that have not been
analyzed in earlier articles on this subject. Specifically, it includes:

  * •

An explicit proof of the fact that the arbitrary CC flow equations are
equivalent to standard CC equations with a specific form of cluster operator.

  * •

The extension of the CC flow algorithms to the time domain. In analogy to the
previous point, we also show that time-dependent CC flow equations are
equivalent to the time-dependent CC equations with a specific form of the
cluster operator.

  * •

The extension of the CC flow equations to the localized basis set. In this
case, we demonstrate that the CC flow equations provide a rigorous definition
(at the level of effective Hamiltonians) of the so-called electron pair. This
type of CC flow in a natural way defines density matrices and higher-rank
excitations for local CC formulations. The introduction of higher-rank
excitations is a well-known problem faced by local formulations of CC methods.

  * •

The extension of CC flows to double unitary CC (DUCC) formulations. We discuss
these flows from the point of view of quantum computing applications, where
DUCC flows can be used to probe configurational spaces of the dimensions
beyond dimensions treated by existing quantum algorithms. Additionally, DUCC
flow for localized orbitals naturally addresses Hamiltonian qubit encoding and
controlling anti-symmetry of the corresponding wave function.

Since the operator algebra involved in the unitary CC methods is non-
commutative, extending canonical SES-CC flows to the DUCC case requires the
utilization of certain approximations. To this end, we will consider backward-
type methods based on the use of approximate Trotter formulas.

We will also discuss the difference between two computational strategies
involving (1) standard approximations based on the selection of cluster
amplitudes and treating them simultaneously (or globally) in numerical
implementations and (2) flow equations where only a portion of selected
amplitudes are processed at the time. While the former computing approach can
take advantage of parallel classical architectures, the latter is ideally
suited for Noisy Intermediate-Scale Quantum devices (NISQ), where a small
subset of fermionic degrees of freedom can be effectively handled. The flow
equation methods also provide a conceptual foundation for introducing certain
approximations classes and eliminating possible problems with their
postulatory character. We will illustrate these advantages on the example of
local CC methods. For simplicity, in this paper we will focus on the CC and
DUCC flow equations for closed-shell systems.

##  II Sub-system embedding sub-algebras CC formalism - stationary and time-
dependent formulations

The SES CC formalism is based on the observation that energy of CC
formulations, $E_{\rm CC}$, in addition to the well know formula
$\langle\Phi|e^{-T}He^{T}|\Phi\rangle$ (where $H$ represents many-body
Hamiltonian), can be obtained through diagonalization of the whole family of
the effective Hamiltonians.Kowalski (2018)

First, let us discuss the basic tenets of the SES-CC formalism. In the exact
wave function limit, the maximum excitation level $m$ included in the cluster
operator $T$ is equal to the number of correlated electrons ($N$) while in the
approximate formulations $m\ll N$. Several typical examples are CCSD
($m=2$),Purvis and Bartlett (1982) CCSDT ($m=3$), Noga and Bartlett (1987,
1988); Scuseria and Schaefer (1988), and CCSDTQ ($m=4$),Kucharski and Bartlett
(1991); Oliphant and Adamowicz (1991) methods. Using the second quantization
language, the $T_{k}$ components of cluster operator producing $k$-tuply
excitations when acting on the reference function can be expressed as

$T_{k}=\frac{1}{(k!)^{2}}\sum_{i_{1},\ldots,i_{k};a_{1}\ldots
a_{k}}t^{i_{1}\ldots i_{k}}_{a_{1}\ldots a_{k}}E^{a_{1}\ldots
a_{k}}_{i_{1}\ldots i_{k}}\;,$ (2)

where indices $i_{1},i_{2},\ldots$ ($a_{1},a_{2},\ldots$) refer to occupied
(unoccupied) spin orbitals in the reference function $|\Phi\rangle$. The
excitation operators $E^{a_{1}\ldots a_{k}}_{i_{1}\ldots i_{k}}$ are defined
through strings of standard creation ($a_{p}^{\dagger}$) and annihilation
($a_{p}$) operators

$E^{a_{1}\ldots a_{k}}_{i_{1}\ldots i_{k}}=a_{a_{1}}^{\dagger}\ldots
a_{a_{k}}^{\dagger}a_{i_{k}}\ldots a_{i_{1}}\;,$ (3)

where creation and annihilation operators satisfy the following anti-
commutation rules

$[a_{p},a_{q}]_{+}=[a_{p}^{\dagger},a_{q}^{\dagger}]_{+}=0\;,$ (4)
$[a_{p},a_{q}^{\dagger}]_{+}=\delta_{pq}\;.$ (5)

The SES-CC approach is based on the particle-hole (p-h) formalism defined with
respect to the reference function $|\Phi\rangle$, where quasi-operators
$b_{p}$ and $b_{p}^{\dagger}$ are defined as

$b_{p}=\begin{cases}a_{p}&{\rm if}\;\;p\in V\\\ a_{p}^{\dagger}&{\rm
if}\;\;p\in O\end{cases}$ (6)

and

$b_{p}^{\dagger}=\begin{cases}a_{p}^{\dagger}&{\rm if}\;\;p\in V\\\ a_{p}&{\rm
if}\;\;p\in O\;,\end{cases}$ (7)

where $O$ and $V$ designate sets of occupied and unoccupied spin orbitals.
Using the p-h formalism we have

$b_{p}|\Phi\rangle=0\;,$ (8)

and

$E^{a_{1}\ldots a_{k}}_{i_{1}\ldots i_{k}}=b_{a_{1}}^{\dagger}\ldots
b_{a_{k}}^{\dagger}b_{i_{k}}^{\dagger}\ldots b_{i_{1}}^{\dagger}\;.$ (9)

Additionally, the $b_{p}/b_{q}^{\dagger}$ operators satisfy the same anti-
commutation relations as $a_{p}/a_{q}^{\dagger}$ operators, i.e.,

$[b_{p},b_{q}]_{+}=[b_{p}^{\dagger},b_{q}^{\dagger}]_{+}=0\;,$ (10)
$[b_{p},b_{q}^{\dagger}]_{+}=\delta_{pq}\;.$ (11)

The p-h formalism significantly simplifies the analysis of the CC equations.
It is also easy to notice that all excitation operators (3) commute, i.e. for

$\displaystyle E^{a_{1}\ldots a_{k}}_{i_{1}\ldots i_{k}}$ $\displaystyle=$
$\displaystyle a_{a_{1}}^{\dagger}\ldots a_{a_{k}}^{\dagger}a_{i_{k}}\ldots
a_{i_{1}}=b_{a_{1}}^{\dagger}\ldots
b_{a_{k}}^{\dagger}b_{i_{k}}^{\dagger}\ldots b_{i_{1}}^{\dagger},$ (12)
$\displaystyle E^{c_{1}\ldots c_{m}}_{j_{1}\ldots j_{m}}$ $\displaystyle=$
$\displaystyle a_{c_{1}}^{\dagger}\ldots a_{c_{m}}^{\dagger}a_{j_{m}}\ldots
a_{j_{1}}=b_{c_{1}}^{\dagger}\ldots
b_{c_{m}}^{\dagger}b_{j_{m}}^{\dagger}\ldots b_{j_{1}}^{\dagger},$ (13)

we have

$[E^{a_{1}\ldots a_{k}}_{i_{1}\ldots i_{k}},E^{c_{1}\ldots c_{m}}_{j_{1}\ldots
j_{m}}]=0\;.$ (14)

After substituting Ansatz (1) into the Schrödinger equation, one gets the
energy-dependent form of the CC equations:

$(P+Q)He^{T}|\Phi\rangle=E(P+Q)e^{T}|\Phi\rangle\;\;,$ (15)

where $P$ and $Q$ are projection operators onto the reference function
($P=|\Phi\rangle\langle\Phi|$) and onto excited configurations (with respect
to $|\Phi\rangle$) generated by the $T$ operator when acting onto the
reference function,

$Q=\sum_{k=1}^{m}\;\sum_{i_{1}<i_{2}<\ldots<i_{k};a_{1}<a_{2}\ldots<a_{k}}|\Phi_{i_{1}\ldots
i_{k}}^{a_{1}\ldots a_{k}}\rangle\langle\Phi_{i_{1}\ldots i_{k}}^{a_{1}\ldots
a_{k}}|\;,$ (16)

where

$|\Phi_{i_{1}\ldots i_{k}}^{a_{1}\ldots a_{k}}\rangle=E^{a_{1}\ldots
a_{k}}_{i_{1}\ldots i_{k}}|\Phi\rangle\;.$ (17)

Diagrammatic analysis Paldus and Li (1999) leads to an equivalent (at the
solution), energy-independent form of the CC equations for cluster amplitudes

$Qe^{-T}He^{T}|\Phi\rangle=Q(He^{T})_{C}|\Phi\rangle=0\;,$ (18)

and energy expression

$E=\langle\Phi|e^{-T}He^{T}|\Phi\rangle=\langle\Phi|(He^{T})_{C}|\Phi\rangle\;,$
(19)

where $C$ designates a connected part of a given operator expression. In the
forthcoming discussion, we refer to $e^{-T}He^{T}$ as a similarity transformed
Hamiltonian $\bar{H}$.

