Dimensionality reduction of many-body problem using coupled-cluster sub-system flow equations: classical and quantum computing perspective
DDimensionality reduction of many-body problem using coupled-clustersub-system flow equations: classical and quantum computing perspective
Karol Kowalski a) Physical Sciences Division, Pacific Northwest National Laboratory, Richland, WA 99354, USA (Dated: February 12, 2021)
We discuss reduced-scaling strategies employing recently introduced sub-system embedding sub-algebras coupled-cluster formalism (SES-CC) to describe many-body systems. These strategies utilize properties of the SES-CCformulations where the equations describing certain classes of sub-systems can be integrated into a computationalflows composed coupled eigenvalue problems of reduced dimensionality. Additionally, these flows can bedetermined at the level of the CC Ansatz by the inclusion of selected classes of cluster amplitudes, which definethe 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 theinformation is passed between the sub-systems in a self-consistent manner. As a special case, we consider localflow formulations where the so-called local character of correlation effects can be closely related to properties ofsub-system embedding sub-algebras employing localized molecular basis. We also generalize flow equations tothe time domain and to downfolding methods utilizing double exponential unitary CC Ansatz (DUCC), wherereduced dimensionality of constituent sub-problems offer a possibility of efficient utilization of limited quantumresources in modeling realistic systems.
I. INTRODUCTION
Over the last few decades, the coupled-cluster (CC)theory has evolved into one of the most accurateand dominant theory to describe various quantum sys-tems across spatial scales hence addressing fundamen-tal problems in nuclear structure theory, quantumchemistry, and material sciences.
Many strengthsof the single-reference CC formalism (SR-CC) originatesin the exponential parametrization of the ground-state wavefunction | Ψ (cid:105) | Ψ (cid:105) = e T | Φ (cid:105) , (1)where T and | Φ (cid:105) correspond to cluster operator and ref-erence function. For example, one can define a hierar-chy of CC approximations by increasing the rank of exci-tations included the cluster operator. Another importantfeature of CC stems from the linked cluster theorem which allows one to build efficient non-iterative algorithmsfor higher-order excitations. When both approximationtechniques are combined, one can define efficient and ac-curate methodologies that can deliver a high-level of ac-curacy in chemical simulations. More recently, CCmethodologies have been integrated with stochastic MonteCarlo methods probing configurational space and leadingto near full-configuration-interaction accuracy of calculatedenergies.
However, the applicability of canonical for-mulations of these theories (especially to large molecularsystems) may be limited by their steep (polynomial) nu-merical scaling. Unfortunately, even with rapid progressin computational technologies, problems with data local-ity, data movement, and polynomial scaling of high-rankcanonical CC methods lead to insurmountable numericalproblems in modeling large systems. Although impressiveprogress has been achieved in the development of localapproaches for CC pair theories, extension of these a) Electronic mail: [email protected] methods to include higher-rank excitations may still requirea significant theoretical effort. Some of these problems maybe addressed by using the mature form of quantum comput-ers; however, due to the limited size of existing quantumregisters this can only be achieved by developing flexible al-gorithms to reduce the dimensionality of quantum problem.These problems have been scrutinized only recently, includ-ing local and reduced-dimensionality quantum computingformulations.
In the light of the above discussion, new high-accuracyCC-based techniques for re-representing quantum many-body problem in reduced-dimensionality spaces are in highdemand. Especially interesting are approaches where theoriginal high-dimensionality problem can be recast in theform of coupled low-dimensionality problems. Also, forquantum computing algorithms, the dimensions of sub-problems coupled into a flow should be tunable to the avail-able quantum computing (QC) resources to provide, bycontrolling the number of parameters processed at a giventime, optimal utilization of computational tools such asVariational Quantum Eigensolvers (VQE).
