# Coupled cluster downfolding techniques: a review of existing applications in
classical and quantum computing for chemical systems

Nicholas P. Bauman<EMAIL_ADDRESS>Pacific Northwest National
Laboratory, Richland, Washington, 99354, USA  Bo Peng<EMAIL_ADDRESS>Pacific Northwest National Laboratory, Richland, Washington, 99354, USA  Karol
Kowalski<EMAIL_ADDRESS>Pacific Northwest National Laboratory,
Richland, Washington, 99354, USA

###### Abstract

In this manuscript, we provide an overview of the recent developments of the
coupled cluster (CC) downfolding methods, where the ground-state problem of a
quantum system is represented through effective/downfolded Hamiltonians
defined using active spaces. All CC downfolding techniques discussed here are
derived from a single-reference exponential ansatz for the ground-state
problem. We discuss several extensions of the non-Hermitian and Hermitian
downfolding approaches to the time domain and the so-called quantum flows. We
emphasize the important role of downfolding formalisms in transitioning
chemical applications from noisy quantum devices to scalable and error-
corrected quantum computers.

##  I Introduction

The coupled cluster (CC) theory coester58_421 ; coester60_477 ; cizek66_4256 ;
paldus72_50 ; purvis82_1910 ; arponen83_311 ; bishop1991overview ;
jorgensen90_3333 ; paldus07 ; crawford2000introduction ; bartlett_rmp  has
evolved into one of the most accurate formulations to describe the correlation
effects in chemistry,paldus07 ; crawford2000introduction ; bartlett_rmp
material sciences, and physics. arponen1987extended1 ; arponen83_311 ;
arponen1987extended ; arponen1991independent ; arponen1993independent ;
robinson1989extended ; arponen1991holomorphic ; emrich1984electron ;
kummel2001post ; PhysRevC.69.054320 ; PhysRevLett.92.132501 ;
PhysRevLett.101.092502  Although the CC formalism originates in the Linked
Cluster Theorem, brandow67_771 ; lindgren12  it has been successfully extended
to describe excited states, properties, and time evolution of the
system.monkhorst77_421 ; jorgensen90_3333 ; arponen83_311 ; bartlett89_57 ;
stanton93_7029 ; xpiecuch ; krylov2008equation ; kknascimento2016linear ;
nascimento2017simulation ; vila2022real ; kvaal2012ab ; pedersen2019symplectic
; sato2018communication ; pedersen2019symplectic ; sato2018communication  Over
the last few decades, a significant effort has been exerted to address the
steep scaling of canonical CC formulations and apply them to realistic
chemical processes. Parallel computing, especially with recently developed
exascale computing architectures, has extended the applicability of
conventional CC methods, but only modestly before encountering prohibitive
costs once more. As a result, there has been much development in recent years
on new reduced-scaling approaches for classical and quantum computing
paradigms to push the envelope of the system sizes tractable by CC formalisms.

Mathematically rigorous formulations for reducing the dimensionality/cost of
quantum formulations are urgently needed to shift the envelope of system-size
tractable by accurate many-body formulations in chemistry, material sciences,
and physics. Among the most successful formulations, one should mention local
coupled cluster (CC) formulations, various partitioning and incremental
schemes, and embedding methods. neese2009efficient ; neese2009efficient2 ;
riplinger2013efficient ; Neese16_024109 ; pavosevic2016 ; pavosevic2017 ;
rolik2013efficient ; nagy2019approaching ; eriksen2021incremental ;
li2010improved  These approaches are driven by various design principles from
the locality of correlation effects in the wave function approaches to
properties of self-energy in correlated systems. Thanks to these formulations,
significant progress has been achieved in describing correlation effects in
large molecular systems allowing for simulations based on the utilization of
modest computational resources.

The dimensionality reduction techniques also play a crucial role in enabling
the early stages of quantum computing driven by noisy intermediate-scale
quantum devices (NISQ). This is associated with the reduction of the qubits
required to represent the quantum problem of interest. As an illustration, one
should mention several techniques developed to take full advantage of the
ubiquitous Variational Quantum Eigensolvers (VQE) approach
peruzzo2014variational ; mcclean2016theory ; romero2018strategies ;
PhysRevA.95.020501 ; Kandala2017 ; kandala2018extending ; PhysRevX.8.011021 ;
huggins2020non ; ryabinkin2018qubit ; cao2019quantum ; ryabinkin2020iterative
; izmaylov2019unitary ; lang2020unitary ; grimsley2019adaptive ;
grimsley2019trotterized ; cerezo2021variational ; mcardle2020quantum ;
bharti2022noisy  in addressing problems beyond the situation where few
electrons are correlated.

In the context of the development of quantum algorithms for quantum chemistry,
the main goal of dimensionality reduction methods is to provide a
mathematically rigorous way of representing interdependencies between static
and dynamical correlation effects. However, while the inclusion of static
effects can be achieved for small-size systems on currently available quantum
hardware, much needed dynamical correlation effects, usually manifesting in a
large number of fermionic degrees of freedom (amplitudes) characterized by
small values, are beyond the reach of current quantum technologies.

The recently introduced downfolding techniques based on the double unitary
coupled cluster Ansatz (DUCC) bauman2019downfolding ; downfolding2020t ;
metcalf2020resource ; bauman2019quantumex ; bauman2020variational ;
baumanpengGFDUCC ; bauman2022coupled ; bauman2022coupled2c  provide one of the
solutions to the above-mentioned problem. The DUCC formalism offers a special
representation of the ground-state wave function that, in analogy to single-
reference sub-system embedding sub-algebras (SES-CC), safkk ; sssc  allows one
to construct effective Hamiltonians that integrate out all out-of-active-space
degrees of freedom usually identified with dynamical amplitudes. Although the
effective Hamiltonian formulations have a long history in quantum history and
physics, especially in dealing with strong correlation effects, there are
notable distinct features of the DUCC and SES-CC formalisms: (1) both
formulations are embedded in the single-reference language employing a
straightforward definition of the excitation domain (i.e., wave function
parameters) in the vain of single-reference formulations, and (2) the
possibility of describing a quantum problem in the form of quantum flows,
i.e., coupled small-dimensionality eigenvalue problems. In this way, one can
probe large sub-spaces of the Hilbert space without unrealistic quantum
resource demands. Since the eigenvalue problems involved in the quantum flow
represent physically well-defined problems (defined by the corresponding
effective Hamiltonians and density matrices), the quantum flow formulation
naturally lent itself to capture possible sparsity characterizing the quantum
system.

This paper provides a compact overview of the main development threads
originating in the single-reference SES-CC formulation (Section 2.1) and its
unitary extension (Section 2.2). Section 3 introduces and discusses the basic
tenets of quantum flows. The extension of the CC downfolding methods to the
time domain and Green’s function formalism is discussed in Sections 4 and 5.
Finally, Section 6 discusses applications of the downfolding formalisms.

##  II Theory

The SES-CC and DUCC formulations have been amply discussed in recent papers
(see Refs. safkk ; bauman2019downfolding ; bauman2022coupled ). Here we
overview only the salient features of these approaches. While the SES-CC
technique forms the basis for non-Hermitian downfolding, the DUCC expansions
provide its Hermitian formulations. In both cases, the ensuing downfolding
procedures are encoded in the properties of exponential ansatzes for the
ground-state wave functions $|\Psi\rangle$:

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

for non-Hermitian formalism given by standard SR-CC expansion, and

$|\Psi\rangle=e^{\sigma_{\rm ext}}e^{\sigma_{\rm int}}|\Phi\rangle\;,$ (2)

for the Hermitian downfolding defined by the DUCC Ansatz. In these equations,
$|\Phi\rangle$ is the so-called reference function usually identified with the
Hartree–Fock determinant, $T$ is the SR-CC cluster operator, and $\sigma_{\rm
ext}$ and $\sigma_{\rm int}$ are the anti-Hermitian external and internal
cluster operators (vide infra).

Both types of downfolding lead to many-body forms of effective or downfolded
Hamiltonians acting in the appropriate active spaces. Although effective
Hamiltonian formulations have a long history in electronic structure theory,
especially in treating strong correlation effects, the present methods have
several unique features compared to the multi-reference effective Hamiltonian
approaches. Among the most distinct, one should mention: (1) the possibility
of developing effective Hamiltonian formalisms using a very simple single-
reference language to define the manifold of excitations used to construct
downfolded Hamiltonians and (2) the concept of the quantum flows (QF), which
boils down to coupling downfolding procedures corresponding to various active
spaces. The former formalism allows for sampling large sub-spaces of the
Hilbert space using reduced-dimensionality eigenvalue problems. The QF
formalism is not only a convenient representation of appropriate many-body
formulations in the form of numerically feasible computational blocks, which
plays a crucial role in the early stages of quantum computing development but
also provides a natural environment for capturing the sparsity characterizing
correlated systems.