The SES-CC formalism hinges upon the notion of excitation sub-algebras of
algebra $\mathfrak{g}^{(N)}$ generated by
$E^{a_{l}}_{i_{l}}=b_{a_{l}}^{\dagger}b_{i_{l}}$ operators in the particle-
hole representation defined with respect to the reference $|\Phi\rangle$. As a
consequence all generators commute, i.e.,
$[E^{a_{l}}_{i_{l}},E^{a_{k}}_{i_{k}}]=0$ and algebra $\mathfrak{g}^{(N)}$
(along with all sub-algebras considered here) is commutative. The SES-CC
formalism utilizes an important class of sub-algebras of commutative
$\mathfrak{g}^{(N)}$ algebra, which contain all possible excitations
$E^{a_{1}\ldots a_{m}}_{i_{1}\ldots i_{m}}$ that excite electrons from a
subset of active occupied orbitals (denoted as $R$) to a subset of active
virtual orbitals (denoted as $S$). These sub-algebras will be designated as
$\mathfrak{g}^{(N)}(R,S)$. In the following discussion, we will use $R$ and
$S$ notation for subsets of occupied and virtual active orbitals
$\\{R_{i},\;i=1,\ldots,x\\}$ and $\\{S_{i},\;i=1,\ldots,y\\}$, respectively
(sometimes it is convenient to use alternative notation
$\mathfrak{g}^{(N)}(x_{R},y_{S})$ where numbers of active orbitals in $R$ and
$S$ orbital sets, $x$ and $y$, respectively, are explicitly called out). Of
special interest in building various approximations are sub-algebras that
include all $n_{v}$ virtual orbitals ($y=n_{v}$) - these sub-algebras will be
denoted as $\mathfrak{g}^{(N)}(x_{R})$. As discussed in Ref.Kowalski (2018)
configurations generated by elements of $\mathfrak{g}^{(N)}(x_{R},y_{S})$
along with the reference function span the complete active space (CAS)
referenced to as the CAS($R,S$) (or eqivalently
CAS($\mathfrak{g}^{(N)}(x_{R},y_{S})$)).

Each sub-algebra $\mathfrak{h}=\mathfrak{g}^{(N)}(x_{R},y_{S})$ induces
partitioning of the cluster operator $T$ into internal ($T_{\rm
int}(\mathfrak{h}$) or $T_{\rm int}$ for short) part belonging to
$\mathfrak{h}$ and external ($T_{\rm ext}(\mathfrak{h}$) or $T_{\rm ext}$ for
short) part not belonging to $\mathfrak{h}$, i.e.,

$T=T_{\rm int}(\mathfrak{h})+T_{\rm ext}(\mathfrak{h})\;.$ (20)

In Ref.Kowalski (2018), it was shown that if the two following criteria are
met: (1) the $|\Psi(\mathfrak{h})\rangle=e^{T_{\rm
int}(\mathfrak{h})}|\Phi\rangle$ is characterized by the same symmetry
properties as $|\Psi\rangle$ and $|\Phi\rangle$ vectors (for example, spin and
spatial symmetries), and (2) the $e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle$
Ansatz generates FCI expansion for the sub-system defined by the CAS
corresponding to the $\mathfrak{h}$ sub-algebra, then $\mathfrak{h}$ is called
a sub-system embedding sub-algebra (SES) for cluster operator $T$. For any SES
$\mathfrak{h}$ we proved the equivalence of two representations of the CC
equations at the solution: (i) standard

$\displaystyle\langle\Phi|\bar{H}|\Phi\rangle$ $\displaystyle=$ $\displaystyle
E\;,$ (21) $\displaystyle Q_{\rm int}\bar{H}\Phi\rangle$ $\displaystyle=$
$\displaystyle 0\;,$ (22) $\displaystyle Q_{\rm ext}\bar{H}|\Phi\rangle$
$\displaystyle=$ $\displaystyle 0\;,$ (23)

and (ii) hybrid

$\displaystyle(P+Q_{\rm int})\bar{H}_{\rm ext}e^{T_{\rm int}}|\Phi\rangle$
$\displaystyle=$ $\displaystyle E(P+Q_{\rm int})e^{T_{\rm
int}}|\Phi\rangle\;,$ (24) $\displaystyle Q_{\rm ext}\bar{H}|\Phi\rangle$
$\displaystyle=$ $\displaystyle 0\;,$ (25)

where

$\bar{H}_{\rm ext}=e^{-T_{\rm ext}}He^{T_{\rm ext}}\;,$ (26)

and the two projection operators $Q_{\rm int}(\mathfrak{h})$ and $Q_{\rm
ext}(\mathfrak{h})$ ($Q_{\rm int}$ and $Q_{\rm ext}$ for short) are spanned by
all excited configurations generated by acting with $T_{\rm
int}(\mathfrak{h})$ and $T_{\rm ext}(\mathfrak{h})$ onto reference function
$|\Phi\rangle$, respectively. The $Q_{\rm int}$ and $Q_{\rm ext}$ projections
operators satisfy the condition

$Q=Q_{\rm int}+Q_{\rm ext}\;.$ (27)

The above equivalence shows that the CC energy can be calculated by
diagonalizing non-Hermitian effective Hamiltonian $H^{\rm eff}$ defined as

$H^{\rm eff}=(P+Q_{\rm int})\bar{H}_{\rm ext}(P+Q_{\rm int})\;$ (28)

in the complete active space corresponding to any SES of CC formulation
defined by cluster operator $T$, i.e.,

$H^{\rm eff}(\mathfrak{h})e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;,\forall_{{\rm SES}\;\mathfrak{h}}\;.$ (29)

Although the idea of effective Hamiltonians has been intensively explored in
the past in many areas of physics and chemistry (see Refs. Bloch (1958); Des
Cloizeaux (1960); Löwdin (1963); Schrieffer and Wolff (1966); Brandow (1967b);
Soliverez (1969); Schucan and Weidenmüller (1972); Jørgensen (1975);
Mukherjee, Moitra, and Mukhopadhyay (1975); Jeziorski and Monkhorst (1981);
Kutzelnigg (1982); Stolarczyk and Monkhorst (1985); Mukherjee (1986); Durand
(1983); Durand and Malrieu (1987); Jeziorski and Paldus (1989); Kaldor (1991);
Rittby and Bartlett (1991); Andersson  _et al._ (1990); Andersson, Malmqvist,
and Roos (1992); Hirao (1992); Finley, Chaudhuri, and Freed (1995); Głazek and
Wilson (1993); Meissner and Nooijen (1995); Meissner (1998); Nakano, Hirao,
and Gordon (1998); Angeli, Cimiraglia, and Malrieu (2001); Angeli  _et al._
(2001); Yanai and Chan (2006); Lyakh  _et al._ (2012); Bravyi, DiVincenzo, and
Loss (2011); Sahinoglu and Somma (2020)) the SES-CC formalism enables on to
build effective Hamiltonians using single-reference formulations. Moreover, it
is an inherent feature of the single reference CC Ansatz, which unlike the
multi-reference formulations does not assume that the wave operator act on the
multi-dimensional model space and where the corresponding effective
Hamiltonian is diagonalized (as an example see the Bloch wave operator
formalism). We also believe that Eq.(29) may be an interesting contribution
from the point of view of recently explored non-Hermitian extensions of
quantum mechanics. Bender and Boettcher (1998); Bender, Brody, and Jones
(2002); Mostafazadeh (2002); Znojil  _et al._ (2009); Bishop and Znojil (2013,
2020)

In contrast to the energy-dependent representation of CC equation (15),
Eqs.(29) represent true eigenvalue problems corresponding to the same
eigenvalue $E$ and $e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle$ as
eigenvectors. One should also notice that: (1) the non-CAS related CC wave
function components (referred here as external degrees of freedom) are
integrated out and encapsulated in the form of $H^{\rm eff}$, and (2) the
internal part of the wave function, $e^{T_{\rm int}}|\Phi\rangle$ is fully
determined by diagonalization of $H^{\rm eff}$ in the corresponding CAS.
Separation of external degrees of freedom in the effective Hamiltonians is a
desired feature, especially for building its reduced-dimensionality
representation for quantum computing (QC). However, a factor that impedes the
use of SES-CC effective Hamiltonians in quantum computing is their non-
Hermitian character. It is also worth mentioning that various CC
approximations are characterized by various SESs, which is a unique
fingerprint of each standard CC approximaiton. For example, for the restricted
Hartree-Fock (RHF) CC formulations the $\mathfrak{g}^{(N)}(1_{R},y_{S})$ and
$\mathfrak{g}^{(N)}(2_{R},y_{S})$ are SESs for CCSD and CCSDTQ approximations
(one should also notice that SES for lower-rank CC approximation is also a SES
of higher-rank CC approximations, i.e. $\mathfrak{g}^{(N)}(1_{R},y_{S})$ is
also a SES for the CCSDTQ approach).

Properties of SESs-induced eigenvalue problems (29) can also be utilized to
design new CC approximations based on various amplitude selections processes
and re-casting CC equations in a different form, which offer interesting
advantages, especially in the way how corresponding equations are solved. This
fact can be illustrated on the example of the flow introduced in Ref.Kowalski
(2018) (see also Fig.1). For now (without a loss of generality) we will focus
on special flow involving computational blocks corresponding to
$\mathfrak{g}^{(N)}(2_{R})$ sub-algebras.

Figure 1: Schematic representation of the CC flow. The entire quantum systems
can be probed with various SES-eigenvalue problems (29) schematically
represented here as $B(\mathfrak{h}_{i})$. These computational blocks can be
coupled into the flow, where information is passed between various
computational blocks $B(\mathfrak{h}_{i})$. Subject to the choice of
particular classes of SESs defining the flow, the CC flow can probe/traverse a
large sub-spaces of entire Hilbert space.