Addition-ally, recent strides made in the development of unitary CCformulations such as their disentangled and adap-tive variants provide tools not only for next-generationVQE-type solvers but also for unlocking properties of uni-tary CC formulations needed in the analysis of reduced-dimensionality methods .In this paper, we will focus on the further extension of re-cently introduced CC sub-system embedding sub-algebrasCC (SES-CC) and double unitary CC downfolding meth-ods (DUCC). In a natural way, these methods allowto calculate ground-state energies as eigenvalues of effec-tive Hamiltonians in pre-defined active spaces describingsub-systems of the whole quantum system. Since in the con-struction of effective Hamiltonians all out-of-active-spacecorrelation effects are integrated out, the CC downfoldingprocedures can be viewed as a natural renormalization tech-niques. The flow equations for single-reference SES-CCcase, utilize this property for each active space involved a r X i v : . [ qu a n t - ph ] F e b in the flow and for this reason can be considered as formula-tions that ”hold the memory” about possible sub-system par-titioning. We will illustrate the ability of these approachesto capture complicated correlation effect and dynamics ofthe system, through traversing large sub-spaces of the en-tire Hilbert space without an unnecessary increase of thesize of the numerical problem to be solved at a given timein the flow algorithm. We will also show that it is possi-ble to define flows that decouple the representation of theSchr¨odinger equation in large sub-spaces of the Hilbert (of-ten defined by net dimensions beyond classical/quantumcomputing capabilities) into smaller problems that are nu-merically tractable.The paper focuses on two essential aspects of SES-CC/DUCC formulations: First, we demonstrate that someproperties of currently developed local CC formulationsare natural consequences of the CC flow memory aboutthe sub-system partitioning. In particular, we show thatwhen the localized occupied molecular basis is used, theproposed formalism introduces in a natural way conceptsof pairs, pairs densities, and higher-rank pair-related exci-tation manifold. This analysis will also demonstrate thatthe local CC formulations are particular manifestations ofmore fundamental properties of the CC theory. Second, theproperties mentioned above beget question if similar typesof flows can be mirrored for quantum computing wherestandard single-reference SES-CC flow is replaced by theflow based on the recently introduced double unitary CCexpansions. In this paper, we discuss UCC flows defined bycoupling Hermitian eigenvalue problems of reduced dimen-sionality (corresponding to various effective/downfoldedHamiltonians) to effectively traverse large sub-spaces ofthe entire Hilbert space. This process allows one to performquantum simulations for larger systems and to simplify thequbit representation of corresponding Hamiltonians. Sincethe operator algebra involved in the unitary CC methodsis non-commutative (in the sense of particle-hole formal-ism), extending canonical SES-CC flows to the DUCC caserequires the utilization of simple approximations. Two ofthese approximations are referred to as the (1) universalamplitude interpretation and (2) backward-type methodsbased on the use of approximate Trotter formulas.We will also discuss the difference between two compu-tational strategies involving (1) standard approximationsbased on the selection of cluster amplitudes and treatingthem simultaneously (or globally) in numerical implemen-tations and (2) flow equations where only a portion of se-lected amplitudes are processed at the time. While theformer computing approach can take advantage of paral-lel classical architectures, the latter is ideally suited forNoisy Intermediate-Scale Quantum devices (NISQ), wherea small subset of fermionic degrees of freedom can be ef-fectively handled. The flow equation methods also providea conceptual foundation for introducing certain approxima-tions classes and eliminating possible problems with theirpostulatory character. We will illustrate these advantageson the example of local CC methods.For simplicity, in this paper we will focus on the CC andDUCC flow equations for closed-shell systems. II. SUB-SYSTEM EMBEDDING SUB-ALGEBRAS CCFORMALISM - STATIONARY ANDTIME-DEPENDENT FORMULATIONS
The SES CC formalism is based on the observation thatenergy of CC formulations, E CC , in addition to well knowformula (cid:104) Φ | e − T He T | Φ (cid:105) (where H represents many-bodyHamiltonian) can be obtained through diagonalization ofthe whole family of the effective Hamiltonians. In the exact wave function limit, the maximum excita-tion level m included in the cluster operator T is equalto the number of correlated electrons ( N ) while in the ap-proximate formulations m (cid:28) N . Several typical examplesare CCSD ( m = CCSDT ( m = , and CCSDTQ( m = methods. Using the second quantization lan-guage, the T k components of cluster operator producing k -tuply excitations when acting on the reference functioncan be expressed as T k = ( k ! ) ∑ i ,..., i k ; a ... a k t i ... i k a ... a k E a ... a k i ... i k , (2)where indices i , i , . . . ( a , a , . . . ) refer to occupied (unoc-cupied) spin orbitals in the reference function | Φ (cid:105) . Theexcitation operators E a ... a k i ... i k are defined through strings ofstandard creation ( a † p ) and annihilation ( a p ) operators E a ... a k i ... i k = a † a . . . a † a k a i