###  II.1 Non-Hermitian CC Downfolding

Active spaces play a central role in the development of CC downfolding
techniques and are defined by the subset $R$ of occupied active orbitals
($R=\\{R_{i},\;i=1,\ldots,x_{R}\\}$) and a subset $S$ of active virtual
orbitals ($S=\\{S_{i},\;i=1,\ldots,y_{s}\\}$). Using many-body language, the
excited Slater determinants spanning the active space (along with the
reference function $|\Phi\rangle$) can be generated by generators
$E^{a_{l}}_{i_{l}}=a_{a_{l}}^{\dagger}a_{i_{l}}$ ($i_{l}\in R$ and $a_{l}\in
S$) acting on the reference function $|\Phi\rangle$. These generators define
the so-called $\mathfrak{g}^{(N)}(R,S)$ sub-algebra. Due to the utilization
the particle-hole formalism all generators $E^{a_{l}}_{i_{l}}$ commute and the
$\mathfrak{g}^{(N)}(R,S)$ is commutative. For the sake of the following
analysis, it is convenient to characterize various types of sub-algebras
$\mathfrak{g}^{(N)}(R,S)$ by specifying the numbers of active occupied
($x_{R}$) and active virtual ($y_{S}$) orbitals, namely,
$\mathfrak{g}^{(N)}(x_{R},y_{S})$.

As shown in Refs. safkk ; downfolding2020t ; kowalski2021dimensionality , each
sub-algebra $\mathfrak{h}=\mathfrak{g}^{(N)}(R,S)$ induces partitioning of the
cluster operator $T$:

$T=T_{\rm int}(\mathfrak{h})+T_{\rm ext}(\mathfrak{h})\;,$ (3)

where $T_{\rm int}(\mathfrak{h})$ belongs to $\mathfrak{h}$ while $T_{\rm
ext}(\mathfrak{h})$ does no belong to $\mathfrak{h}$. If the expansion
$e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle$ produces all Slater determinants
(of the same symmetry as the $|\Phi\rangle$ state) in the active space, we
call $\mathfrak{h}$ the sub-system embedding sub-algebra for the CC
formulation defined by the $T$ operator. In Ref. safkk , we showed that each
standard CC approximation has its own class of SESs.

The existence of the SESs for standard CC approximations provides alternative
ways for calculating CC energies, which can be obtained in contrast to the
standard CC energy $E$ expression

$E=\langle\Phi|e^{-T}He^{T}|\Phi\rangle\;,$ (4)

as an eigenvalue of the active-space non-Hermitian eigenproblem

$H^{\rm eff}(\mathfrak{h})e^{T_{\rm int}(\mathfrak{h})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;.$ (5)

where

$H^{\rm eff}(\mathfrak{h})=(P+Q_{\rm int}(\mathfrak{h}))\bar{H}_{\rm
ext}(\mathfrak{h})(P+Q_{\rm int}(\mathfrak{h}))\;$ (6)

and

$\bar{H}_{\rm ext}(\mathfrak{h})=e^{-T_{\rm ext}(\mathfrak{h})}He^{T_{\rm
ext}(\mathfrak{h})}\;.$ (7)

The above result is known as the SES-CC Theorem. In Eq. (6), $P$ stands for
the projection operator onto reference function and $Q_{\rm
int}(\mathfrak{h})$ is a projection operator onto all excited Slater
determinants with respect to the reference function $|\Phi\rangle$ that
correspond to $\mathfrak{h}$. One should also mention that the standard energy
expression given by Eq. (4) can be reproduced from Eq. (5) when $\mathfrak{h}$
represents the simplest case when there are no generators (both sets $R$ and
$S$ are empty). When SES-CC Theorem is applied to the exact cluster operator
corresponding to the full coupled cluster approach, the lowest eigenvalues of
effective Hamiltonians correspond to the exact ground-state full configuration
interaction (FCI) energy.

Standard CC approximations (such as CCSD, CCSDT, CCSDTQ, etc. methods) are
characterized by specific classes of SESs. For example, typical CCSD SESs are
$\mathfrak{g}^{(N)}(1_{R},y_{S})$ or $\mathfrak{g}^{(N)}(x_{R},1_{S})$ sub-
algebras. For the CCSDTQ approach, corresponding SESs are of the
$\mathfrak{g}^{(N)}(2_{R},y_{S})$ or $\mathfrak{g}^{(N)}(x_{R},2_{S})$ form.
From these definitions, it is easy to see that lower-rank CC approximations
are SESs for high-rank approaches. For example, the CCSD SESs
$\mathfrak{g}^{(N)}(1_{R},y_{S})$ are SESs for the CCSDTQ formalism.

The SES-CC Theorem is flexible in the choice of active spaces, providing a
number of alternative ways for calculating CC energies using effective
Hamiltonians corresponding to various SESs. For example, for the CCSD
approximation with a fixed molecular orbital basis, where the SES
$\mathfrak{g}^{(N)}(R,S)$ defined at the orbital levelsssc  contains either
one occupied active orbital or one virtual active orbital, the number of
different SESs $S_{\rm CCSD}$ and corresponding effective Hamiltonians that
upon diagonalization reproduce the standard CCSD energy is

$S_{\rm CCSD}=n_{o}(2^{n_{v}}-1)+n_{v}(2^{n_{o}}-1)-n_{o}n_{v}\;.$ (8)

This formula is a consequence of binomial expansion, in which $k$ active
virtual/occupied orbitals can be chosen in
$\binom{n_{v}}{k}$/$\binom{n_{o}}{k}$ different ways, where $n_{o}$ and
$n_{v}$ stand for the numbers of correlated occupied and virtual orbitals,
respectively.

The validity of the SES-CC Theorem has recently been confirmed numerically on
the example of several benchmark systems. In addition to the standard spatial
orbital-based definition of SES-generated active spaces, it was shown that the
SES-CC Theorem also holds for the active spaces defined by a non-trivial
number of active spin orbitals. In the extreme case, we demonstrated that the
SES-CC Theorem is also satisfied for the active space describing one active
electron ”correlated” in two active $\alpha$-type spin-orbitals.sssc

###  II.2 Hermitian CC Downfolding

The Hermitian form of the downfolded Hamiltonian is obtained as a consequence
of utilizing DUCC representation of the wave function bauman2019downfolding ;
downfolding2020t

$|\Psi\rangle=e^{\sigma_{\rm ext}(\mathfrak{h})}e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle\;,$ (9)

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

$\displaystyle\sigma_{\rm int}^{\dagger}(\mathfrak{h})$ $\displaystyle=$
$\displaystyle-\sigma_{\rm int}(\mathfrak{h})\;,$ (10)
$\displaystyle\sigma_{\rm ext}^{\dagger}(\mathfrak{h})$ $\displaystyle=$
$\displaystyle-\sigma_{\rm ext}(\mathfrak{h})\;.$ (11)

In analogy to the non-Hermitian case, the $\sigma_{\rm int}(\mathfrak{h})$
operator is defined by parameters carrying only active spin-orbital labels and
$\sigma_{\rm ext}(\mathfrak{h})$ operators are defined by parameters with at
least one in-active spin-orbital label.

The use of the DUCC Ansatz (9), in analogy to the SES-CC case, leads to an
alternative way of determining energy, which can be obtained by solving
active-space Hermitian eigenvalue problem:

$H^{\rm eff}(\mathfrak{h})e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle=Ee^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle,$ (12)

where

$H^{\rm eff}(\mathfrak{h})=(P+Q_{\rm int}(\mathfrak{h}))\bar{H}_{\rm
ext}(\mathfrak{h})(P+Q_{\rm int}(\mathfrak{h}))$ (13)

and

$\bar{H}_{\rm ext}(\mathfrak{h})=e^{-\sigma_{\rm
ext}(\mathfrak{h})}He^{\sigma_{\rm ext}(\mathfrak{h})}\;.$ (14)

When the external cluster amplitudes are known (or can be effectively
approximated), the energy (or its approximation) can be calculated by
diagonalizing the Hermitian effective/downfolded Hamiltonian (13) in the
active space using various quantum or classical diagonalizers.