While the CCSD equations cannot be represented as a union of equations
corresponding to Eqs.(29) for various CCSD’s SESs
$\mathfrak{g}^{(N)}(1_{R},y_{S})$ (there are no SES in the CCSD case that
would embrace doubly excited amplitudes $t^{ij}_{ab}$ where spinorbitals $i$
and $j$ correspond to distinct orbitals), there exist formalisms which can
probe a significant portion of Hilbert space and are defined by the set of
equations that correspond to a union of non-symmetric eigenvalue problems of
the type (29) for various SESs. For example, the SCSAF-CCSD(2) approach of
Ref.Kowalski (2018) uses cluster operator $T$ defined as

$T\simeq T_{1}+T_{2}+\sum_{I}T_{{\rm
int},3}(\mathfrak{g}^{(N)}(2_{R_{I}}))+\sum_{I}T_{{\rm
int},4}(\mathfrak{g}^{(N)}(2_{R_{I}}))$ (30)

where $T_{1}$ and $T_{2}$ are singly and doubly excited cluster operators and
$T_{{\rm int},3}(\mathfrak{g}^{(N)}(2_{R_{I}}))$ and $T_{{\rm
int},4}(\mathfrak{g}^{(N)}(2_{R_{I}}))$ contain triple and quadruple
excitations corresponding to SES $\mathfrak{g}^{(N)}(2_{R_{I}})$. Summation
over $I$ in (30) runs over all possible SESs $\mathfrak{g}^{(N)}(2_{R})$. It
can be shown (see Appendix A) that in this case, the global set of CC
equations

$Q(e^{-T}He^{T})|\Phi\rangle=0\;,$ (31)

where all equations are processed simultaneously in the iterative process of
finding the solution, can be re-cast (at the CC solution) in the form of
coupled equations (29) of the form

$H^{\rm eff}(\mathfrak{h})e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;,\forall_{\mathfrak{h}=\mathfrak{g}^{(N)}(2_{R_{I}})}\;.$
(32)

Figure 2: Two types of flow CC formulations for
$\mathfrak{g}^{(N)}(2_{R_{I}})$ sub-algebras (panel (a) represents serial
executions; panel (b) corresponds to the parallel processing of computational
blocks (see text for details)).

As shown in Fig.2, the solution process of equations (31) can be organized in
the form of flow where the algebraic form of each computational blocks
$B(\mathfrak{g}^{(N)}(2_{R_{I}}))$ represents eigenvalue problem (29) for sub-
algebra $\mathfrak{g}^{(N)}(2_{R_{I}})$. In panel (a) we represent particular
flow where particular computational blocks $B(\mathfrak{g}^{(N)}(2_{R_{I}}))$
are communicating in serial. For this purpose, we first establish an ordering
of active spaces defined by $\mathfrak{g}^{(N)}(2_{R_{I}})$, in the way that
reflects their importance (for example, the first active space contains the
most important effects related to the sought for electronic state). Then we
define a protocol for passing information between
$B(\mathfrak{g}^{(N)}(2_{R_{I}}))$’s including ”shared” cluster amplitudes
between various blocks. This problem is caused by the fact that two distinct
SESs $\mathfrak{g}^{(N)}(2_{R_{I}})$ and $\mathfrak{g}^{(N)}(2_{R_{J}})$ can
share a single orbital and effectively share all single excitations from this
orbital and double excitations exciting $\alpha$ and $\beta$ electrons from
the shared orbital. This redundancy is very small compared to the total number
of excitations defining distinct sub-algebras $\mathfrak{g}^{(N)}(2_{R_{I}})$
and $\mathfrak{g}^{(N)}(2_{R_{J}})$. For example , there is no overlap with
the largest classes of excitations corresponding to triple and quadruple
excitations. At the solution, these redundancies are irrelevant because the
equations for shared amplitudes are the same irrespective of the SESs
eigenvalue problem (32) they are part of. To control this effect, common pool
of amplitudes obtained in previous $K$ steps of $i$-th iteration (denoted as
$CP(i,K)$) is passed to $K+1$ computational block and external amplitudes
(needed to construct $\mathfrak{g}^{(N)}(2_{R_{K+1}})$ effective Hamiltonian)
as well as shared amplitudes that correspond to excitations in
$\mathfrak{g}^{(N)}(2_{R_{K+1}})$ enter computational block
$B(\mathfrak{g}^{(N)}(2_{R_{K+1}}))$ as known parameters. In this case, the
algebraic form of $B(\mathfrak{g}^{(N)}(2_{R_{K+1}}))$ still takes the form of
eigenvalue problem of smaller size

$\displaystyle(P+Q_{\rm int}^{\rm X})[e^{-T_{\rm int}^{\rm
CP}(\mathfrak{h})}H^{\rm eff}(\mathfrak{h})e^{T_{\rm int}^{\rm
CP}(\mathfrak{h})}]e^{T_{\rm int}^{\rm X}(\mathfrak{h})}|\Phi\rangle$
$\displaystyle=$ $\displaystyle Ee^{T_{\rm int}^{\rm
X}(\mathfrak{h})}|\Phi\rangle\;,\;\;\;\mathfrak{h}=\mathfrak{g}^{(N)}(2_{R_{K+1}})\;,$
$\displaystyle T_{\rm int}(\mathfrak{h})=T_{\rm int}^{\rm
CP}(\mathfrak{h})+T_{\rm int}^{\rm X}(\mathfrak{h})\;,\;\;\;$
$\displaystyle\mathfrak{h}=\mathfrak{g}^{(N)}(2_{R_{K+1}})\;,$ (33)

where $T_{\rm int}^{\rm CP}(\mathfrak{h})$ is a part of $T_{\rm
int}(\mathfrak{h})$ determined by shared amplitudes from the common pool of
amplitudes and $T_{\rm int}^{\rm X}(\mathfrak{h})$ is a part of $T_{\rm
int}(\mathfrak{h})$ that is determined in the $K+1$ computational blocks.
After Eq.(33) is solved , the $CP(i,K)$ is update for the $T_{\rm int}^{\rm
X}(\mathfrak{h})$ amplitudes defining in this way $CP(i,K+1)$ and process is
continued with the $K+2$ block. After $M$ steps, the $CP(i,M)$ is used as a
starting common pool of amplitudes for the $i+1$ iteration. In Eq.(33), the
$Q_{\rm int}^{\rm X}$ operator is the projection operator onto excited
configurations generated by $T_{\rm int}^{\rm X}(\mathfrak{h})$ when acting
onto reference function $|\Phi\rangle$. In panel (b) we see an alternative
”parallel” flow where computational blocks are independent and corresponds to
the original eigenproblems (29) for SESs $\mathfrak{g}^{(N)}(2_{R_{I}})$. This
step is followed by a sync-up of all shared amplitudes by various SESs. More
details on the CC flows can be found in Appendices A,B,C, and D, where we
discuss the equivalence of global representation and coupled computational
blocks involved in the flow, the time domain extension, approximate solvers
for computational blocks, and general algorithmic structure of the practical
CC flow realization.

The CC flow equations (29) or (32) can also be viewed as a configurational (or
more aptly - sub-space) version of the Aufbau principle, which is a
consequence of the fact that each problem corresponding to some SES
$\mathfrak{h}$ provides a rigorous mechanism for extending the sub-space
probed in a flow. In other words, the spaces probed in each SES problem (29)
are additive. This fact is a unique feature, which should be referred to as
the sub-system ”memory” of the CC wave function. Additionally, CC flow assures
size-consistency of the calculated ground-state energies. Similar flows cannot
be easily constructed using configuration interaction type methods or standard
many-body perturbation techniques. We believe that this is yet another
argument in favor of non-perturbative analysis of CC equations.

In general, flow-based CC formulations are very flexible and allows one to use
sub-algebras $\mathfrak{g}^{(N)}(x_{R},y_{S})$ defined by various $x$ and $y$
parameters. This property may be used to introduce selective groups of higher
excitations and tune the cost of flow equations to available computing
resources. Although the numerical implementation of flow-based formalism may
be numerically less efficient than the implementation based on the global
representation, its advantage lies in the fact that computational blocks
contributing to flow are physically interpretable in terms of Schrödinger-type
equations for sub-systems described by relevant SESs. This fact has profound
consequences and allows one to construct more justified (or equivalently less
”postulated”) and better-controlled approximations based on the flow
equations. An interesting illustration of this fact will be CC flow equations
for localized orbitals (see Section II.2).

Summarizing, the flow equations as shown in Fig.3 can be represented in the
form of the ”global” connected CC equations with the cluster operator $T$
defined as a union of unique internal excitations of all sub-systems included
in the flow. The inverse statement is also true: for specific choices of the
excitation domain included in the cluster operator, the CC equations can be
represented in the form of coupled eigenvalue problems corresponding to
various sub-systems embedding algebras. This statement is a foundation for the
reduced-scaling formulations discussed in this paper.

Figure 3: Schematic representation of the equivalence between the ”global”
representation of the CC equations and coupled SES eigenvalue problems for
properly defined cluster operators (see text for details).

###  II.1 Time-dependent CC flows

The extension of the CC methods to the time domain keeps attracting much
attention in various fields of chemistry and physics. Several developments in
this area, including Arponen’s seminal papers on this subject Arponen (1983)
(see also Refs.Arponen, Bishop, and Pajanne (1987a, b); Arponen and Bishop
(1991a, 1993a, 1993b); Kvaal (2012); Pedersen and Kvaal (2019); Kvaal,
Laestadius, and Bodenstein (2020); Sato  _et al._ (2018)) paved the way for
mature applications of these formulations to describe time-dependent physical
and chemical processes.