k . . . a i , (3)where creation and annihilation operators satisfy the fol-lowing anti-commutation rules [ a p , a q ] + = [ a † p , a † q ] + = , (4) [ a p , a † q ] + = δ pq . (5)The SES-CC approach is based on the particle-hole (p-h)formalism defined with respect to the reference function | Φ (cid:105) , where quasi-operators b p and b † p are defined as b p = (cid:40) a p if p ∈ Va † p if p ∈ O (6)and b † p = (cid:40) a † p if p ∈ Va p if p ∈ O , (7)where O and V designate sets of occupied and unoccupiedspin orbitals. Using the p-h formalism we have b p | Φ (cid:105) = , (8)and E a ... a k i ... i k = b † a . . . b † a k b † i k . . . b † i . (9)Additionally, b p / b † q satisfy the same anti-commutation re-lations as a p / a † q operators, i.e., [ b p , b q ] + = [ b † p , b † q ] + = , (10) [ b p , b † q ] + = δ pq . (11)The p-h formalism significantly simplifies the analysis ofthe CC equations. It is also easy to notice that all excitationoperators (3) commute, i.e. for E a ... a k i ... i k = a † a . . . a † a k a i k . . . a i = b † a . . . b † a k b † i k . . . b † i , (12) E c ... c m j ... j m = a † c . . . a † c m a j m . . . a j = b † c . . . b † c m b † j m . . . b † j , (13)we have [ E a ... a k i ... i k , E c ... c m j ... j m ] = . (14)After substituting Ansatz (1) into the Schr¨odinger equa-tion one gets the energy-dependent form of the CC equa-tions: ( P + Q ) He T | Φ (cid:105) = E ( P + Q ) e T | Φ (cid:105) , (15)where P and Q are projection operators onto the refer-ence function ( P = | Φ (cid:105)(cid:104) Φ | ) and onto excited configurations(with respect to | Φ (cid:105) ) generated by the T operator whenacting onto the reference function, Q = m ∑ k = ∑ i < i <...< i k ; a < a ...< a k | Φ a ... a k i ... i k (cid:105)(cid:104) Φ a ... a k i ... i k | , (16)where | Φ a ... a k i ... i k (cid:105) = E a ... a k i ... i k | Φ (cid:105) . (17)Diagrammatic analysis leads to an equivalent ( at the so-lution ), energy-independent form of the CC equations forcluster amplitudes Qe − T He T | Φ (cid:105) = Q ( He T ) C | Φ (cid:105) = , (18)and energy expression E = (cid:104) Φ | e − T He T | Φ (cid:105) = (cid:104) Φ | ( He T ) C | Φ (cid:105) , (19)where C designates a connected part of a given operatorexpression. In the forthcoming discussion, we refer to e − T He T as a similarity transformed Hamiltonian ¯ H .The SES-CC formalism hinges upon the notion of excita-tion sub-algebras of algebra g ( N ) generated by E a l i l = b † a l b i l operators in the particle-hole representation defined withrespect to the reference | Φ (cid:105) . As consequence all generatorscommute, i.e., [ E a l i l , E a k i k ] = g ( N ) (along withall sub-algebras considered here) is commutative. The SES-CC formalism utilizes an important class of sub-algebrasof commutative g ( N ) algebra, which contain all possibleexcitations E a ... a m i ... i m that excite electrons from a subset ofactive occupied orbitals (denoted as R ) to a subset of activevirtual orbitals (denoted as S ). These sub-algebras will bedesignated as g ( N ) ( R , S ) . In the following discussion, wewill use R and S notation for subsets of occupied and vir-tual active orbitals { R i , i = , . . . , x } and { S i , i = , . . . , y } ,respectively (sometimes it is convenient to use alternativenotation g ( N ) ( x R , y S ) where numbers of active orbitals in R and S orbital sets, x and y , respectively, are explicitly calledout). Of special interest in building various approximationswere sub-algebras that include all n v virtual orbitals ( y = n v ) - these sub-algebras will be denoted as g ( N ) ( x R ) . As dis-cussed in Ref. configurations generated by elements of g ( N ) ( x R , y S ) along with the reference function span the com-plete active space (CAS) referenced to as the CAS( R , S ) (oreqivalently CAS( g ( N ) ( x R , y S ) )).Each sub-algebra h = g ( N ) ( x R , y S ) induces partitioningof the cluster operator T into internal ( T int ( h ) or T int forshort) part belonging to h and external ( T ext ( h ) or T ext forshort) part not belonging to h , i.e., T = T int ( h ) + T ext ( h ) . (20)In Ref. , it was shown that if the two following criteriaare met: (1) the | Ψ ( h ) (cid:105) = e T int ( h ) | Φ (cid:105) is characterized by thesame symmetry properties as | Ψ (cid:105) and | Φ (cid:105) vectors (for ex-ample, spin and spatial symmetries), and (2) the e T int ( h ) | Φ (cid:105) Ansatz generates FCI expansion for the sub-system definedby the CAS corresponding to the h sub-algebra, then h iscalled a sub-system embedding sub-algebra (SES) for clus-ter operator T . For any SES h we proved the equivalenceof two representations of the CC equations at the solution:(i) standard (cid:104) Φ | ¯ H | Φ (cid:105) = E , (21) Q int ¯ H Φ (cid:105) = , (22) Q ext ¯ H | Φ (cid:105) = , (23)and (ii) hybrid ( P + Q int ) ¯ H ext e T int | Φ (cid:105) = E ( P + Q int ) e T int | Φ (cid:105) , (24) Q ext ¯ H | Φ (cid:105) = , (25)where ¯ H ext = e − T ext He T ext (26)and the two projection operators Q int ( h ) and Q ext ( h ) ( Q int and Q ext for short) are spanned by all excited configurationsgenerated by acting with T int ( h ) and T ext ( h ) onto referencefunction | Φ (cid:105) , respectively. The Q int and Q ext projectionsoperators satisfy the condition Q = Q int + Q ext . (27)The above equivalence shows that the CC energy can be cal-culated by diagonalizing effective Hamiltonian H eff definedas H eff = ( P + Q int ) ¯ H ext ( P + Q int ) (28)in the complete active space corresponding to any SES ofCC formulation defined by cluster operator T , i.e., H eff ( h ) e T int ( h ) | Φ (cid:105) = Ee T int ( h ) | Φ (cid:105) , ∀ SES h . (29)One should also notice that: (1) the non-CAS related CCwave function components (referred here as external de-grees of freedom) are integrated out and encapsulated in theform of H eff , and (2) the internal part of the wave function, e T int | Φ (cid:105) is fully determined by diagonalization of H eff inthe corresponding CAS. Separation of external degrees offreedom in the effective Hamiltonians is a desired feature, 𝐵 𝔥 ! 𝐵 𝔥 " 𝐵 𝔥 …𝐵 𝔥 $ Quantum System