For quantum computing applications second-quantized representation of $H^{\rm
eff}(\mathfrak{h})$ is required. In the light of the non-commuting character
of components defining the $\sigma_{\rm ext}(\mathfrak{h})$ operator, one has
to rely on the finite-rank commutator expansions, i.e.,

$H^{\rm eff}(\mathfrak{h})\simeq(P+Q_{\rm
int}(\mathfrak{h}))(H+\sum_{i=1}^{l}\frac{1}{i!}[\ldots[H,\sigma_{\rm
ext}(\mathfrak{h})],\ldots],\sigma_{\rm ext}(\mathfrak{h})]_{i}(P+Q_{\rm
int}(\mathfrak{h}))\;.$ (15)

Due to the numerical costs associated with the contractions of multi-
dimensional tensors and the rapidly expanding number of terms in this
expansion, only approximations based on the inclusion of low-rank commutators
are feasible. In recent studies, approximations based on single, double, and
part of triple commutators were explored where one- and two-body interactions
were retained in the second quantized form of $H^{\rm eff}(\mathfrak{h})$ were
retained. In practical applications, one also has to determine the approximate
form of $\sigma_{\rm ext}(\mathfrak{h})$. For practical reasons, we used the
following approximation

$\sigma_{\rm ext}(\mathfrak{h})\simeq T_{\rm ext}(\mathfrak{h})-T_{\rm
ext}(\mathfrak{h})^{\dagger}\;,$ (16)

where $T_{\rm ext}$ were defined through the external parts of the $T_{1}$ and
$T_{2}$ operators obtained in CCSD calculations.

##  III Quantum Flows

###  III.1 Non-Hermitian CC Flows

In the case of non-Hermitian downfolding, the SES-CC Theorem can be used to
form computational frameworks (quantum flow) that integrate eigenvalue
problems kowalski2021dimensionality ; bauman2022coupled

$H^{\rm eff}(\mathfrak{h}_{i})e^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle=Ee^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle\;(i=1,\ldots,M_{\rm SES})\;,$ (17)

where $M_{\rm SES}$ is the total number of SESs or active space problems
included in the flow. In Ref. kowalski2021dimensionality ; bauman2022coupled ,
we demonstrated that a problem defined in this way is equivalent (at the
solution) to the standard CC equations with cluster operator defined as a
combination of all unique excitations included in $T_{\rm
int}(\mathfrak{h}_{i})$ $(i=1,\ldots,M_{\rm SES})$ operators, i.e.,

$T=\bigcup_{i=1}^{M}T_{\rm int}(\mathfrak{h}_{i})$ (18)

and

$\displaystyle Qe^{-T}He^{T}|\Phi\rangle=0\;,$ (19)
$\displaystyle\langle\Phi|e^{-T}He^{T}|\Phi\rangle=E\;,$ (20)

where the $Q$ operator is a projection operator onto a sub-space of excited
Slater determinants generated by the action of $T$ operator of Eq. (18) onto
the reference function. The discussed equivalence is known as the Equivalence
Theorem. Initially, as discussed in Ref. safkk , the quantum flows were
introduced as a form of the invariance of the SES-CC Theorem upon separate
rotations of occupied and virtual orbitals.

Although the form given by Eqs. (19) and (20) is generally better suited in
canonical calculations to take advantage of parallel computing architectures,
the representation given by Eq. (17) is well-poised to capture a general type
of the sparsity characterizing quantum systems. This is because Eq. (17)
represent reduced-dimensionality computational blocks representing quantum
problems defined by non-Hermitian Hamiltonians $H^{\rm
eff}(\mathfrak{h}_{i})$. In analogy to the bi-variational CC formulations
arponen83_311  one can introduce the left eigenvectors of active-space
Hamiltonians $H^{\rm eff}(\mathfrak{h}_{i})$, using either CC-$\Lambda$

$\langle\Phi|(1+\Lambda_{\rm int}(\mathfrak{h}_{i}))\;\;(i=1,\ldots,M_{\rm
SES})\;,$ (21)

or the extended CC formalism

$\langle\Phi|e^{S_{\rm int}(\mathfrak{h}_{i})}\;\;(i=1,\ldots,M_{\rm SES})\;,$
(22)

where $\Lambda_{\rm int}(\mathfrak{h}_{i})$ and $S_{\rm
int}(\mathfrak{h}_{i})$ are de-excitation operators acting in the
corresponding active spaces, to form one-particle reduced density matrices
${\bf\gamma}(\mathfrak{h}_{i})$. For the $\Lambda$-CC formalism the matrix
elements of the ${\bf\gamma}(\mathfrak{h}_{i})$ are given by the formula

$\displaystyle\gamma^{p}_{q}(\mathfrak{h}_{i})$ $\displaystyle=$
$\displaystyle\langle\Phi|(1+\Lambda_{\rm
int}(\mathfrak{h}_{i}))a_{p}^{\dagger}a_{q}e^{T_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle\;,\;$ (24) $\displaystyle
a_{p}^{\dagger}a_{q}\in\mathfrak{h_{i}}\;,\;\forall_{i=1,\ldots,M_{\rm
SES}}\;.$

The above construct, in contrast to the existing local CC formulations where
the one-particle reduced density matrices are postulated, allows one to
introduce them in a natural way. A more detailed analysis of the local
formulations stemming from the Equivalence Theorem is discussed in Refs.
kowalski2021dimensionality ; bauman2022coupled . This procedure can also be
extended to systems driven by different types of interactions, such as in
nuclear structure theory or quantum lattice models, where the extension of the
standard local CC formulations as used in quantum chemistry may be less
obvious.

###  III.2 Hermitian CC Flows

Using non-Hermitian formulation as a guide, the idea of quantum flow can be
generalized to the DUCC case. We start our analysis by assuming that we would
like to perform DUCC effective simulations for SES $\mathfrak{h}$ problem
expressed in Eq. (12) for an active space that is too big to be handled either
in classical or quantum computing. We will assume that external amplitudes
$\sigma_{\rm ext}(\mathfrak{h})$ can be effectively approximated. 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 $(P+Q_{\rm int}(\mathfrak{h}))$ space (in the simplest
case it can be just the $(P+Q_{\rm int}(\mathfrak{h}))H(P+Q_{\rm
int}(\mathfrak{h}))$ operator). We will denote $A(\mathfrak{h})$ simply by
$A$. Next, we assume that excitations in $\mathfrak{h}$ that are relevant to
the state of interest can be captured by excitation sub-algebras:
$\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$, $\ldots$, $\mathfrak{h}_{M}$, 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 simulations.

The $A(\mathfrak{h})$ Hamiltonian and the $(P+Q_{\rm int}(\mathfrak{h}))$
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.,

$A^{\rm eff}(\mathfrak{h}_{i})e^{\sigma_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle=Ee^{\sigma_{\rm
int}(\mathfrak{h}_{i})}|\Phi\rangle\;\;(i=1,\ldots,M)\;$ (25)

or in the VQE-type variational representation as

$\min_{{\bm{\theta}}(\mathfrak{h}_{i})}\langle\Psi({\bm{\theta}}(\mathfrak{h}_{i}))|A^{\rm
eff}(\mathfrak{h}_{i})|\Psi({\bm{\theta}}(\mathfrak{h}_{i}))\rangle\;\;(i=1,\ldots,M)\;,$
(26)

where $|\Psi({\bm{\theta}}(\mathfrak{h}_{i}))\rangle$ approximates
$e^{\sigma_{\rm int}(\mathfrak{h}_{i})}|\Phi\rangle$. Each $A^{\rm
eff}(\mathfrak{h}_{i})$ is defined as

$A^{\rm eff}(\mathfrak{h}_{i})=(P+Q_{\rm int}(\mathfrak{h}_{i}))\bar{A}_{\rm
ext}(\mathfrak{h}_{i})(P+Q_{\rm int}(\mathfrak{h}_{i}))$ (27)

and

$\bar{A}_{\rm ext}(\mathfrak{h}_{i})=e^{-\sigma_{\rm
ext}(\mathfrak{h}_{i})}Ae^{\sigma_{\rm ext}(\mathfrak{h}_{i})},$ (28)