In Ref.Kowalski and Bauman (2020), we demonstrated that the SES-based
downfolding techniques could also be extended to the time-dependent
Schrödinger equation when all orbitals and the reference functions
$|\Phi\rangle$ are assumed to be time-independent. As in the stationary case,
we will assume a general partitioning of the time-dependent cluster operator
$T(t)$ into its internal ($T_{\rm int}(\mathfrak{h},t)$) and external ($T_{\rm
ext}(\mathfrak{h},t)$) parts, i.e,

$|\Psi(t)\rangle=e^{T_{\rm ext}(\mathfrak{h},t)}e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,\forall\mathfrak{h}\in SES\;.$ (34)

For generality, we also include phase factor $T_{0}(\mathfrak{h},t)$ in the
definition of the $T_{\rm int}(\mathfrak{h},t)$ operator. After substituting
(34) into time-dependent Schrödinger equation and utilizing properties of SES
algebras, we demonstrated that the ket-dynamics of the sub-system wave
function $e^{T_{\rm int}(\mathfrak{h},t)}|\Phi\rangle$ corresponding to
arbitrary SES $\mathfrak{h}$

$i\hbar\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h},t)e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,$ (35)

where

$H^{\rm eff}(\mathfrak{h},t)=(P+Q_{\rm int}(\mathfrak{h}))\bar{H}_{\rm
ext}(\mathfrak{h},t)(P+Q_{\rm int}(\mathfrak{h}))\;$ (36)

and

$\bar{H}_{\rm ext}(\mathfrak{h},t)=e^{-T_{\rm ext}(\mathfrak{h},t)}He^{T_{\rm
ext}(\mathfrak{h},t)}\;.$ (37)

In analogy to the stationary cases, various sub-systems computational blocks
can be integrated into a flow enabling sampling of large sub-spaces of Hilbert
space through a number of coupled reduced-dimensionality problems. For
example, the time-dependent variant of the the SCSAF-CCSD(2) approach uses the
time-dependent cluster operator $T(t)$ in the form

$\displaystyle T(t)$ $\displaystyle\simeq$ $\displaystyle
T_{1}(t)+T_{2}(t)+\sum_{I}T_{{\rm int},3}(\mathfrak{g}^{(N)}(2_{R_{I}}),t)$
(38) $\displaystyle+\sum_{I}T_{{\rm int},4}(\mathfrak{g}^{(N)}(2_{R_{I}}),t))$

which can be equivalently represented as coupled time-evolution problems for
sub-systems

$i\hbar\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h},t)e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,\forall_{\mathfrak{h}=\mathfrak{g}^{(N)}(2_{R_{I}})}\;,$
(39)

(for details see Appendix B). These equations can be solved using similar
flows as shown in Fig.2 with the difference that now the iterative cycles for
converging amplitudes/energy correspond to elementary time steps with
increment corresponding to $\Delta t$. As in the stationary case, the time-
dependent CC flow equations represent an extension of the ”sub-space Aufbau”
principle mentioned earlier, where each SES-problem (39) extends the space
probed in time-dependent CC formalism. In the view of deep analogies between
stationary CC flow equations based on the localized orbitals and local CC
formulations developed in the last few decades in quantum chemistry (see the
next Subsection), the flow described by Eq.(39) can be considered as a
reduced-scaling variant of the time-dependent CC formulations.

###  II.2 CC flows for localized orbital basis and localized sub-systems

The SES-CC flow formalism also systematizes and further extends the notion of
”sub-system” composed of orbital pairs. This problem has intensively been
studied in early non-orthogonal pseudo-natural orbitals based formulations of
CI Meyer (1973), coupled electron pair approximation (CEPA) Meyer and Rosmus
(1975), and their extensions to local CEPA/CC methods based on the local pair
natural orbitals (LPNO) and their domain-based LPNO variant (DLPNO).Neese,
Wennmohs, and Hansen (2009); Neese  _et al._ (2009) To analyze SES-CC
approximations, let us, in the analogy to the DLPNO-CC formulations, assume
that the set of selected orbital pairs (for simplicity, we will focus on the
closed-shell formulations) $\cal{P}$=$\\{(i,j)\\}$ that significantly
contribute to correlation energy is known. These pairs are also employed to
define PNO spaces and corresponding CCSD cluster amplitudes. In the standard
pair-driven DLPNO-CCSD approximation orbital pairs $(i,j)$ (including pairs
where $i=j$, i.e., $(i,i)$) along with $(i,j)$-specific natural virtual
orbitals are used to select PNO space and relevant single and double
excitations. The $(i,j)$-specific density matrix defined as MP2-type density
matrix ${\bf D}^{ij}$ (see Ref.Riplinger and Neese (2013)) is used to
determine virtual PNOs in the way that only natural orbitals characterized by
occupation numbers greater than the user-defined threshold are retained. It
leads to a significant reduction in the size of pair-specific PNOs spaces and
consequently to a significant reduction of the number of pair-specific singly
($t^{i}_{a_{ii}}$) and doubly ($t^{ij}_{a_{ij}b_{ij}}$) excited amplitudes,
where virtual indices $a_{ii}$, $a_{ij}$, and $b_{ij}$ are defined by reduced-
size PNOs. It should be noted that each pair $(i,j)$ introduces its own set of
PNO virtual orbitals, which may not be orthogonal to PNOs corresponding to
distinct pairs.

The CC $\mathfrak{g}^{(N)}(2_{R})$ flow employing localized occupied orbitals
in a natural way introduces several elements underlying standard DLPNO-CCSD
design. Our analysis is based on the observation that there is a a natural
correspondence between orbital pairs from $\cal{P}$ (along with all virtual
orbitals) with $\mathfrak{g}^{(N)}(2_{R})$ sub-algebras, where

$R\equiv(i,j)\;,\;\;(i,j)\in\cal{P}\;.$ (40)

For short we will refer to these SESs as $\mathfrak{g}^{(N)}(2_{ij})$
($(i,j)\in\cal{P}$). Additionally, each Schrödinger-type equation in (29) or
(32)

$H^{\rm eff}(\mathfrak{h})e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;,\mathfrak{h}=\mathfrak{g}^{(N)}(2_{ij})\;\;(i,j)\in\cal{P}\;.$
(41)

naturally defines corresponding one-body density matrix
$\rho(\mathfrak{g}^{(N)}(2_{ij}))$ and its PNOs (or DLPNOs) without any
additional assumptions regarding the form and the origin of the pair density
matrix. One can readily notice that the ${\bf D}^{ij}$ density matrices used
in original DLPNO-CC papers is its low-order approximation. This feature of
SES equations (41) can be viewed as a CC-derived systematization of the sub-
system (or pair) concepts discussed in original works of Sinanoğlu Sinanoğlu
(1962); Sinanoglu (1964) and Meyer Meyer (1971, 1973); Meyer and Rosmus
(1975).

It is worth mentioning that in contrast to the DLPNO-CCSD formalism, the
$\mathfrak{g}^{(N)}(2_{ij})$ include specific classes of triple and quadruple
excitations. This feature may be a possible way to define the balanced
inclusion of higher-rank excitations in the local CC formulations. For this
purpose one can also envision CC flows based on $\mathfrak{g}^{(N)}(x_{R})$
sub-algebras with $x>2$ that employ localized orbitals.

Since each SES $(\mathfrak{g}^{(N)}(2_{ij}))$ computational block contributing
to flow (41) defines its own set of PNOs, in analogy to DLPNO-CC approaches,
it can be re-expressed through its own set of PNOs and set of pre-selected
internal amplitudes (utilizing predefined thresholds). However, introducing
threshold in the flow CC equations can be performed less abruptly than in the
existing DLPNO-CC methods. In fact, the amplitude selection process is
equivalent of selecting sub-algebra $\mathfrak{g}^{(N)}(2_{ij},y_{S})$ of
$\mathfrak{g}^{(N)}(2_{ij})$ in the way that $S$ is the set of PNOs
corresponding to occupation number greater than the predefined threshold and
$y$ is the total number of virtual PNOs selected this way. This selection
induces a natural partitioning of the $T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij}))$ into a part belonging to
$\mathfrak{g}^{(N)}(2_{ij},y_{S})$ ($T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij},y_{S}))$ and remaining ”neglegible” part of
excitations ($\Delta T_{\rm int}(\mathfrak{g}^{(N)}(2_{ij}))$)

$T_{\rm int}(\mathfrak{g}^{(N)}(2_{ij}))=T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij},y_{S}))+\Delta T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij}))\;\;.$ (42)

The controlled version of the selection step is achieved by noticing that the
effect of the ”neglegible” amplitudes $\Delta T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij}))$ can still be absorbed (using another
downfolding step within the $\mathfrak{g}^{(N)}(2_{ij})$ CAS space) in the
form of additional similarity transformation:

$(P+Q_{\rm int}(\mathfrak{f}_{ij}))[e^{-\Delta T_{\rm
int}(\mathfrak{h}_{ij})}H^{\rm eff}(\mathfrak{h}_{ij})e^{\Delta T_{\rm
int}(\mathfrak{h}_{ij})}]e^{T_{\rm
int}(\mathfrak{f}_{ij})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{f}_{ij})}|\Phi\rangle\;,\;\mathfrak{f}_{ij}=\mathfrak{g}^{(N)}(2_{ij},y_{S})\;,\;\mathfrak{h}_{ij}=\mathfrak{g}^{(N)}(2_{ij})$
(43)

where $Q_{\rm int}(\mathfrak{g}^{(N)}(2_{ij},y_{S}))$ is a projection operator
onto excited configurations generated by the $T_{\rm
int}(\mathfrak{g}^{(N)}(2_{ij},y_{S}))$ when acting onto reference function
$|\Phi\rangle$.

The CC flow equations can also be naturally linked to approximate CC schemes
used in studies of spin systems. In Refs.Bishop, Parkinson, and Xian (1991);
Bishop and Li (2011) Bishop et al. considered a hierarchy of approximations
where excitation manifolds are defined using the so-called sub-systems defined
by contiguous lattice sites, each of which is nearest neighbors to at least
one other in the sub-system. These sub-systems can be naturally identified
with the active spaces, making CC flows equations similar to the SUB$n$-$m$
and LSUB$m$ schemes discussed in Refs.Bishop, Parkinson, and Xian (1991);
Bishop and Li (2011) As shown in the previous paragraphs, the application of
CC flows to the spin systems offers an interesting way of defining reduced-
scaling methods for quantum lattice models. This can be achieved by selecting
the essential class of excitations (for a given sub-system) using density
matrices corresponding to sub-system’s downfolded Hamiltonians. This approach
can address problems associated with high numerical overheads of highly
accurate SUB$n$-$m$ and LSUB$m$ approaches. The related developments will be
discussed in forthcoming papers.