Figure 1. Schematic representation of the CC flow. The entirequantum systems can be probed with various SES-eigenvalueproblems (29) schematically represented here as B ( h i ) . Thesecomputational blocks can be coupled into the flow, where infor-mation is passed between various computational blocks B ( h i ) .Subject to the choice of particular classes of SESs defining theflow, the CC flow can probe/traverse a large sub-spaces of entireHilbert space. especially for building its reduced-dimensionality repre-sentation for quantum computing (QC). However, a factorthat impedes the use of SES-CC effective Hamiltonians inquantum computing is their non-Hermitian character. Itis also worth mentioning that various CC approximationsare characterized by various SESs, which is a unique fin-gerprint of each standard CC approximaiton. For example,for the restricted Hartree-Fock (RHF) CC formulations the g ( N ) ( R , y S ) and g ( N ) ( R , y S ) are SESs for CCSD and CCS-DTQ approximations (one should also notice that SES forlower-rank CC approximation is also a SES of higher-rankCC approximations, i.e. g ( N ) ( R , y S ) is also a SES for theCCSDTQ approach).Properties of SESs-induced eigenvalue problems (29) canalso be utilized to design new CC approximations basedon various amplitude selections processes and re-castingCC equations in a different form, which offer interestingadvantages, especially in the way how corresponding equa-tions are solved. This fact can be illustrated on the exampleof the flow introduced in Ref. (see also Fig.1), whichrenders all singly and doubly and a subset of triply andquadruply excited amplitudes. While the CCSD equationscannot be represented as a union of equations correspond-ing to Eqs.(29) for various CCSD’s SESs g ( N ) ( R , y S ) (thereare no SES in the CCSD case that would embrace doublyexcited amplitudes t i jab where spinorbitals i and j corre-spond to distinct orbitals), there exist formalisms whichcan probe a significant portion of Hilbert space and aredefined by the set of equations that correspond to a unionof non-symmetric eigenvalue problems of the type (29) forvarious SESs. For example, the SCSAF-CCSD(2) approachof Ref. uses cluster operator T defined as T (cid:39) T + T + ∑ I T int , ( g ( N ) ( R I )) + ∑ I T int , ( g ( N ) ( R I )) (30)where T and T are singly and doubly excited cluster opera-tors and T int , ( g ( N ) ( R I )) and T int , ( g ( N ) ( R I )) contain tripleand quadruple excitations corresponding to SES g ( N ) ( R I ) . !( ! (2 " ! )) !( ! (2 " " )) !( ! (2 " )) Initial guesses of amplitudes (0-th cycle) ...
Amplitude synchronization for the (&+ () -th cycle) ! -th cycle !( ! (2 " ! )) !( ! (2 " " )) !( ! (2 " )) !"($,&) !"((,&) !"()−$,&) ! + -th cycle Initial guesses of amplitudes (0-th cycle) ... (a)(b)
Figure 2. Two types of flow CC formulations for g ( N ) ( R I ) sub-algebras (panel (a) represents serial executions; panel (b) corre-sponds to the parallel processing of computational blocks (see textfor details)). Summation over I in (30) runs over all possible SESs g ( N ) ( R ) . It can be shown that in this case, the global set of CC equations Q ( e − T He T ) | Φ (cid:105) = , (31)where all equations are processed simultaneously in theiterative process of finding the solution, can be re-cast (atthe CC solution) in the form of coupled equations (29) ofthe form H eff ( h ) e T int ( h ) | Φ (cid:105) = Ee T int ( h ) | Φ (cid:105) , ∀ h = g ( N ) ( RI ) . (32)As shown in Fig.2, the solution process of equations (31)can be organized in the form of flow where the algebraicform of each computational blocks B ( g ( N ) ( R I )) representseigenvalue problem (29) for sub-algebra g ( N ) ( R I ) . In panel(a) we represent particular flow where particular compu-tational blocks B ( g ( N ) ( R I )) are communicating in serial.For this purpose, we first establish an ordering of activespaces defined by g ( N ) ( R I ) , in the way that reflects theirimportance (for example, the first active space contains themost important effects related to the sought for electronicstate). Then we define a protocol for passing informationbetween B ( g ( N ) ( R I )) ’s to avoid double counting” or ”shar-ing” cluster amplitudes between various com. This problemis caused by the fact that two distinct SESs g ( N ) ( R I ) and g ( N ) ( R J ) can share a single orbital and effectively shareall single excitations from this orbital and double excita-tions exciting α and β electrons from the shared orbital.This redundancy is very small compared to total numberof excitations defining distinct sub-algebras g ( N ) ( R I ) and g ( N ) ( R J ) . In particular, there is not overlap with the largestclasses of excitations corresponding to triple and quadru-ple excitations. At the solution, these redundancies areirrelevant because the equations for shared amplitudes arethe same irrespective of