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 to active spin orbitals defining $\mathfrak{h}$ and
not belonging to the set of active spin orbtitals defining
$\mathfrak{h}_{i}$). In other words, sub-algebras $\mathfrak{h}_{i}$ generate
active sub-spaces in the larger active space $\mathfrak{h}$, i.e., $(P+Q_{\rm
int}(\mathfrak{h}_{i}))\subset(P+Q_{\rm int}(\mathfrak{h}))$. Due to the non-
commutativity of components defining $\sigma$-operators, connecting DUCC
computational blocks given by Eq. (25) or Eq. (26) directly into a flow is a
rather challenging task. To address these issues (see Ref.
kowalski2021dimensionality ) 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.,

$\sigma_{\rm int}(\mathfrak{h})\simeq\sum_{i=1}^{M}\sigma_{\rm
int}(\mathfrak{h}_{i})+X(\mathfrak{h},\mathfrak{h}_{1},\ldots,\mathfrak{h}_{M})\;,$
(29)

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

$\sigma_{\rm int}(\mathfrak{h})=\sigma_{\rm
int}(\mathfrak{h}_{i})+R(\mathfrak{h}_{i})\;\;(i=1,\ldots,M)\;,$ (30)

where

$R(\mathfrak{h}_{i})=^{(i)}\sum_{j=1}^{M}\;\sigma_{\rm
int}(\mathfrak{h}_{j})+X$ (31)

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

$e^{\sigma_{\rm int}(\mathfrak{h})}|\Phi\rangle=e^{\sigma_{\rm
int}(\mathfrak{h}_{i})+R(\mathfrak{h}_{i})}|\Phi\rangle\;\;(i=1,\ldots,M)\;.$
(32)

Using the Trotter formula, we can approximate the right-hand side of Eq. (32)
for a given $j$ as

$e^{\sigma_{\rm
int}(\mathfrak{h})}|\Phi\rangle\simeq(e^{R(\mathfrak{h}_{i})/N}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N})^{N}|\Phi\rangle\;.$ (33)

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

$G^{(N)}_{i}=(e^{R(\mathfrak{h}_{i})/N}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N})^{N-1}e^{R(\mathfrak{h}_{i})/N}\;\;(i=1,\ldots,M)\;,$
(34)

the ”internal” wave function (32) can be expressed as

$e^{\sigma_{\rm int}(\mathfrak{h})}|\Phi\rangle\simeq
G^{(N)}_{i}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N}|\Phi\rangle\;\;(i=1,\ldots,M)\;,$ (35)

where $G^{(N)}_{i}$ 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, using this expression, one can define the practical way of
determining computational blocks for flow equations. To this end, let us
introduce the expansion in Eq. (35) to Eq. (12) (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

$(P+Q_{\rm int}(\mathfrak{h}_{i}))[G^{(N)}_{i}]^{-1}AG^{(N)}_{i}e^{\sigma_{\rm
int}(\mathfrak{h}_{i})/N}|\Phi\rangle\simeq Ee^{\sigma_{\rm
int}(\mathfrak{h})_{i})/N}|\Phi\rangle\;(i=1,\ldots,M)\;.$ (36)

These equations define computational blocks for the DUCC flow. To make
practical use of Eqs. (36) 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 in Eq. (36) by an
optimization procedures described by Eq. (26) which also offer an easy way to
deal with ”shared” amplitudes. Namely, if, in analogy to SR-CC flow, we
establish an ordering of $\mathfrak{h}_{i}$ sub-algebras, with
$\mathfrak{h}_{1}$ corresponding to the importance of active spaces with
respect to the wave function of interest. Then in the $\mathfrak{h}_{i}$
problem we partition set of parameters ${\bm{\theta}}_{N}(\mathfrak{h}_{i})$
into subset ${\bm{\theta}}_{N}^{\rm CP}(\mathfrak{h}_{i})$ that refers to
common pool of amplitudes determined in preceding steps (say, for
$\mathfrak{h}_{j}\;(j=1,\ldots,i-1)$) and subset ${\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i})$ that is uniquely determined in the $\mathfrak{h}_{i}$
minimization step, i.e,

$\min_{{\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i})}\langle\Psi({\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i}),{\bm{\theta}}_{N}^{\rm
CP}(\mathfrak{h}_{i}))|\Gamma_{i}^{(N)}(pc)|\Psi({\bm{\theta}}_{N}^{\rm
X}(\mathfrak{h}_{i}),{\bm{\theta}}_{N}^{\rm
CP}(\mathfrak{h}_{i}))\rangle\;\;(i=1,\ldots,M)\;,$ (37)

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 discussed
formalism introduces a broad class of control parameters defining 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.

An essential feature of the DUCC flow equation is associated with the fact
that each computational block (37) can be encoded using a much smaller number
of qubits compared to the qubits requirement associated with the original
problem. 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 (for
quantum algorithms exploiting locality of interactions, see Refs.
mcclean2014exploiting ; otten2022localized ).

##  IV Time-Dependent CC Extensions

The SES-CC-based downfolding techniques could also be extended to the time-
dependent domain.downfolding2020t ; bauman2022coupled  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 (we also assume that the employed
molecular orbitals are time-independent), i.e,

$|\Psi(t)\rangle=e^{T_{\rm ext}(\mathfrak{h},t)}e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,\forall\mathfrak{h}\in SES\;.$ (38)

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
(38) 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 $\mathfrak{h}$

$i\hbar\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h},t)e^{T_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,$ (39)

where

$H^{\rm eff}(\mathfrak{h},t)=(P+Q_{\rm int}(\mathfrak{h}))\bar{H}_{\rm
ext}(\mathfrak{h},t)(P+Q_{\rm int}(\mathfrak{h}))\;$ (40)

and

$\bar{H}_{\rm ext}(\mathfrak{h},t)=e^{-T_{\rm ext}(\mathfrak{h},t)}He^{T_{\rm
ext}(\mathfrak{h},t)}\;.$ (41)

If $T_{\rm ext}(\mathfrak{h},t)$ operator is known or can be efficiently
approximated, then the dynamics of the entire system can be described by
effective Hamiltonian $H^{\rm eff}(\mathfrak{h},t)$.

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 through a number of coupled reduced-dimensionality problems
(time-dependent quantum flows), i.e.,

$i\hbar\frac{\partial}{\partial t}e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle=H^{\rm eff}(\mathfrak{h}_{i},t)e^{T_{\rm
int}(\mathfrak{h}_{i},t)}|\Phi\rangle\;,\;(i=1,\ldots,M_{\rm SES})\;.$ (42)

Given analogies between stationary CC flow equations based on the localized
orbitals and local CC formulations developed in the last few decades in
quantum chemistry, time-dependent flow equations given by Eq. (42) can be
utilized to design reduced-scaling variants of the time-dependent CC
formulations.

The time-dependent variant of the DUCC Ansatz is represented by the normalized
time-dependent wave function $|\Psi_{\rm DUCC}(\mathfrak{h},t)\rangle$,

$|\Psi_{\rm DUCC}(\mathfrak{h},t)\rangle=e^{\sigma_{\rm
ext}(\mathfrak{h},t)}e^{\sigma_{\rm
int}(\mathfrak{h},t)}|\Phi\rangle\;,\forall\mathfrak{h}\in SES\;,$ (43)

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

$\displaystyle\sigma_{\rm int}(\mathfrak{h},t)^{\dagger}$ $\displaystyle=$
$\displaystyle-\sigma_{\rm int}(\mathfrak{h},t)\;,$ (44)
$\displaystyle\sigma_{\rm ext}(\mathfrak{h},t)^{\dagger}$ $\displaystyle=$
$\displaystyle-\sigma_{\rm ext}(\mathfrak{h},t)\;.$ (45)

Again, as in the SES-CC case, the dynamics of the entire system are given by
the active-space time-dependent effective Hamiltonian $H^{\rm eff}(\sigma_{\rm
ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm ext}(\mathfrak{h},t)}{\partial
t})$

$H^{\rm eff}(\sigma_{\rm ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm
ext}(\mathfrak{h},t)}{\partial t})=(P+Q_{\rm int})\\{\bar{H}_{\rm
ext}(\mathfrak{h},t)-i\hbar A(\sigma_{\rm
ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm ext}(\mathfrak{h},t)}{\partial
t})\\}(P+Q_{\rm int})\;.$ (46)

where anti-Hermitian operator $A(\sigma_{\rm
ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm ext}(\mathfrak{h},t)}{\partial
t})$ is expressed as