Summarizing, several basic threads of DLPNO-CCSD equations are consequences of
CC flow equations defined by $\mathfrak{g}^{(N)}(2_{ij})$ sub-algebras. The
local character of correlation effects is a net effect of the local character
of the basis set used, asymptotic properties of one- and two-electron
interactions, and fundamental properties of CC formalism associated with the
CC sub-system memory. This feature allows one to construct, in a rigorous way,
Schrödinger-type equations for sub-systems (in this case, a pair of orbitals)
defining the flow. Additionally, a rigorous definition of sub-system and
associated wave function lead to a natural definition of the sub-system
density matrix and natural orbitals. We believe that the CC flow equations
based on the SES formalism are an interesting tool for constructing various
approximations for correlated systems. This formulation can be universally
used in stationary formulations of canonical and local CC formulations and be
extended to the time-dependent CC equations. The general CC flow formalism is
not limited to types of interactions that are considered molecular systems and
can also be extended to other types of many-body interactions encountered in
chemical and physical applications.

##  III Sub-system flows based on the double unitary CC representations of
wave function

We find properties of SR-CC flows very appealing from the point of view of
quantum computing. Instead of considering expensive ”global” space approaches
(as done in the majority of existing QC formalisms) that require too many
parameters to be optimized at the same time, one could partition the problem
into smaller computational sub-problems that can be tuned to available systems
of qubits. For this reason, we would like to adapt the SR-CC ideas from
previous sections to double unitary CC Ansatz (DUCC; see Ref.Bauman  _et al._
(2019)). While the DUCC formalism mirrors some properties of the SES-CC
formalism and additionally assures the Hermitian character of the effective
Hamiltonians in CAS($\mathfrak{h}$), due to the non-commutative nature of the
anti-Hermitian cluster operators employed by this formalism, coupling various
DUCC problems into a flow requires several approximations, described in the
following subsection.

The DUCC formalism discussed in Refs.Bauman  _et al._ (2019); Kowalski and
Bauman (2020) uses a composite unitary CC Ansatz to represent the exact wave
function $|\Psi\rangle$, i.e.,

$|\Psi\rangle=e^{\sigma_{\rm ext}(\mathfrak{h})}e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;,$ (44)

where $\sigma_{\rm ext}(\mathfrak{h})$ and $\sigma_{\rm int}(\mathfrak{h})$
are general-type anti-Hermitian operators

$\displaystyle\sigma_{\rm int}^{\dagger}(\mathfrak{h})$ $\displaystyle=$
$\displaystyle-\sigma_{\rm int}(\mathfrak{h})\;,$ (45)
$\displaystyle\sigma_{\rm ext}^{\dagger}(\mathfrak{h})$ $\displaystyle=$
$\displaystyle-\sigma_{\rm ext}(\mathfrak{h})\;.$ (46)

All cluster amplitudes defining $\sigma_{\rm int}$ cluster operator carry
active indices only (or indices of active orbitals defining given
$\mathfrak{h}$). The external part $\sigma_{\rm ext}(\mathfrak{h})$ is defined
by amplitudes carrying at least one inactive orbital index. In contrast to the
SR-CC approach, internal/external parts of anti-Hermitian cluster operators
are not defined in terms of excitations belonging explicitly to a given sub-
algebra but rather by indices defining active/inactive orbitals specific to a
given $\mathfrak{h}$. Therefore $\mathfrak{h}$ will be used here in the
context of CAS’s generator.

When the external cluster amplitudes are known (or can be effectively
approximated), in analogy to single-reference SES-CC formalism, the energy (or
its approximation) can be calculated by diagonalizing Hermitian
effective/downfolded Hamiltonian in the active space using various quantum or
classical diagonalizers. An important step towards developing practical
computational schemes is to simplify the infinite expansions defining both
cluster amplitudes and non-terminating commutator expansions defining
downfolded Hamiltonians. A legitimate approximation of $\sigma_{\rm
ext}(\mathfrak{h})$ and $\sigma_{\rm int}(\mathfrak{h})$ in Eq.(44) for well-
defined active spaces is to retain lowest-order terms only, i.e.,

$\displaystyle\sigma_{\rm int}(\mathfrak{h})$ $\displaystyle\simeq$
$\displaystyle T_{\rm int}(\mathfrak{h})-T_{\rm
int}(\mathfrak{h})^{\dagger}\;,$ (47) $\displaystyle\sigma_{\rm
ext}(\mathfrak{h})$ $\displaystyle\simeq$ $\displaystyle T_{\rm
ext}(\mathfrak{h})-T_{\rm ext}(\mathfrak{h})^{\dagger}\;,$ (48)

which has been discussed in Ref.Bauman  _et al._ (2019). In particular,
$T_{\rm ext}(\mathfrak{h})$ can be approximated by SR-CCSD amplitudes that
carry at least one external spinorbital index. Other possible sources for
obtaining external cluster amplitudes are higher-rank SR-CC methods and
approximate unitary CC formulations such as UCC(n) methods Bartlett,
Kucharski, and Noga (1989); Taube and Bartlett (2006).

Using DUCC representation (44) it can be shown that in analogy to the SR-CC
case, the energy of the entire system (once the exact form of $\sigma_{\rm
ext}(\mathfrak{h})$ operator is known) can be calculated through the
diagonalization of the effective/downfolded Hamiltonian in SES-generated
active space, i.e.,

$H^{\rm eff}(\mathfrak{h})e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle=Ee^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle,$ (49)

where

$H^{\rm eff}(\mathfrak{h})=(P+Q_{\rm int}(\mathfrak{h}))\bar{H}_{\rm
ext}(\mathfrak{h})(P+Q_{\rm int}(\mathfrak{h}))$ (50)

and

$\bar{H}_{\rm ext}(\mathfrak{h})=e^{-\sigma_{\rm
ext}(\mathfrak{h})}He^{\sigma_{\rm ext}(\mathfrak{h})}.$ (51)

Figure 4: Schematic representation of the DUCC flow. It is assumed that the
most important classes of excitations required to describe a state of interest
are captured by the target active space (or virtual space) too large for
direct QC simulations and that the target active space (corresponding to sub-
algebra $\mathfrak{h}$) can be ”approximated” by excitations included in
smaller yet computationally feasible active spaces corresponding to sub-
algebras $\mathfrak{h}_{1}$, …, $\mathfrak{h}_{M}$ (see text). The DUCC flow
combines computational blocks that correspond to variational problems
associated with each sub-algebra $\mathfrak{h}_{i}\;(i=1,\ldots,M)$. The green
dashed line represents Fermi level.

Typical approximations for downfolded Hamiltonian utilize: (1) various sources
for evaluation of the $T_{\rm ext}(\mathfrak{h})$ operator in (48), (2)
various length of commutator expansion defining the $\bar{H}_{\rm
ext}(\mathfrak{h})$ operator, (3) various excitation-ranks in the many-body
expansion of the $\bar{H}_{\rm ext}(\mathfrak{h})$ operator, and (4) various
molecular basis choices.

Recently, applications of QPE and VQE quantum algorithms to evaluate
eigenvalues of downfolded Hamiltonians $H^{\rm eff}(\mathfrak{h})$ became a
subject of intensive studies. In the case of the VQE method, the energy
functional

$\min_{{\bm{\theta}}(\mathfrak{h})}\langle\Psi({\bm{\theta}}(\mathfrak{h}))|H^{\rm
eff}(\mathfrak{h})|\Psi({\bm{\theta}}(\mathfrak{h}))\rangle$ (52)

is optimized with respect to variational parameters
${\bm{\theta}}(\mathfrak{h})$ where
$|\Psi({\bm{\theta}}(\mathfrak{h}))\rangle$ approximates $e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle$

$|\Psi({\bm{\theta}}(\mathfrak{h}))\rangle\simeq e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle$ (53)

at the level of quantum circuit. This approach turned out to be very
efficient, especially when ”correlated” natural orbitals are employed. The
advantage of using the VQE approach is the possibility of extracting the
information about cluster amplitudes defining $\sigma_{\rm int}(\mathfrak{h})$
from the optimized parameters ${\bm{\theta}}(\mathfrak{h})$. This feature
plays a vital role in designing DUCC sub-system flows and assures mechanism of
quantum information passing between various computational blocks. In the
following analysis we will assume that the variational parameters
${\bm{\theta}}(\mathfrak{h})$ correspond to cluster amplitudes in $\sigma_{\rm
int}(\mathfrak{h})$ expansion.

###  III.1 DUCC flow equations: applications in quantum computing

The DUCC flow idea is very interesting from the point of view of its
applications in quantum computing, where a quantum computer can process
computational blocks (either corresponding to energy functional minimization
or diagonalization of the downfolded Hamiltonians). In this section, we will
extend the idea of the SR-CC sub-system flow to the DUCC formalism, and we
will highlight the similarities and differences between these two approaches.
The main differences between SR-CC and DUCC should be attributed to the non-
commutative nature of many-body components defining anti-Hermitian DUCC
cluster operators. This fact, in the case of the DUCC approach, significantly
impedes the analysis of the equations and partitioning them into separate
computational blocks that can be integrated into a sub-system flow equations.
However, this can be achieved with a sequence of approximations we describe
below.