the SESs eigenvalue problem (32)they are part of. To control this effect, common pool ofamplitudes obtained in previous K steps of i -th iteration (denoted as CP ( i , K ) ) is passed to K + g ( N ) ( R K + ) enter computational block B ( g ( N ) ( R K + )) as known parameters. In this case, the algebraic form of B ( g ( N ) ( R K + )) still takes the form of eigenvalue problemof smaller size ( P + Q Xint )[ e − T CPint ( h ) H eff ( h ) e T CPint ( h ) ] e T Xint ( h ) | Φ (cid:105) = Ee T Xint ( h ) | Φ (cid:105) , h = g ( N ) ( R K + ) , T int ( h ) = T CPint ( h ) + T Xint ( h ) , h = g ( N ) ( R K + ) , (33)where T CPint ( h ) is a part of T int ( h ) determined by amplitudesfrom the common pool of amplitudes and T Xint ( h ) is partof T int ( h ) that is determined in the K + Q Xint operator is the projection opera-tor onto excited configurations generated by T Xint ( h ) whenacting onto reference function | Φ (cid:105) . In panel (b) we see analternative ”parallel” flow where computational blocks areindependent and corresponds to the original eigenproblems(29) for SESs g ( N ) ( R I ) . This step is followed by a sync-upof all shared amplitudes by various SESs.The CC flow equations (29) or (32) can also be viewedas a configurational (or more aptly - sub-space) version ofthe Aufbau principle , which is a consequence of the factthat each problem corresponding to some SES h providesa rigorous mechanism for extending the sub-space probedin a flow. In other words, the spaces probed in each SESproblem (29) are additive. This fact is a unique feature,which should be referred to as the ”sub-system” memory ofthe CC wave function. Additionally, CC flow assure size-consistency of the calculated ground-state energies. Similarflows cannot be easily constructed using configuration in-teraction type methods or standard many-body perturbationtechniques. We believe that this is yet another argument infavor of non-perturbative analysis of CC equations.In general, flow-based CC formulations are very flexibleand allows one to use sub-algebras g ( N ) ( x R , y S ) defined byvarious x and y parameters. This property may be used tointroduce selective groups of higher excitations and tunethe cost of flow equations to available computing resources.Although the numerical implementation of flow-based for-malism may be numerically less efficient than the imple-mentation based on the global representation, its advantagelies in the fact that computational blocks contributing toflow are physically interpretable in terms of Schr¨odinger-type equations for sub-systems described by relevant SESs.This fact has profound consequences and allows one to con-struct more justified (or equivalently less ”postulated”) andbetter-controlled approximations based on the flow equa-tions. An interesting illustration of this fact will be CC flowequations for localized orbitals (see Section II B) A. Time-dependent CC flows
In Ref. , we demonstrated that the SES-based downfold-ing techniques could also be extended to the time-dependent Schr¨odinger equation in the simplest case when all orbitalsand the reference functions | Φ (cid:105) are assumed to be time-independent. As in the stationary case, we will assume ageneral partitioning of the time-dependent cluster opera-tor T ( t ) into its internal ( T int ( h , t ) ) and external ( T ext ( h , t ) )parts, i.e, | Ψ ( t ) (cid:105) = e T ext ( h , t ) e T int ( h , t ) | Φ (cid:105) , ∀ h ∈ SES . (34)For generality, we also include phase factor T ( h , t ) in thedefinition of the T int ( h , t ) operator. After substituting (34)into time-dependent Schr¨odinger equation and utilizingproperties of SES algebras, we demonstrated that the ket-dynamics of the sub-system wave function e T int ( h , t ) | Φ (cid:105) cor-responding to arbitrary SES h i ¯ h ∂∂ t e T int ( h , t ) | Φ (cid:105) = H eff ( h , t ) e T int ( h , t ) | Φ (cid:105) , (35)where H eff ( h , t ) = ( P + Q int ( h )) ¯ H ext ( h , t )( P + Q int ( h )) (36)and ¯ H ext ( h , t ) = e − T ext ( h , t ) He T ext ( h , t ) . (37)In analogy to the stationary cases, various sub-systems com-putational blocks can be integrated into a flow enabling sam-pling of large sub-spaces of Hilbert space through througha number of coupled reduced-dimensionality problems. Forexample, the time-dependent variant of the the SCSAF-CCSD(2) approach uses the time-dependent cluster opera-tor T ( t ) in the form T ( t ) (cid:39) T ( t ) + T ( t ) + ∑ I T int , ( g ( N ) ( R I ) , t )+ ∑ I T int , ( g ( N ) ( R I ) , t )) (38)which can be equivalently represented as coupled time-evolution problems for sub-systems i ¯ h ∂∂ t e T int ( h , t ) | Φ (cid:105) = H eff ( h , t ) e T int ( h , t ) | Φ (cid:105) , ∀ h = g ( N ) ( RI ) . (39)These equations can be solved using similar flows as shownin Fig.2 with the difference that now the iterative cyclesfor converging amplitudes/energy correspond to elemen-tary time steps with increment corresponding to ∆ t . As inthe stationary case, the time-dependent CC flow equationsrepresent a correlated version 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 flowequations based on the localized orbitals and local CC for-mulations developed in the last few decades in quantumchemistry (see the next Subsection), the flow described byEq.(39) can be considered as a reduced-scaling variant ofthe time-dependent CC formulations. B. CC flows for localized orbital basis