$A(\sigma_{\rm ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm
ext}(\mathfrak{h},t)}{\partial
t})=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)!}I_{k}(\sigma_{\rm
ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm ext}(\mathfrak{h},t)}{\partial
t})$ (47)

and

$I_{k}(X(\mathfrak{h},t),\frac{\partial X(\mathfrak{h},t)}{\partial
t})=\underbrace{[X(\mathfrak{h},t),[X(\mathfrak{h},t),\ldots[X(\mathfrak{h},t),[X(\mathfrak{h},t),\frac{\partial
X(\mathfrak{h},t)}{\partial t}]]\ldots]]}_{k\;{\rm times}}\;.$ (48)

If the fast-varying in time part of the wave function (or $\sigma_{\rm
ext}(\mathfrak{h},t)$-dependent part of the wave function) is known or can be
efficiently approximated, then the slow-varying dynamic (captured by the
proper choice of the active space and $\sigma_{\rm int}(\mathfrak{h},t)$
operator) of the entire system can be described as a sub-system dynamics
generated by the Hermitian $H^{\rm eff}(\sigma_{\rm
ext}(\mathfrak{h},t),\frac{\partial\sigma_{\rm ext}(\mathfrak{h},t)}{\partial
t})$ operator. This decoupling of various time regimes (slow- vs. fast-varying
components) is analogous to decoupling high- and low-energy Fermionic degrees
of freedom in stationary formulations of the SES-CC and DUCC formalisms.

##  V Green’s Function Applications

The CC Green’s function formulations have recently evolved into important
formulations to describe spectral functions in various energy regimes
nooijen92_55 ; nooijen93_15 ; meissner93_67 ; schirmer2004intermediate ;
mertins96_2140 ; kkgfcc2 ; kkgfcc3  and as high-accuracy solvers for quantum
embedding formulations. Following original formulations based on the CC bi-
variational approach, the corresponding frequency-dependent Green’s function
for an $N$-particle system can be expressed as

$\displaystyle G_{pq}(\omega)=$
$\displaystyle\langle\Phi|(1+\Lambda)e^{-T}a_{q}^{\dagger}(\omega+(H-E)-\text{i}\eta)^{-1}a_{p}e^{T}|\Phi\rangle+$
$\displaystyle\langle\Phi|(1+\Lambda)e^{-T}a_{p}(\omega-(H-E)+\text{i}\eta)^{-1}a_{q}^{\dagger}e^{T}|\Phi\rangle\;,$
(49)

where $\omega$ denotes the frequency parameter, and the imaginary part $\eta$
is often called a broadening factor. The cluster operator $T$ and de-
excitation operator $\Lambda$ define correlated ket ($|\Psi\rangle$) and bra
($\langle\Psi|$) ground-state wave functions for $N$-electron system

$\displaystyle|\Psi\rangle$ $\displaystyle=$ $\displaystyle
e^{T}|\Phi\rangle\;,$ (50) $\displaystyle\langle\Psi|$ $\displaystyle=$
$\displaystyle\langle\Phi|(1+\Lambda)e^{-T}\;.$ (51)

The ground-state energy $E_{0}$, and the amplitudes defining $T$ and $\Lambda$
operators are obtained from the following sequence of CC equations.

To combine GFCC and DUCC formalisms we replace $T$, $\Lambda$, and $H$
operators in Eq. (49) by cluster ($\tilde{T}_{\rm int}$), de-excitation
($\tilde{\Lambda}_{\rm int}$), and Hermitian $H^{\rm eff}(\mathfrak{h})$ [Eq.
(13), which is further denoted as $\Gamma$] operators acting in the some
active space generated by $\mathfrak{h}$ (for the notational simplicity we
also skip the $\mathfrak{h}$ symbol).baumanpengGFDUCC  We will also consider
the case when the set of active orbitals consists of all occupied orbitals and
a small subset of active virtual orbitals (containing $n_{v}^{\rm act}$ active
virtual orbitals), where, in general, $n_{v}^{\rm act}\ll n_{v}$, where
$n_{v}$ designates the total number of virtual orbitals. The standard CC
equations for $T$, CC energy $E$, and $\Lambda$ are are replaced by their
“active” counterparts

$\displaystyle Q_{\rm int}e^{-\widetilde{T}_{\rm int}}\Gamma
e^{\widetilde{T}_{\rm int}}|\Phi\rangle$ $\displaystyle=$ $\displaystyle
0\leavevmode\nobreak\ ,$ (52) $\displaystyle\langle\Phi|e^{-\widetilde{T}_{\rm
int}}\Gamma e^{\widetilde{T}_{\rm int}}|\Phi\rangle$ $\displaystyle=$
$\displaystyle E_{0}^{\rm int}\leavevmode\nobreak\ ,$ (53)
$\displaystyle\langle\Phi|(1+\widetilde{\Lambda}_{\rm
int})e^{-\widetilde{T}_{\rm int}}\Gamma e^{\widetilde{T}_{\rm int}}Q_{\rm
int}$ $\displaystyle=$ $\displaystyle E_{0}^{\rm
int}\langle\Phi|(1+\widetilde{\Lambda}_{\rm int})Q_{\rm
int}\leavevmode\nobreak\ .$ (54)

The coupled cluster Green’s function employing the DUCC Hamiltonian $\Gamma$
can be expressed for active orbitals as follows

$\displaystyle G_{PQ}^{\rm DUCC}(\omega)$ $\displaystyle=$
$\displaystyle\langle\Phi|(1+\widetilde{\Lambda}_{\rm
int})e^{-\widetilde{T}_{\rm int}}a_{Q}^{\dagger}(\omega+(\Gamma-E_{0}^{\rm
int})-\text{i}\eta)^{-1}a_{P}e^{\widetilde{T}_{\rm int}}|\Phi\rangle+$ (55)
$\displaystyle\langle\Phi|(1+\widetilde{\Lambda}_{\rm
int})e^{-\widetilde{T}_{\rm int}}a_{P}(\omega-(\Gamma-E_{0}^{\rm
int})+\text{i}\eta)^{-1}a_{Q}^{\dagger}e^{\widetilde{T}_{\rm
int}}|\Phi\rangle\;,$

where indices $P,Q,\ldots$ designate active spin orbitals. Again, applying the
resolution of identity $e^{-\widetilde{T}_{\rm int}}e^{\widetilde{T}_{\rm
int}}$ in the above equation, one gets the following expressions for DUCC
Green’s function matrix elements

$\displaystyle G_{PQ}^{\rm DUCC}(\omega)=$
$\displaystyle\langle\Phi|(1+\widetilde{\Lambda}_{\rm
int})\overline{a_{Q}^{\dagger}}^{\rm
int}(\omega+\overline{\Gamma}_{N}-\text{i}\eta)^{-1}\overline{a_{P}}^{\rm
int}|\Phi\rangle+$ (56) $\displaystyle\langle\Phi|(1+\widetilde{\Lambda}_{\rm
int})\overline{a_{P}}^{\rm
int}(\omega-\overline{\Gamma}_{N}+\text{i}\eta)^{-1}\overline{a_{Q}^{\dagger}}^{\rm
int}|\Phi\rangle\;,$

where we used the following definitions:

$\displaystyle\overline{\Gamma}$ $\displaystyle=$ $\displaystyle
e^{-\widetilde{T}_{\rm int}}\Gamma\leavevmode\nobreak\ e^{\widetilde{T}_{\rm
int}}\;,$ (57) $\displaystyle\overline{\Gamma}_{N}$ $\displaystyle=$
$\displaystyle\overline{\Gamma}-E_{0}^{\rm int}\;,$ (58)
$\displaystyle\overline{a_{P}}^{\rm int}$ $\displaystyle=$ $\displaystyle
e^{-\widetilde{T}_{\rm int}}a_{P}\leavevmode\nobreak\ e^{\widetilde{T}_{\rm
int}},$ (59) $\displaystyle\overline{a_{Q}^{\dagger}}^{\rm int}$
$\displaystyle=$ $\displaystyle e^{-\widetilde{T}_{\rm
int}}a_{Q}^{\dagger}\leavevmode\nobreak\ e^{\widetilde{T}_{\rm int}}.$ (60)

In the active-space driven DUCC-GFCC approach, the $X_{p}(\omega)$ and
$Y_{q}(\omega)$ operators are replaced by $X_{P}^{\rm int}(\omega)$ and
$Y_{Q}^{\rm int}(\omega)$, respectively, which are given by the following
expressions:

$\displaystyle X_{P}^{\rm int}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{I}x^{I}(P,\omega)^{\rm
int}a_{I}+\sum_{I<J,A}x^{IJ}_{A}(P,\omega)^{\rm
int}a_{A}^{\dagger}a_{J}a_{I}+\ldots\leavevmode\nobreak\ ,$ (61)
$\displaystyle Y_{Q}^{\rm int}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{A}y_{A}(Q,\omega)^{\rm
int}a_{A}^{\dagger}+\sum_{I,A<B}y^{I}_{AB}(Q,\omega)^{\rm
int}a_{A}^{\dagger}a_{B}^{\dagger}a_{I}+\ldots\leavevmode\nobreak\ ,$ (62)

where indices $I,J,\ldots$ and $A,B,\ldots$ refer to active occupied and
unoccupied spin orbitals indices, respectively (again, in the present
discussion, we assume that all occupied spin orbitals are treated as active).
These operators satisfy

$\displaystyle(\omega+\overline{\Gamma}_{N}-\text{i}\eta)X_{P}^{\rm
int}(\omega)|\Phi\rangle=\overline{a_{P}}^{\rm int}|\Phi\rangle\;,$ (63)
$\displaystyle(\omega-\overline{\Gamma}_{N}+\text{i}\eta)Y_{Q}^{\rm
int}(\omega)|\Phi\rangle=\overline{a_{Q}^{\dagger}}^{\rm int}|\Phi\rangle\;,$
(64)

and the $G_{PQ}^{\rm DUCC}(\omega$) is given by the expression

$\displaystyle G_{PQ}^{\rm DUCC}(\omega)=$
$\displaystyle\langle\Phi|(1+\Lambda_{\rm int})\overline{a_{Q}^{\dagger}}^{\rm
int}X_{P}^{\rm int}(\omega)|\Phi\rangle+$ (65)
$\displaystyle\langle\Phi|(1+\Lambda_{\rm int})\overline{a_{P}}^{\rm
int}Y_{Q}^{\rm int}(\omega)|\Phi\rangle\;.$

We demonstrated that the combined GFCC and DUCC frameworks reproduce the main
features of the standard GFCCSD spectral function. In a series of test
calculations, we demonstrated that increasing active space size leads to
monotonic improvements in the location of peaks obtained with the DUCC-GFCCSD
approach with respect to the full GFCCSD results. We attribute this behavior
to the presence of dynamical (out-of-active-space) correlation effects
encapsulated in each of DUCC effective Hamiltonians. In contrast to the DUCC-
GFCCSD formalism, the utilization of active space bare Hamiltonians leads to
less consistent results for the peak positions. The utilization of the DUCC
effective Hamiltonians can also significantly reduce the cost of the GFCC
calculations for the energy regime embraced by the corresponding active space.

##  VI Review of Applications

This section briefly reviews several exemplary application areas and numerical
studies involving various CC downfolding formalisms.

###  VI.1 Numerical Validation of the SES-CC Theorem

The SES-CC Theorem has recently been validated on the example of several
benchmark systems (H4, H6, H8 models) used to test CC methodologies in
situations corresponding to the presence of weak and strong correlation
effects (see Ref. sssc ). We numerically verified the SES-CC Theorem using
various active spaces corresponding to physically meaningful active spaces,
capturing the most important correlation for the ground-state wave function
description in the valence region as well as for the active spaces that are
remotely related to the ground-state correlation effects. To this end, we used
two approaches, CCSD and CCSDTQ, to calculate the eigenvalues of
effective/downfolded Hamiltonians. In all cases considered in Ref. sssc , we
were able to reproduce the CCSD or CCSDTQ energies, obtained using the
standard expression for the energy, as lowest-energy eigenvalues of the
effective Hamiltonian. In the extreme case, we used a spin-orbital-based
definition of active to correlate a single electron, which also resulted in
reproducing standard CC energies.

###  VI.2 Approximations Based on Quantum Flows

In Ref. safkk , we introduced QFs based on the ordered flow of the
$\mathfrak{g}^{(N)}(2_{R})$ sub-algebras:

$\mathfrak{g}^{(N)}(2_{R_{1}})\stackrel{{\scriptstyle\mathrm{\rm
passingT}}}{{\longrightarrow}}\mathfrak{g}^{(N)}(2_{R_{2}})\stackrel{{\scriptstyle\mathrm{\rm
passingT}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle\mathrm{\rm
passingT}}}{{\longrightarrow}}\mathfrak{g}^{(N)}(2_{R_{\rm final}})$ (66)

where SESs are ordered according to some importance criterium, for example,
corresponding to a descending order with respect to the sum of orbital
energies ($\epsilon_{k_{i}}+\epsilon_{l_{i}}$) corresponding to orbitals
included in $R_{i}$ sets. The Equivalence Theorem states that such a flow (in
the discussed case, all $\mathfrak{g}^{(N)}(2_{R})$-generated active space
problems are integrated) is equivalent to the CC formalism defined by the
following cluster operator $T$:

$T\simeq T_{1}+T_{2}+\sum_{R}T_{\rm
int,3}(\mathfrak{g}^{(N)}(2_{R}))+\sum_{R}T_{\rm
int,4}(\mathfrak{g}^{(N)}(2_{R}))\;.$ (67)

This approach is further referred to as the self-consistent sub-algebra flow
CC method (SCSAF-CC). In contrast to the class of the so-called active-space
CC approaches (see Refs. piecuchactive  and references therein), this
formulation includes classes of triply and quadruply excitations (in addition
to all possible single and double excitations) corresponding to triply and
quadruply excited amplitudes corresponding to active-space problems included
in the flow.

The performance of the SCSAF-CC methods was evaluated on the examples
involving single and double bond-breaking processes. For example, for the F2
benchmark system, the non-parallel error (NPE) of the SCSAF-CC formalism given
by Eq. (67) in describing ground-state potential energy surface is comparable
to the NPSs yielded by the 4-reference reduced multi-reference CCSD(T)
approach (see Ref. safkk  for more details). As demonstrated in Ref. safkk ,
the SCSAF-CC formalism provided an efficient way for perturbative corrections
due to triple or quadruple excitations not included in the iterative SCSAF-CC
formulation given by Eq. 67). It was shown that perturbative SCSAD-CC methods
could bypass typical problems with perturbative inclusion of higher-rank
clusters in situations with strong correlation effects.

Similarly to the ground-state case, the SCSAF-CC methods can be extended to
their equation-of-motion CC (EOMCC) bartlett89_57 ; bartlett93_414 ;
stanton93_7029  formulations using a similar manifold of excitations as in the
ground-state case. For example, the state-specific excitation operator for
$K$-th state $X(K)$ corresponding to the $\mathfrak{g}^{(N)}(2_{R})$ flow as
in Eq. (67) is given by the expansion:

$\displaystyle X(K)$ $\displaystyle\simeq$ $\displaystyle
X(K)_{0}+X(K)_{1}+X(K)_{2}+\sum_{R}X(K)_{\rm
int,3}(\mathfrak{g}^{(N)}(2_{R}))$ (68) $\displaystyle+\sum_{R}X(K)_{\rm
int,4}(\mathfrak{g}^{(N)}(2_{R}))\;.$

These EOMCC-type extensions have been shown to properly capture excited-state
correlation effects for singly excited states and a more challenging class of
excited states dominated by double excitations.

###  VI.3 Quantum Computing

The Hermitian CC downfolding plays a vital role in realizing quantum computing
applications in computational chemistry with limited resources defined by
Noisy Intermediate-State Quantum (NISQ) devices. In particular, a significant
effort has been expanded to provide frameworks that significantly reduce the
size of the virtual space. Using these techniques, we could adequately
reproduce total ground-state energies for systems described by 50-70 molecular
basis functions employing small-size active spaces (5-15 molecular orbitals)
and various types of solvers, including VQE and QPE quantum algorithms. One
should stress that quantum simulations for systems described by 50-70 orbitals
are currently beyond reach, which is a net effect of the required logical
qubits, quantum errors, quantum circuit depth, and large numbers of fermionic
degrees of freedom (amplitudes) to be included to achieve the necessary level
of accuracy. The last factor translates into a massive number of quantum
measurements concerning VQE class of methods.

Special consideration is required when constructing approximate downfolded
Hamiltonians is required to enable downfolding methods in quantum computing.
The key factors that need to be taken care of when approximating infinite
expansions for the $H^{\rm eff}(\mathfrak{h})$ operator, Eq. (14), are as
follows:

  * •

Rank of the commutator expansion. All approximations introduced in past years
are based on finite-rank approximations. In the most accurate approximations,
first-, second-, and classes of third-rank commutators are included.