We will start our analysis from assuming that we would like to perform DUCC
effective simulations for SES $\mathfrak{h}$ problem (49) which is, for
whatever reason, too complex or too big for quantum processing. We will assume
that external amplitudes $\sigma_{\rm ext}(\mathfrak{h})$ can be effectively
evaluated using perturbative formulations. For simplicity we will introduce a
new DUCC Hermitian Hamiltonian $A(\mathfrak{h})$ which is defined as $H^{\rm
eff}(\mathfrak{h})$ or its approximation in the $(P+Q(\mathfrak{h}))$ space
(in the simplest case it can be just the
$(P+Q(\mathfrak{h}))H(P+Q(\mathfrak{h}))$ operator). We will denote
$A(\mathfrak{h})$ simply by $A$. We will also assume the situation where
excitations from $\mathfrak{h}$ that are relevant to state of interest can be
captured by excitation sub-algebras: $\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$,
$\ldots$, $\mathfrak{h}_{M}$ (see Fig.4), where, in analogy to the SR-CC case,
we admit the possibility of ”sharing” excitations/de-excitations between these
sub-algebras. We also assume that the number of excitations belonging to each
$\mathfrak{h}_{i}$ $(i=1,\ldots,M)$ is significantly smaller than the number
of excitations in $\mathfrak{h}$ and therefore numerically tractable in
quantum simulations. Below we will discuss the challenges and approximations
that are needed to obtain well defined DUCC flow equations.

The $A(\mathfrak{h})$ Hamiltonian and the $(P+Q(\mathfrak{h}))$ space can be
treated as a starting point for the secondary DUCC decompositions generated by
sub-system algebras $\mathfrak{h}_{i}$ $(i=1,\ldots,M)$ defined above, i.e.,

$A^{\rm eff}(\mathfrak{h}_{i})e^{\sigma_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle=Ee^{\sigma_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle\;\;(i=1\ldots,M)\;.$ (54)

or in the VQE-type variational representation as

$\min_{{\bm{\theta}}(\mathfrak{h}_{i})}\langle\Psi({\bm{\theta}}(\mathfrak{h}_{i}))|A^{\rm
eff}(\mathfrak{h}_{i})|\Psi({\bm{\theta}}(\mathfrak{h}_{i}))\rangle\;\;(i=1\ldots,M)\;.$
(55)

Each $A^{\rm eff}(\mathfrak{h}_{i})$ is defined as

$A^{\rm eff}(\mathfrak{h}_{i})=(P+Q_{\rm int}(\mathfrak{h}_{i}))\bar{A}_{\rm
ext}(\mathfrak{h}_{i})(P+Q_{\rm int}(\mathfrak{h}_{i}))$ (56)

and

$\bar{A}_{\rm ext}(\mathfrak{h}_{i})=e^{-\sigma_{\rm
ext}(\mathfrak{h}_{i})}Ae^{\sigma_{\rm ext}(\mathfrak{h}_{i})}.$ (57)

where we defined external $\sigma_{\rm ext}(\mathfrak{h}_{i})$ operator with
respect to $\mathfrak{h}$ or $(P+Q_{\rm int}(\mathfrak{h}))$ space
(i.e.cluster amplitudes defining $\sigma_{\rm ext}(\mathfrak{h}_{i})$ must
carry at last one index belonging do active spin orbitals defining
$\mathfrak{h}$ and not belonging to set of active spin orbtitals defining
$\mathfrak{h}_{i}$). In other words, sub-algebras $\mathfrak{h}_{i}$ generate
active sub-spaces in larger active space $\mathfrak{h}$, i.e., $(P+Q_{\rm
int}(\mathfrak{h}_{i}))\subset(P+Q(\mathfrak{h}))$. However, connecting DUCC
computational blocks (54) or (55) directly into a flow is a rather challenging
task. In contrast to the SR-CC sub-system flows where cluster amplitudes are
universal for all sub-algebras induced problems (i.e., given amplitude carries
the same value across all computational blocks), the same is no longer valid
for DUCC flows. Again, this is a consequence of the non-commutativity of the
anti-Hermitian operators defining DUCC representation of the wave function.
For example, the internal amplitude for some $\mathfrak{h}_{i}$ problem may
assume a different value as the same amplitude being an internal amplitude for
a different problem corresponding to sub-algebra $\mathfrak{h}_{j}$ $(i\neq
j)$, which means that DUCC amplitudes explicitly depend on the sub-algebra
index $\mathfrak{h}_{i}$ (as opposed to the SR-CC flow formalism, where values
of particular amplitudes were independent of the sub-algebra index). Similar
effects could be observed in ADAPT-VQE formulations,Grimsley  _et al._ (2019b)
where amplitudes from the excitation poll may appear multiple times carrying
various values in the wavefunction expansion. An additional problem is related
to the fact that while for the SR-CC flows, effective Hamiltonians
corresponding for various SESs can be constructed exactly, for the DUCC case
$A^{\rm eff}(\mathfrak{h}_{i})$ can be constructed only in an approximate way,
and therefore their ground-state eigenvalues may not be exactly equal.

To address these issues and define practical DUCC flow we will discuss the
algorithm that combines secondary downfolding steps with Trotterization of the
unitary CC operators. Let us assume that the $\sigma_{\rm int}(\mathfrak{h})$
operator can be approximated by $\sigma_{\rm
int}(\mathfrak{h}_{i})(i=1,\ldots,M)$, i.e.,

$\sigma_{\rm int}(\mathfrak{h})\simeq\sum_{i=1}^{M}\sigma_{\rm
int}(\mathfrak{h}_{i})+X(\mathfrak{h},\mathfrak{h}_{1},\ldots,\mathfrak{h}_{M})$
(58)

where the $X(\mathfrak{h},\mathfrak{h}_{1},\ldots,\mathfrak{h}_{M})$ operator
(or $X$ for short) eliminates possible overcounting of the ”shared”
amplitudes. It enables to re-express $\sigma_{\rm int}(\mathfrak{h})$ as

$\sigma_{\rm int}(\mathfrak{h})=\sigma_{\rm
int}(\mathfrak{h}_{i})+R(\mathfrak{h}_{i})\;\;(i=1,\ldots,M)\;,$ (59)

where

$R(\mathfrak{h}_{i})=^{(i)}\sum_{j=1}^{M}\;\sigma_{\rm
int}(\mathfrak{h}_{j})+X$ (60)

and ${}^{(i)}\sum_{j=1}^{M}$ designates the sum where the $i$-th element is
neglected. Consequently, we get

$e^{\sigma_{\rm int}(\mathfrak{h})}|\Phi\rangle=e^{\sigma_{\rm
int}(\mathfrak{h}_{i})+R(\mathfrak{h}_{i})}|\Phi\rangle\;\;(i=1,\ldots,M)\;.$
(61)

Using Trotter formula we can approximate right hand side of (61) for a given
$j$ as

$e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle\simeq(e^{R(\mathfrak{h}_{i})/N}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N})^{N}|\Phi\rangle\;.$ (62)

Introducing auxiliary operator $G^{(N)}_{i}$

$G^{(N)}_{i}=(e^{R(\mathfrak{h}_{i})/N}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N})^{N-1}e^{R(\mathfrak{h}_{i})/N}\;\;(i=1,\ldots,M)\;,$
(63)

the ”internal” wave function (61) can be expressed as

$e^{\sigma_{\rm int}(\mathfrak{h})}|\Phi\rangle\simeq
G^{(N)}_{i}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N}|\Phi\rangle\;\;(i=1,\ldots,M)\;.$ (64)

One should remember that $G^{(N)}_{i}$ is a complicated function of all
$\sigma_{\rm int}(\mathfrak{h}_{j})\;(j=1,\ldots,M)$ and the above expression
does not decouple $\sigma_{\rm int}(\mathfrak{h}_{i})$ from the $G^{(N)}_{i}$
term. However, this expression may help define the practical way for
determining computational blocks for flow equations. To see this, let us
introduce expansion (64) to Eq.(49) (with $H^{\rm eff}(\mathfrak{h})$ replaced
by the $A$ operator), pre-multiply both sides by $[G^{(N)}_{i}]^{-1}$, and
project onto $(P+Q_{\rm int}(\mathfrak{h}_{i}))$ sub-space, which leads to
non-linear eigenvalue problems

$(P+Q_{\rm int}(\mathfrak{h}_{i}))[G^{(N)}_{i}]^{-1}AG^{(N)}_{i}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N}|\Phi\rangle\simeq Ee^{\sigma_{\rm
int}(\mathfrak{h})_{i})/N}|\Phi\rangle\;(i=1,\ldots,M)\;.$ (65)

We will utilize these equations as a computational blocks for the DUCC flow.
To make a practical use of Eqs. (65) let us linearize them by defining the
downfolded Hamiltonian $\Gamma_{i}^{(N)}$, $\Gamma_{i}^{(N)}=(P+Q_{\rm
int}(\mathfrak{h}_{i}))[G^{(N)}_{i}]^{-1}AG^{(N)}_{i}(P+Q_{\rm
int}(\mathfrak{h}_{i}))$ as a function of all $\sigma_{\rm
int}(\mathfrak{h}_{j})\;(j=1,\ldots,M)$ from the previous flow cycle(s)
($pc$). We will symbolically designate this fact by using special notation for
$\Gamma_{i}^{(N)}$ effective Hamiltonian, i.e., $\Gamma_{i}^{(N)}(pc)$
Hamiltonian. Now, we replace eigenvalue problems (65)) by optimization
procedures described by Eqs.(55) which also offer an easy way to deal with
”shared” amplitudes. Namely, if in analogy to SR-CC sub-system flow we
establish an ordering of $\mathfrak{h}_{i}$ sub-algebras, with
$\mathfrak{h}_{1}$ corresponding to the CAS closest to the wave function of
interest, then in the $\mathfrak{h}_{i}$ problem we partition (in analogy to
Eq.(33)) set of parameters ${\bm{\theta}}_{N}(\mathfrak{h}_{i})$ into sub-set
${\bm{\theta}}_{N}^{\rm CP}(\mathfrak{h}_{i})$ that refers to common pool of
amplitudes determined in preceding steps (say, for
$\mathfrak{h}_{j}\;(j=1,\ldots,i-1)$) and sub-set ${\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i})$ that is uniquely determined in the $\mathfrak{h}_{i}$
minimization step, i.e,