The SES-CC flow formalism also systematizes and fur-ther extends the notion of ”sub-system” composed of orbitalpairs. This problem has intensively been studied in earlynon-orthogonal pseudo-natural orbitals based formulationsof CI , coupled electron pair approximation (CEPA) ,and their extensions to local CEPA/CC methods based onthe local pair natural orbitals (LPNO) and their domain-based LPNO variant (DLPNO). To analyze SES-CCapproximations, let us, in the analogy to the DLPNO-CCformulations, assume that the set of selected orbital pairs(for simplicity, we will focus on the closed-shell formula-tions) P = { ( i , j ) } that significantly contribute to correlationenergy is known. These pairs are also employed to definePNO spaces and corresponding CCSD cluster amplitudes.In the standard pair-driven DLPNO-CCSD approximationorbital pairs ( i , j ) (including pairs where i = j , i.e., ( i , i ) )along with ( i , j ) -specific natural virtual orbitals are used toselect PNO space and relevant single and double excitations.The ( i , j ) -specific density matrix defined as MP2-type den-sity matrix D i j is used to determine virtual PNOs in the waythat only natural orbitals characterized by occupation num-bers greater than the user-defined threshold are retained. Itleads to a significant reduction in the size of pair-specificPNOs spaces and consequently to a significant reduction ofthe number of pair-specific singly ( t ia ii ) and doubly ( t i ja ij b ij )excited amplitudes, where virtual indices a ii , a i j , and b i j aredefined by reduced-size PNOs. It should be noted that eachpair ( i , j ) introduces its own set of PNO virtual orbitals,which may not be orthogonal to PNOs corresponding todistinct pairs.The CC g ( N ) ( R ) flow employing localized occupiedorbitals in a natural way introduces several elements un-derlying standard DLPNO-CCSD design. Our analysis isbased on the observation that there is a a natural correspon- dence between orbital pairs from P (along with all virtualorbitals) with g ( N ) ( R ) sub-algebras, where R ≡ ( i , j ) , ( i , j ) ∈ P . (40)For short we will refer to these SESs as g ( N ) ( i j ) ( ( i , j ) ∈ P ). Additionally, each Schr¨odinger-type equation in (29)or (32) H eff ( h ) e T int ( h ) | Φ (cid:105) = Ee T int ( h ) | Φ (cid:105) , h = g ( N ) ( i j ) ( i , j ) ∈ P . (41)naturally defines corresponding one-body density matrix ρ ( g ( N ) ( i j )) and its PNOs (or DLPNOs) without any ad-ditional assumptions regarding the form and the origin ofthe pair density matrix. One can readily notice that the D i j density matrices used in original DLPNO-CC papersis its low-order approximation. This feature of SES equa-tions (41) can be viewed as a CC-derived systematizationof the sub-system concepts discussed in original works ofSinano˘glu and Meyer .It is worth mentioning that in contrast to the DLPNO-CCSD formalism, the g ( N ) ( i j ) include specific classes oftriple and quadruple excitations. This feature may be apossible way to define the balanced inclusion of higher-rank excitations in the local CC formulations. For thispurpose one can also envision CC flows based on g ( N ) ( x R ) sub-algebras with x > ( g ( N ) ( i j )) computational block con-tributing to flow (41) defines its own set of PNOs, in analogyto DLPNO-CC approaches, it can be re-expressed throughits own set of PNOs and set of pre-selected internal ampli-tudes (utilizing predefined thresholds). However, introduc-ing threshold in the flow CC equations can be performedless abruptly than in the existing DLPNO-CC methods. Infact, the amplitude selection process is equivalent of se-lecting sub-algebra g ( N ) ( i j , y S ) of g ( N ) ( i j ) in the way that S is the set of PNOs corresponding to occupation num-ber greater than the predefined threshold and y is the totalnumber of virtual PNOs selected this way. This selection in-duces a natural partitioning of the T int ( g ( N ) ( i j )) into a partbelonging to g ( N ) ( i j , y S ) ( T int ( g ( N ) ( i j , y S )) and remaining”neglegible” part of excitations ( ∆ T int ( g ( N ) ( i j )) ) T int ( g ( N ) ( i j )) = T int ( g ( N ) ( i j , y S )) + ∆ T int ( g ( N ) ( i j )) . (42)The controlled version of the selection step is achieved bynoticing that in analogy to Eqs.(33) the effect of the ”ne-glegible” amplitudes ∆ T int ( g ( N ) ( i j )) can still be absorbedin the form of additional similarity transformation: ( P + Q int ( f i j ))[ e − ∆ T int ( h ij ) H eff ( h i j ) e ∆ T int ( h ij ) ] e T int ( f ij ) | Φ (cid:105) = Ee T int ( f ij ) | Φ (cid:105) , f i j = g ( N ) ( i j , y S ) , h i j = g ( N ) ( i j ) (43)where Q int ( g ( N ) ( i j , y S )) is a projection operator onto ex-cited configurations generated by the T int ( g ( N ) ( i j , y S )) when acting onto reference function | Φ (cid:105) . We also believethat the eigenvalues of effective Hamiltonians correspond-ing to g ( N ) ( R , y S ) , where R correspond to a single occupied orbital (or a pair ( i , i ) ) may provide much-needed accuracydiagnostic for the DLPNO-CCSD-type methods.Summarizing, several basic threads of DLPNO-CCSDequations are consequences of CC flow equations definedby g ( N ) ( i j ) sub-algebras. The local character of correla-tion effects is a net effect of the local character of the basisset used, asymptotic properties of one- and two-electronproperties, and fundamental properties of CC formalismassociated with the CC sub-system memory. This featureallows one to construct, in a rigorous way, Schr¨odinger-typeequations for sub-systems (in this case, a pair of orbitals)defining the flow. Additionally, a rigorous definition ofsub-system and associated wave function lead to a natu-ral definition of the sub-system density matrix and naturalorbitals. We believe that the CC flow equations based onthe SES formalism are an interesting tool for constructingvarious approximations for correlated systems. This formu-lation can be universally used in stationary formulationsof canonical and local CC formulations and be extendedto the time-dependent CC equations. The general CC flowformalism is not limited to types of interactions that areconsidered in quantum chemistry and can also be extendedto other types of many-body interactions encountered innuclear physics. III. SUB-SYSTEM FLOWS BASED ON THEDOUBLE UNITARY CC REPRESENTATIONS OFWAVE FUNCTION