  * •

Source of the $\sigma_{\rm ext}(\mathfrak{h}$ amplitudes. As in the Eq. (16),
the $\sigma_{\rm ext}(\mathfrak{h})$ are approximated in a UCC way. In
practical realizations considered so far, $T_{\rm ext}(\mathfrak{h})$ are
extracted from the converged CCSD amplitudes.

  * •

Rank of many-body interactions in $H^{\rm eff}(\mathfrak{h})$. Currently,
downfolded Hamiltonians are constructed to include one- and two-body effective
interactions.

  * •

Perturbative consistency. Many-body perturbation theory (MBPT) allows one to
better balance the correlation effects in the expansion in Eq. (14) (see
extensive discussion of MBPT expansions for UCC theories in Refs. unitary1 ;
unitary2 ). For perturbative ”consistency” in some cases, we include Fock
operator ($F_{N}$) dependent terms.

Various approximations for downfolded Hamiltonians discussed in Ref.
bauman2022coupled2c  are collected in Table 1.

Table 1: Various approximations for the $\bar{H}_{\rm ext}$ operator. Special notation is used to designate the perturbative structure of expansions employed. For example, $X^{(2)}$ and $X^{(3)}$ terms on the right-hand side designate parts of the operator $X$ correct up to the second and third order of MBPT. For the case of the natural orbitals, the same expressions are used with the full non-diagonal form of the Fock operator $F_{N}$. The normal product form of the electronic Hamiltonian $H_{N}$ is defined as $H_{N}=H-\langle\Phi|H|\Phi\rangle$.  A(1) | $\bar{H}_{\rm ext}^{\rm A(1)}=H$
---|---
A(2) | $\bar{H}_{\rm ext}^{\rm A(2)}=H+[H_{N},\sigma_{\rm ext}]^{(2)}+\frac{1}{2}[[F_{N},\sigma_{\rm ext}],\sigma_{\rm ext}]^{(2)}$
A(3) | $\bar{H}_{\rm ext}^{\rm A(3)}=H+[H_{N},\sigma_{\rm ext}]$
A(4) | $\bar{H}_{\rm ext}^{\rm A(4)}=H+[H_{N},\sigma_{\rm ext}]+\frac{1}{2}[[F_{N},\sigma_{\rm ext}],\sigma_{\rm ext}]$
A(5) | $\bar{H}_{\rm ext}^{\rm A(5)}=H+[H_{N},\sigma_{\rm ext}]^{(3)}+\frac{1}{2}[[H_{N},\sigma_{\rm ext}],\sigma_{\rm ext}]^{(3)}$
| $+\frac{1}{6}[[[F_{N},\sigma_{\rm ext}],\sigma_{\rm ext}],\sigma_{\rm
ext}]^{(3)}$
A(6) | $\bar{H}_{\rm ext}^{\rm A(6)}=H+[H_{N},\sigma_{\rm ext}]+\frac{1}{2}[[H_{N},\sigma_{\rm ext}],\sigma_{\rm ext}]$
A(7) | $\bar{H}_{\rm ext}^{\rm A(7)}=H+[H_{N},\sigma_{\rm ext}]+\frac{1}{2}[[H_{N},\sigma_{\rm ext}],\sigma_{\rm ext}]$
| $+\frac{1}{6}[[[F_{N},\sigma_{\rm ext}],\sigma_{\rm ext}],\sigma_{\rm ext}]$

Obtaining Hermitian downfolded Hamiltonians, especially those including
higher-rank commutators, is usually associated with the inclusion of hundreds
or thousands of Hugenholtz-type diagrams, which requires developing and
utilizing specialized symbolic tools to derive and efficiently implement the
corresponding algebraic expressions.

The Hermitian versions of the downfolded Hamiltonians have been integrated
with various VQE and QPE solvers. The efficiency of these workflows has been
illustrated in the example of bond-stretching processes for typical benchmark
systems such as H2, LiH, Li2, N2, H2O, and C2H4 systems.metcalf2020resource ;
bauman2020variational ; bauman2022coupled2c  In all cases, downfolded
Hamiltonians significantly improve the accuracy of diagonalization of the bare
Hamiltonians in active spaces and provides results much closer to the exact
(or nearly exact) results obtained when all orbitals are correlated.

##  VII Conclusions

The CC downfolding techniques are a relatively new tool to analyze/derive new
properties of CC methods. One of the most appealing ones is the possibility of
calculating CC energies through the diagonalization of the
effective/downfolded Hamiltonians in broad classes of active spaces
corresponding to standard CC approximations. These observations have been
extended to the time-domain and quantum flows, which provide an alternative
way to rigorously encapsulate the inherent sparsity/sparsities of quantum
systems.

Aside from interesting new properties of CC methodology, the downfolding
techniques have been used as a design principle to select classes of
amplitudes in the vain of the Equivalence Theorem. These methods proved
efficient in treating strong correlation effects in ground and excited states.

The Hermitian form of the downfolding formalism is a promising extension for
quantum computing applications. Due to the possibility of reducing the
dimensionality of the quantum problem, it enables simulations for effective
representations of Hamiltonians in basis sets that would be beyond the reach
of current simulators and hardware if direct approaches are used. Since the
current quantum algorithm, such as the VQE methodologies, can effectively
handle only a relatively small number of wave function parameters, usually
corresponding to the so-called static correlation effects, the active-space-
driven downfolding is a potential tool to extend the area of applications to
larger systems and larger basis sets. As a next frontier, our group is
intensively developing quantum algorithms based on the unitary CC flow
equations. These methods can traverse larger sub-spaces of Hilbert spaces than
currently possible using modest quantum computing resources associated with
the size of the maximum active space involved in the flow. This is a
consequence of the fact that in the global representation of the quantum
problem, requiring a full qubit register to assure the antisymmetry of the
wave function, is replaced/approximated by flows (or computable reduced
dimensionality eigenvalue problems) where global antisymmetry problems no
longer exist.

New computational paradigms associated with the emergence and broad
utilization of machine learning techniques offer an exciting avenue for
utilizing Hermitian CC downfolding to extract the analytical form of effective
inter-electron interactions. These ”phenomenological” interactions are ideal
candidates to be integrated with the low-rank formulations such as
Hartree–Fock, Density Functional Theory, and various types of multi-
configurational self-consistent field methods as described in Ref.
bauman2022coupled .

##  VIII Acknowledgement

The main part of 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). NPB and BP acknowledge the support from “Embedding Quantum
Computing into Many-body Frameworks for Strongly Correlated Molecular and
Materials Systems” project, which is funded by the U.S. Department of Energy
(DOE), Office of Science, Office of Basic Energy Sciences, the Division of
Chemical Sciences, Geosciences, and Biosciences. All work was performed at
Pacific Northwest National Laboratory (PNNL) operated for the U.S. Department
of Energy by the Battelle Memorial Institute under Contract DE-
AC06-76RLO-1830. One of the authors of this review (KK) would like to express
his deep gratitude to all his colleagues, friends, and mentors he had the
honor to work with and learn from during his early ”CC days.” Countless
discussions with the quantum chemistry pioneers in Poland have inspired part
of the presented material.

## References

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# On Implications of Scaling Laws on Feature Superposition

Pavan Katta∗

###### Abstract

Using results from scaling laws, this theoretical note argues that the
following two statements cannot be simultaneously true:

  1. 1.

Superposition hypothesis where sparse features are linearly represented across
a layer is a complete theory of feature representation.

  2. 2.

Features are universal, meaning two models trained on the same data and
achieving equal performance will learn identical features.

11footnotetext: Correspondence to<EMAIL_ADDRESS>
##  1 Introduction

Scaling laws for language models give us a relation for a model’s macroscopic
properties such as cross entropy loss $L$, Amount of Data $D$ used and Number
of non-embedding parameters $N$ in the model (Kaplan et al., 2020).

$L(N,D)=\left[\left(\frac{N_{c}}{N}\right)^{\frac{\alpha_{N}}{\alpha_{D}}}+\left(\frac{D_{c}}{D}\right)^{\alpha_{D}}\right]^{\alpha_{D}}$
(1)

where $N_{c}$, $D_{c}$, $\alpha_{N}$, and $\alpha_{D}$ are constants for a
given task such as language modeling.