$\min_{{\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i})}\langle\Psi({\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i}),{\bm{\theta}}_{N}^{\rm
CP}(\mathfrak{h}_{i}))|\Gamma_{i}^{(N)}(pc)|\Psi({\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i}),{\bm{\theta}}_{N}^{\rm
CP}(\mathfrak{h}_{i}))\rangle\;\;(i=1,\ldots,M)\;,$ (66)

where $|\Psi({\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i}),{\bm{\theta}}_{N}^{\rm CP}(\mathfrak{h}_{i}))\rangle$
approximates $e^{\sigma_{\rm int}(\mathfrak{h}_{i})/N}|\Phi\rangle$. In this
way, each computational block coupled into a flow corresponds to a
minimization procedure that optimizes parameters ${\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i})$ using quantum algorithms such as the VQE approach. At
the end of the iterative cycle, once all amplitudes are converged, in contrast
to the SR-CC flows, the energy is calculated using $\mathfrak{h}_{1}$ problem
as an expectation value of the $\Gamma_{1}^{(N)}$ operator. The DUCC flow is
composed of classical computing steps where approximate second-quantized form
of the $\Gamma_{i}^{(N)}(pc)$ operators (at the cost of additional similarity
transformations or their approximate variants in small-size active space) are
calculated and quantum computing steps, where cluster amplitudes are
determined using the VQE algorithm. The discussed formalism introduces a broad
class of control parameters, which define each computational step’s
dimensionality. These are the numbers of occupied/unoccupied active orbitals
defining $\mathfrak{h}_{i}$ sub-algebras $x_{R_{i}}$/$y_{S_{i}}$,
respectively. Similar results can be obtained by using the Zassenhaus formula.
Moreover, the present formalisms, in analogy to the SR-CC flows, can be
extended to the time-domain.

An essential feature of the DUCC flow equation is associated with the fact
that each computational block (66) can be encoded using a much smaller number
of qubits compared to the full size of the global problem. In fact, the
maximum size of the qubit register ($QR(\mathfrak{m}_{\rm max})$) required in
DUCC quantum flow is associated with the maximum size of the sub-system and
not with the size of the entire quantum system of interest

$QR(\mathfrak{m}_{\rm max})\ll QR(\mathfrak{g}^{(N)})\;,$ (67)

where $QR(\mathfrak{g}^{(N)})$ is the total number of qubits required to
describe whole system. This observation significantly simplifies the qubit
encoding of the effective Hamiltonians included in quantum DUCC flows,
especially in formulations based on the utilization of localized molecular
basis set as discussed in Section II.C (for early quantum algorithms
exploiting locality of interactions see Ref.McClean  _et al._ (2014)).

##  IV Conclusions

This paper discussed the properties of SR-CC sub-system flow equations
stemming from the SES-CC formalism for RHF reference functions. It was shown
that flow equations define an alternative (to the canonical formulations) way
of introducing selected classes of higher-rank excitations based on system
partitioning or choice of sub-system excitation sub-algebras corresponding to
various active spaces. An essential feature of the SR-CC flow lies in the fact
that flow equations can be built upon $\mathfrak{g}^{(N)}(x_{R},y_{S})$ sub-
algebras with $x_{R}$ and $y_{S}$ chosen in a way that makes the flow tunable
to available computational resources. We also demonstrated that the idea of CC
flow naturally extends to the time-domain, offering a possibility of
performing calculations for the quantum system’s time evolution affordably.
Interestingly, the ideas behind SES-CC and SR-CC sub-system flows can also
provide a deeper understanding of local CC formulations and the concept of
”locality” of correlation effects. As explained in section II.B, the SR-CC
flows based on the utilization of local molecular orbitals provide a rigorous
way of defining sub-system through the effective Hamiltonian corresponding to
$(i,j)$-determined SES, $\mathfrak{g}^{(N)}(2_{ij})$. The $(i,j)$-pair density
matrix can be further used to calculate pair-natural orbitals and select
leading excitation as postulated in the DLPNO-CCSD formulations. We believe
that the SR-CC flows defined by larger active spaces also provide a natural
way of introducing higher-rank excitations, although maintaining linear-
scaling of the resulting local CC formulation may not be possible. On the
other hand, following the SR-CC flow philosophy for localized orbitals,
although numerically more expensive, may help in re-establishing the desired
level of accuracy in perturbative (non-iterative) energy corrections due to
the higher-rank cluster excitations.

Due to the non-commutative nature of the general type unitary CC formulations,
the direct extension of SR-CC sub-system flows to DUCC-type flow is a rather
challenging endeavor. However, utilizing Trotter formula in downfolding
procedures lead to computationally feasible algorithms. In the quantum
computing variant, the flow represents a sequence of coupled Hermitian
eigenproblems, where diagonalization is replaced by the VQE-type optimization
to obtain a corresponding sub-set of amplitudes. In this formulation, the flow
of quantum information corresponding to shared/external amplitudes (defined
for a given sub-system) can be easily implemented at the level of the quantum
circuit. In analogy to the SR-CC flows, the DUCC-flows can be tuned to the
available quantum resources. As such, the DUCC flows offer an interesting
possibility of decomposing large-dimensionality problems into a collection of
reduced dimensionality computational blocks. We believe that the DUCC flow
methods can significantly push the envelope of system-size tractable in
quantum simulations. It should also be stressed that the SR-CC/DUCC flow
methods allow one to enlarge the size of probed space systematically while
retaining the size-extensivity of the calculated in this way energies. This
feature is especially important in applications of DUCC flows to chemical
reactions and extended systems.

An exciting feature of the SES-CC formalism and CC flows is their universal
character irrespective of the particular form of interactions defining
correlated many-body systems.

##  V acknowledgement

This work was supported by the Quantum Science Center (QSC), a National
Quantum Information Science Research Center of the U.S. Department of Energy
(DOE). Part of this work was supported by the “Embedding QC into Many-body
Frameworks for Strongly Correlated Molecular and Materials Systems” project,
which is funded by the U.S. Department of Energy, Office of Science, Office of
Basic Energy Sciences (BES), the Division of Chemical Sciences, Geosciences,
and Biosciences.

##  Appendix A

In this Appendix we will analyze properties of the CC flow equations. Let us
assume that flow equations involve $M$ eigenvalue problems defined by sub-
algebras $\\{\mathfrak{h}_{i}\\}_{i=1}^{M}$ (see Eqs.(32))

$H^{\rm eff}(\mathfrak{h}_{i})e^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle\;\;(i=1,\ldots,M)\;.$ (68)

The $i$-th computational block can be written in the following form

$(P+Q_{\rm int}(\mathfrak{h}_{i}))[e^{-T_{\rm
ext}(\mathfrak{h}_{i})}He^{T_{\rm ext}(\mathfrak{h}_{i})}-E]e^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle=0\;,$ (69)

where the $T_{\rm ext}(\mathfrak{h}_{i})$ operator in the above equation
contains excitations from all remaining cluster operators $T_{\rm
int}(\mathfrak{h}_{j})\;\;(j=1,\ldots,M\;;j\neq i)$ not belonging to the set
of excitations defining the $T_{\rm int}(\mathfrak{h}_{i})$ operator. After
introducing the resolution of identity

$e^{-T_{\rm int}(\mathfrak{h}_{i})}e^{T_{\rm int}(\mathfrak{h}_{i})}=1$ (70)

right to the projection operator $(P+Q_{\rm int}(\mathfrak{h}_{i}))$ in
Eq.(69) one obtains

$(P+Q_{\rm int}(\mathfrak{h}_{i}))e^{T_{\rm
int}(\mathfrak{h}_{i})}[e^{-T}He^{T}-E]|\Phi\rangle=0\;,$ (71)

where

$T=T_{\rm int}(\mathfrak{h}_{i})+T_{\rm ext}(\mathfrak{h}_{i})\;.$ (72)

It should be stressed that the $T$ operator is not carrying index of any sub-
algebra, and $T$ is defined as a sum of unique excitations defining $\\{T_{\rm
int}(\mathfrak{h}_{i})\\}_{i=1}^{M}$ operators. We will symbolically represent
$T$ as a union of unique excitations originating in various $T_{\rm
int}(\mathfrak{h}_{i})$

$T=\bigcup_{i=1}^{M}T_{\rm int}(\mathfrak{h}_{i})\;.$ (73)

This is a consequence of the CC flow definition. Given the fact that, the
operator $e^{T_{\rm int}(\mathfrak{h}_{i})}$ is non-singular and that

$(P+Q_{\rm int}(\mathfrak{h}_{i}))e^{T_{\rm int}}=(P+Q_{\rm
int}(\mathfrak{h}_{i}))e^{T_{\rm int}}(P+Q_{\rm int}(\mathfrak{h}_{i}))\;,$
(74)

the eigenvalue problem is equivalent at the solution to

$(P+Q_{\rm int}(\mathfrak{h}_{i}))[e^{-T}He^{T}-E]|\Phi\rangle=0$ (75)

or in ”standard” form of CC equations

$\displaystyle(P+Q_{\rm int}(\mathfrak{h}_{i}))[e^{-T}He^{T}]|\Phi\rangle=0$
(76) $\displaystyle E=\langle\Phi|e^{-T}He^{T}|\Phi\rangle\;.$ (77)

Using Eqs.(76-77) one can draw the following conclusions:

  * •

Conclusion 1: The CC flow equations (68) at the solution are equivalent to the
standard connected CC equations for the $T$ cluster operator.

  * •

Conclusion 2: All ground-state eigenvalues of computational blocks given by
Eq.(69) are equal and at the solution assume the value of CC energy calculated
using standard formulation for the $T$ cluster operator.