We find properties of SR-CC flows very appealing fromthe point of view of quantum computing. Instead of con-sidering expensive ”global” space approach (as done in themajority of existing QC formalisms) that requires too manyparameters to be optimized at the same time, one could par-tition the problem into smaller computational sub-problemsthat can be tuned to available systems of qubits. For this rea-son, we would like to adapt the SR-CC ideas from previoussections to double unitary CC Ansatz (DUCC; see Ref. ).While the DUCC formalism mirrors some properties of theSES-CC formalism and additionally assures the Hermitiancharacter of the effective Hamiltonians in CAS( h ), due tothe non-commutative nature of the anti-Hermitian clusteroperators employed by this formalism, coupling variousDUCC problems into a flow requires several approxima-tions, described in the following subsection.The DUCC formalism discussed in Refs. uses a com-posite unitary CC Ansatz to represent the exact wave func-tion | Ψ (cid:105) , i.e., | Ψ (cid:105) = e σ ext ( h ) e σ int ( h ) | Φ (cid:105) , (44)where σ ext ( h ) and σ int ( h ) are general-type anti-hermitian σ †int ( h ) = − σ int ( h ) , (45) σ †ext ( h ) = − σ ext ( h ) . (46)and all cluster amplitudes defining σ int cluster amplitudescarry active indices only (or indices of active orbitals defin-ing given h ). The external part σ ext ( h ) is defined by am-plitudes carrying at least one inactive orbital index. Incontrast to the SR-CC approach, internal/external parts ofanti-Hermitian operators are not defined in terms of ex-citations belonging explicitly to a given sub-algebra butrather by indices defining active/inactive orbitals specific toa given h . When the external cluster amplitudes are known (orcan be effectively approximated), in analogy to single-reference SES-CC formalism, the energy (or its approx-imation) can be calculated by diagonalizing Hermitian ef-fective/downfolded Hamiltonian in the active space usingvarious quantum or classical diagonalizers. An importantstep towards developing practical computational schemes isto simplify the infinite expansions defining both cluster am-plitudes and non-terminating commutator expansions defin-ing downfolded Hamiltonians. A legitimate approximationof σ ext ( h ) and σ int ( h ) in Eq.(44) is to retain lowest-orderterms only, i.e., σ int ( h ) (cid:39) T int ( h ) − T int ( h ) † , (47) σ ext ( h ) (cid:39) T ext ( h ) − T ext ( h ) † , (48)which has been discussed in Ref. . In particular, T ext ( h ) can be approximated by SR-CCSD amplitudes that carry atleast one external spinorbital index. Other possible sourcesfor obtaining external cluster amplitudes are higher-rankSR-CC methods and approximate unitary CC formulationssuch as UCC(n) methods .Using DUCC representation (44) it can be shown thatin analogy to the SR-CC case, the energy of the entiresystem (once the exact form of σ ext ( h ) operator is known)can be calculated through the diagonalization of the ef-fective/downfolded Hamiltonian in SES-generated activespace, i.e., H eff ( h ) e σ int ( h ) | Φ (cid:105) = Ee σ int ( h ) | Φ (cid:105) , (49)where H eff ( h ) = ( P + Q int ( h )) ¯ H ext ( h )( P + Q int ( h )) (50)and ¯ H ext ( h ) = e − σ ext ( h ) He σ ext ( h ) . (51)Typical approximations for downfolded Hamiltonian uti-lize: (1) various sources for evaluation of the T ext ( h ) oper-ator in (48), (2) various length of commutator expansiondefining the ¯ H ext ( h ) operator, (3) various excitation-ranksin the many-body expansion of the ¯ H ext ( h ) operator, and(4) various molecular basis choice.Recently, applications of QPE and VQE quantum algo-rithms to evaluate eigenvalues of downfolded Hamiltonians H eff ( h ) became a subject of intensive studies. In the caseof the VQE method, the energy functionalmin θ ( h ) (cid:104) Ψ ( θ ( h )) | H eff ( h ) | Ψ ( θ ( h )) (cid:105) (52)is optimized with respect to variational parameters θ ( h ) where | Ψ ( θ ( h )) (cid:105) approximates e σ int ( h ) | Φ (cid:105)| Ψ ( θ ( h )) (cid:105) (cid:39) e σ int ( h ) | Φ (cid:105) (53)at the level of quantum circuit. This approach turned out tobe very efficient, especially when ”correlated” natural or-bitals were employed. The advantage of the VQE approachis the possibility of extracting the information (if some formof the unitary CC expansion is used to represent | Ψ ( θ ( h )) (cid:105) )about the leading excitations in e σ int ( h ) | Φ (cid:105) , which playsa vital role in designing DUCC sub-system flows and as-sures mechanism of quantum information passing betweenvarious computational blocks. Target space ( ! )Full space Orbital energies ⋮⋮⋮⋮ ! ! ! " ⋯⋯ Figure 3. Schematic representation of the DUCC flow. It isassumed that the most important classes of excitations requiredto describe a state of interest are captured by the target activespace (too large for direct QC simulations) and that target activespace (corresponding to sub-algebra h ) can be ”approximated”by excitations included in smaller yet computationally feasibleactive spaces corresponding to sub-algebras h , ..., h M (see text).The DUCC flow combines computational blocks that correspondto variational problems associated with each sub-algebra h i ( i = ,..., M ) . The green dashed line represents Fermi level. A. DUCC flow equations: applications in quantumcomputing