The scaling laws are not mere empirical observations and can be seen as
predictive laws on limits of language model performance. During training of
GPT-4, OpenAI was able to predict the final loss of GPT-4 early in the
training process using scaling laws with high accuracy (OpenAI, 2023).

An important detail is that the relation is expressed in terms of the number
of parameters. It’s natural to think of a model’s computational capacity in
terms of parameters, as they are the fundamental independent variables that
the model can tune during learning. The amount of computation that a model
performs in FLOPs for each input is also estimated to be $2N$ (Kaplan et al.,
2020).

Let’s compare this with Interpretability, where the representation of a
feature is defined in terms of neurons or groups of neurons. At first glance,
it might seem unnecessary to distinguish between computational capacity and
feature representational capacity, as parameters are connections between
neurons after all. However, we can change the number of neurons in a model
while keeping the number of parameters constant. Kaplan et al. found that
Transformer performance depends very weakly on the shape parameters
$n_{layer}$ (number of layers), $n_{heads}$ (number of attention heads), and
$d_{ff}$ (feedforward layer dimension) when we hold the total non-embedding
parameter count $N$ fixed (Kaplan et al., 2020). The paper reports that the
aspect ratio (the ratio of number of neurons per layer to the number of
layers) can vary by more than an order of magnitude, with performance changing
by less than 1

In this paper, we assume the above to be true and consider the number of
parameters to be the true limiting factor, and we can achieve similar model
performance for a range of aspect ratios. We then apply this as a postulate to
the superposition hypothesis, our current best and successful theory of
feature representation, and explore the implications.

The superposition hypothesis states that models can pack more features than
the number of neurons they have (Elhage et al., 2022). There will be
interference between the features as they can’t be represented orthogonally,
but when the features are sparse enough, the benefit of representing a feature
outweighs the cost of interference. Concretely, given a layer of activations
of $m$ neurons, we can decompose it linearly into activations of $n$ features,
where $n>m$, as:

$activation_{layer}=x_{f_{1}}W_{f_{1}}+x_{f_{2}}W_{f_{2}}+\cdots+x_{f_{n}}W_{f_{n}}$
(2)

where $activation_{layer}$ and $W_{f_{i}}$ are vectors of size $m$, and
$x_{f_{i}}$ represents the magnitude of activation of the $i$-th feature.
Sparsity means that for a given input, only a small fraction of features are
active, which means $x_{f_{i}}$ is non-zero for only a few values of $i$.

##  2 Case study on changing Aspect Ratio

Table 1: Macroscopic Properties of Transformer Models with Different Aspect Ratios but Equal Parameters | Model A | Model B
---|---|---
Total Parameters | $d_{model}^{2}n_{layer}$ | $d_{model}^{2}n_{layer}$
Neurons per Layer | $d_{model}$ | $2d_{model}$
Number of Layers | $n_{layer}$ | $\frac{n_{layer}}{4}$
Total Number of Neurons | $d_{model}n_{layer}$ | $\frac{d_{model}n_{layer}}{2}$
Total Number of Features Learned | $F$ | $F$
Number of Features per Layer | $\frac{F}{n_{layer}}$ | $\frac{4F}{n_{layer}}$
Features per Neuron | $\frac{F}{d_{model}n_{layer}}$ | $\frac{2F}{d_{model}n_{layer}}$

Let’s consider two Transformer models, Model A and Model B, having the same
macroscopic properties. Both have an equal number of non-embedding parameters,
are trained on the same dataset, and achieve similar loss according to scaling
laws (Kaplan et al., 2020). However, their shape parameters differ. Using the
same notation as Kaplan et al., let’s denote the number of layers as
$n_{layer}$, and number of neurons per layer as $d_{model}$. Model B has twice
the number of neurons per layer compared to A. As the number of parameters is
approximated by $d_{model}^{2}n_{layer}$, Model B must have $\dfrac{1}{4}$ the
number of layers to maintain the same number of parameters as Model A. This
means Model B has 8 times the aspect ratio ($\dfrac{d_{model}}{n_{layer}}$) of
A which falls under the reported range in Kaplan et al.

The total number of neurons in a model is calculated by multiplying the number
of neurons per layer by the number of layers. As a result, Model B has half
the total number of neurons compared to Model A.

Now, let’s apply the superposition hypothesis, which states that features can
be linearly represented in each layer. Since both models achieve equal loss on
the same dataset, it’s reasonable to assume that they have learned the same
features (Olah et al., 2020). Let’s denote the total number of features
learned by both models as $F$.

The above three paragraphs are summarized in Table 1.

The average number of features per neuron is calculated by dividing the number
of features per layer by the number of neurons per layer. In Model B, this
value is twice as high as in Model A, which means that Model B is effectively
compressing twice as many features per neuron, in other words, there’s a
higher degree of superposition. However, superposition comes with a cost of
interference between features, and a higher degree of superposition requires
more sparsity.

Elhage et al.(Elhage et al., 2022) show that, using lower bounds of compressed
sensing (Ba et al., 2010), if we want to recover $n$ features compressed in
$m$ neurons (where $n>m$), the bound is

$m=\Omega(-n(1-S)\log(1-S)),$ (3)

where $1-S$ is the sparsity of the features. For example, if a feature is non-
zero only 1 in 100 times, then $1-S$ equals 0.01. We can define the degree of
superposition as

$\frac{n}{m}=\frac{1}{(1-S)\log(1-S)}$ (4)

which is a function of sparsity, inline with our theoretical understanding.

So Model B, with higher degree of superposition, should have sparser features
compared to Model A. But, sparsity of a feature is a property of the data
itself, and the same feature can’t be sparser in Model B if both models are
trained on the same data. This might suggest that they are not the same
features, which breaks our initial assumption of two models learning the same
features. So either our starting assumption of feature representation through
superposition or feature universality needs revision. In the next section, we
discuss how we might modify our assumptions.

##  3 Discussion

To recap, we started with the postulate that model performance is invariant
over a wide range of aspect ratios and arrived at the inconsistency between
superposition and feature universality. Though we framed the argument through
the lens of superposition, the core issue is that the model’s computational
capacity is a function of parameters whereas the model’s representational
capacity is a function of total neurons.

A useful, though non-rigorous analogy, is to visualize a solid cylinder of
radius $d_{model}$ and height $n_{layer}$. The volume (parameters) of the
cylinder can be thought of as computational capacity whereas features are
represented on the surface (neurons). We can change the aspect ratio of the
cylinder while keeping the volume constant by stretching or squashing it. This
changes the surface area accordingly. Though this analogy doesn’t include
sparsity, it captures the essentials of the argument in a simple way.

Figure 1: A visual representation of the cylinder analogy, showing the
relationship between the number of neurons per layer and the number of layers.
The volume of the cylinder represents the total number of parameters, while
the surface area corresponds to the total number of neurons available for
feature representation.

Coming to solutions, I do not have one that’s consistent with scaling laws,
superposition hypothesis, and feature universality, but will speculate on what
a possible one might look like.

###  3.1 Schemes of Compression Alternative to Superposition

A crude and simple way to convert the total number of features into a function
of parameters is to add a square term to compressed sensing bounds so it
becomes $n=m^{2}.f(1-S)$. But this would require a completely new compression
scheme compared to superposition. Methods such as Dictionary learning which
disentangle features assuming superposition hypothesis have been successful
for extracting interpretable features (Bricken et al., 2023). So it’s not
ideal to ignore it, representation schemes whose first-order approximation
looks like superposition might be more viable.

This isn’t to say there’s nothing we can improve on in the superposition
hypothesis. Although dictionary learning features in Bricken et al. are much
more mono-semantic than individual neurons, the lower activation levels in
these features still look quite polysemantic.

###  3.2 Cross Layer Superposition

Previously, we used to look for features in a single neuron (Radford et al.,
2017), now we extended it to a group of neurons in a layer. A natural
progression is to look for features localizing to neurons across multiple
layers. But Model B from the above section, has half the number of neurons as
A and the same inconsistencies would arise if features grow linearly on the
number of neurons. Number of features represented across two or more layers by
cross-layer superposition should grow superlinearly if Model B were to
compensate for fewer neurons and still have the same representational
capacity.

## Acknowledgements

I’m thankful to Jeffrey Wu and Tom McGrath for their helpful feedback on this
topic. Thanks to Vinay Bantupalli for providing feedback on the draft. Earlier
version of this work was supported by an Open Philanthropy research grant.

## References

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