The final remark concerns the size of the sub-space sampled by our flow
equations, which is defined by excitations included in the $T$ cluster
operator. For the CC flow defined by all possible
$\mathfrak{g}^{(N)}(2_{R_{i}})$ sub-algebras, the $T$ operator contains all
singles, doubles, and subsets of triple and quadruple exictations as shown in
Eq.(30). For the CC flow defined by all $\mathfrak{g}^{(N)}(3_{R_{i}})$ sub-
algebras, the $T$ operator contains all singles, doubles, and triples, and
subsets of quadruples, pentuples, and hextuples. The CC flows equations also
offer a flexibility in choosing sub-algebras involved in the flow, for
example, the flow can involve various
$\mathfrak{g}^{(N)}(x_{R_{i}},y_{S_{i}})$ sub-algebras. As long as $T$ is
defined by unique excitations of the internal cluster operators corresponding
to the sub-algebras involved in the flow, conclusions 1 and 2 are still valid.

##  Appendix B

The analysis in Appendix A can be extended to the CC flows in the time domain,
where the flow is composed of coupled time-dependent CC equations

$i\hbar\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h}_{i},t)e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle\;,(i=1,\ldots,M)\;.$ (78)

For a given sub-algebra $\mathfrak{h}_{i}$ time-dependent equations can be
cast in the form (we assume that spinorbitals are time-independent)

$i\hbar(P+Q_{\rm int}(\mathfrak{h}_{i}))\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h}_{i},t)e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle$ (79)

By expanding $H^{\rm eff}(\mathfrak{h}_{i},t)$, Eq.(79) can be rewritten as

$(P+Q_{\rm int}(\mathfrak{h}_{i}))e^{T_{\rm int}(\mathfrak{h}_{i},t)}(P+Q_{\rm
int}(\mathfrak{h}_{i}))\\{i\hbar\frac{\partial}{\partial
t}T(t)-e^{-T(t)}He^{T(t)}\\}|\Phi\rangle=0\;,$ (80)

where $T(t)$ is defined as

$T(t)=\bigcup_{i=1}^{M}T_{\rm int}(\mathfrak{h}_{i},t)\;.$ (81)

Using matrix representation of the $T_{\rm int}(\mathfrak{h}_{i},t)$ operator
in the $\mathfrak{h}_{i}$ generated CAS space denoted as $\bm{T}_{\rm
int}(\mathfrak{h}_{i},t)$ we get

${\rm det}(e^{\bm{T}_{\rm int}(\mathfrak{h}_{i},t)})=e^{{\rm Tr}(\bm{T}_{\rm
int}(\mathfrak{h}_{i},t))}=1$ (82)

for arbitrary time $t$. This is a consequence of the fact that $\bm{T}_{\rm
int}(\mathfrak{h}_{i},t)$ is a lower diagonal matrix with zeros on the
diagonal. Therefore, Eq.(83) is equivalent to standard time-dependent
equations

$(P+Q_{\rm int}(\mathfrak{h}_{i}))\\{i\hbar\frac{\partial}{\partial
t}T(t)-e^{-T(t)}He^{T(t)}\\}|\Phi\rangle=0\;.$ (83)

Following the same reasoning as in Appendix A, we can state that:

  * •

Conclusion 3: The time-dependent CC flow approach (78) is equivalent to
standard representation of the time-dependent CC equations defined by the
$T(t)$.

.

##  Appendix C

In this appendix we discuss properties of the CC flows defined by approximate
methods for solving computational blocks (68). First, let us assume that the
$T_{\rm int}(\mathfrak{h}_{i})$ (defined by excitation level $m_{i}$) is
approximated by operator $T_{\rm int}^{(A)}(\mathfrak{h}_{i})$ which by
defined by excitations od maximum rank $m(A)_{i}$ ($m(A)_{i}<m_{i}$). In this
case, to obtain working equations for $T_{\rm int}^{(A)}(\mathfrak{h}_{i})$
amplitudes, we will project Eq.(69) for a given $\mathfrak{h}_{i}$ onto
$P+Q_{\rm int}^{(A)}(\mathfrak{h}_{i})$, where $Q_{\rm
int}^{(A)}(\mathfrak{h}_{i})$ is a projection operator onto excitations
generated by $T_{\rm int}^{(A)}(\mathfrak{h}_{i})$ when acting on
$|\Phi\rangle$, i.e.,

$(P+Q_{\rm int}^{(A)}(\mathfrak{h}_{i}))[H^{\rm
eff}(\mathfrak{h}_{i})-E]e^{T_{\rm
int}^{\rm(A)}(\mathfrak{h}_{i})}|\Phi\rangle=0\;,$ (84)

where the full projection operator $Q_{\rm int}(\mathfrak{h}_{i})$ can be
partitioned as

$Q_{\rm int}(\mathfrak{h}_{i})=Q_{\rm int}^{(A)}(\mathfrak{h}_{i})+R_{\rm
int}^{(A)}(\mathfrak{h}_{i})\;.$ (85)

Eq.(84) can be now re-written in the form

$(P+Q_{\rm int}^{(A)}(\mathfrak{h}_{i}))[H^{\rm
eff}(\mathfrak{h}_{i})-E](Q_{\rm int}^{(A)}(\mathfrak{h}_{i})+R_{\rm
int}^{(A)}(\mathfrak{h}_{i}))e^{T_{\rm
int}^{\rm(A)}(\mathfrak{h}_{i})}|\Phi\rangle=0\;,$ (86)

which clearly shows that truncating the full form of $T_{\rm
int}(\mathfrak{h}_{i})$ results in equations that due to the presence of
$R_{\rm int}^{(A)}(\mathfrak{h}_{i})$-dependent term are no longer
representing eigenvalue problem. Similar form of non-linear eigenvalue
representation of CC has been analyzed by Živković and Monkhorst in
Ref.Živković and Monkhorst (1978). Nevertheless, using similar reasoning as in
the Appendix A, it can be shown that even in the case of standard
approximations, Eq.(84) can be recast in the standard connected form

$\displaystyle(P+Q_{\rm
int}^{\rm(A)}(\mathfrak{h}_{i}))e^{-T^{\rm(A)}}He^{T^{\rm(A)}}|\Phi\rangle=0$
$\displaystyle\;,$ (87) $\displaystyle
E=\langle\Phi|e^{-T^{\rm(A)}}He^{T^{\rm(A)}}|\Phi\rangle\;,$ (88)

where

$T^{(A)}=\bigcup_{i=1}^{M}T_{\rm int}^{\rm(A)}(\mathfrak{h}_{i})\;.$ (89)

##  Appendix D

In this Appendix we focus on details of the numerical realization of the CC
flow shown in Fig.2. Without a loss of generality, we will focus on the
”serial” flow shown in Fig.2(a). The following steps are involved in the
iterative process:

  1. 1.

Defining sub-algebras/active spaces forming the flow: In this step we define a
set of active spaces corresponding to sub-algebras $\mathfrak{h}_{i}$
$i=1,\ldots,M$. In the flow we solve for cluster operator $T$,
$T=\bigcup_{i=1}^{M}T_{\rm int}(\mathfrak{h}_{i})$.

  2. 2.

Ordering of active spaces: In serial flows, an important step is associated
with the establishing the importance of active spaces. For example, this can
be achieved using values of the MBPT(2) (second-order of many-body
perturbation theory) correlation energy contributions in active spaces
included in the flow.

  3. 3.

Initialization of the $T$ operator: For this purpose we can use simple
perturbative or low-rank CC approximations (CCSD).

  4. 4.

Solving eigenvalue problems for active spaces: In this step we solve for
$T_{\rm int}(\mathfrak{h}_{i})$ update by diagonalizing $e^{-T_{\rm
ext}(\mathfrak{h}_{i})}He^{T_{\rm ext}(\mathfrak{h}_{i})}$ in the
corresponding active space ($T_{\rm ext}(\mathfrak{h}_{i})=T-T_{\rm
int}(\mathfrak{h}_{i})$). In this step, resulting CI-type coefficients have to
be transformed , using cluster analysis, to the $T_{\rm
int}(\mathfrak{h}_{i})$ amplitudes.

  5. 5.

Update of the global $T$ operator: All $T_{\rm int}(\mathfrak{h}_{i})$
$i=1,\ldots,M$ define a new one $T$ operator.

  6. 6.

Convergence check: If $T$ operator satisfies convergence criteria, final value
of correlation energy is calculated using either standard CC energy expression
or diagonalizing any of the effective Hamiltonians involved in the flow. If
does not, we repeat procedure from step (2).

The peak computational cost of the CC flow is defined by maximum-size active-
space problem ($C_{\rm max}$). Therefore the cost per iteration is
proportional to $M\times C_{\rm max}$. The CC flow equations approach offers a
flexibility in the choice of the number of active spaces ($M$) and their size
($C_{\rm max}$) therefore providing a framework where the resulting
computational model is tuned to available computational resources.

It is also instructive to analyze the numerical cost of solving each
computational block with approximate type methods such as CCSDT or CCSDTQ (see
Appendix C). Without loss of generality, let us assume a CC flow defined by
$\mathfrak{g}^{(N)}(3_{R_{i}},y_{S_{i}})$ $i=1,\ldots,M$ problems. We assume
that for each $i=1,\ldots,M$ the number of active virtual orbitals is the same
and equals $y$. For the CCSDT solver the upper-bound for the numerical cost of
the flow, $FC({\rm CCSDT})$, is given by the formula, which is obtained by
analizing the contributions from the the most expensive terms:

$FC({\rm CCSDT})\leq\alpha\times M\times{y\choose 3}\times n_{v}^{2}\;.$ (90)

Using the CCSDTQ solver for the same type of flow the numerical cost upper-
bound, $FC({\rm CCSDTQ})$, is given by the formula

$FC({\rm CCSDTQ})\leq\beta\times M\times{y\choose 4}\times n_{v}^{2}\;.$ (91)

In (90) and (91), $\alpha$ and $\beta$ are constant pre-factors. Both upper-
bounds depends on $M$ and $y$, whose values can be chosen to match available
computational resources and/or provide the desired level of accuracy.

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