The DUCC flow idea is very interesting from the pointof view of its applications in quantum computing, where aquantum computer can process computational blocks (ei-ther corresponding to energy functional minimization ordiagonalization of the downfolded Hamiltonians). In thissection, we will extend the idea of the SR-CC sub-systemflow to the DUCC formalism, and we will highlight the sim-ilarities and differences between these two approaches. Themain differences between SR-CC and DUCC should be at-tributed to the non-commutative nature of many-body com-ponents defining anti-Hermitian DUCC cluster operators.This fact, in the case of the DUCC approach, significantlyimpedes the analysis of the equations and partitioning theminto separate computational blocks that can be integratedinto a sub-system flow equations. However, this can beachieved with a sequence of approximations we describebelow.We will start our analysis from assuming that we wouldlike to perform DUCC effective simulations for SES h problem (49) which is, for whatever reason, too complexor too big for quantum/classical processing. We will as-sume that external amplitudes σ ext ( h ) can be effectivelyevaluated using perturbative formulations. For simplicitywe will introduce a new Hamiltonian A ( h ) which is definedas H eff ( h ) or its approximation in the ( P + Q ( h )) space (in the simplest case it can be just the ( P + Q ( h )) H ( P + Q ( h )) operator). We will use denote A ( h ) simply by A . We willalso assume the situation where excitations from h thatare relevant to state of interest can be captured by exci-tation sub-algebras: h , h , . . . , h M (see Fig.3), where,in analogy to the SR-CC case, we admit the possibilityof ”sharing” excitations/de-excitations between these sub-algebras. We also assume that the number of excitationsbelonging to each h i ( i = , . . . , M ) is significantly smallerthan the excitations in h and therefore numerically tractablein quantum/classical simulations. Below we will discussthe challenges and approximations that are needed to obtainwell defined DUCC flow equations.In the discussed algorithm, the A ( h ) Hamiltonian andthe ( P + Q ( h )) space can be treated as a starting point forsecondary DUCC decompositions generated by sub-systemalgebras h i ( i = , . . . , M ) defined above, i.e., A eff ( h i ) e σ int ( h i ) | Φ (cid:105) = Ee σ int ( h i ) | Φ (cid:105) ( i = . . . , M ) . (54)or in the VQE-type variational representation asmin θ ( h i ) (cid:104) Ψ ( θ ( h i )) | A eff ( h i ) | Ψ ( θ ( h i )) (cid:105) ( i = . . . , M ) . (55)Each A eff ( h i ) is defined as A eff ( h i ) = ( P + Q int ( h i )) ¯ A ext ( h i )( P + Q int ( h i )) (56)and ¯ A ext ( h i ) = e − σ ext ( h i ) Ae σ ext ( h i ) . (57)where we defined external σ ext ( h i ) operator with respectto h or ( P + Q int ( h )) space (i.e.cluster amplitudes defining σ ext ( h i ) must carry at last one index belonging do activespin orbitals defining h and not belonging to set of activespin orbtitals defining h i ). In other words, sub-algebras h i generate active sub-spaces in larger active space h , i.e., ( P + Q int ( h i )) ∈ ( P + Q ( h )) . However, connecting DUCCcomputational blocks (54) or (55) directly into a flow isa rather challenging task. In contrast to the SR-CC sub-system flows where cluster amplitudes are universal forall sub-algebras induced problems (i.e., given amplitudecarries the same value across all computational blocks),the same is no longer valid for DUCC flows. Again, thisis a consequence of the non-commutativity of the anti-Hermitian operators defining DUCC representation of thewave function. For example, the internal amplitude forsome h i problem may assume a different value as the sameamplitude being classified as external amplitude for a differ-ent problem corresponding to sub-algebra h j ( i (cid:54) = j ) , whichmeans that DUCC amplitudes explicitly depend on the sub-algebra index h i (as opposed to the SR-CC flow formalism,where values of particular amplitudes were independent ofthe sub-algebra index). Similar effects could be observed inADAPT-VQE formulations, where amplitudes from theexcitation poll may appear multiple times carrying variousvalues in the wavefunction expansion. The most straight-forward approximation to enable DUCC flows would beto assume the universality of all DUCC amplitudes clusteramplitudes in the flow. An additional problem is related tothe fact that while for the SR-CC flows, effective Hamilto-nians corresponding for various SESs can be constructedexactly, for the DUCC case A eff ( h i ) can be constructed onlyin an approximate way, and therefore their ground-stateeigenvalues may not be exactly equal.To address these issues, we assume the following approx-imations: (1) we assume that all amplitudes are transferableand universal across various h i problems, (2) we replaceeigenvalue problems 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 h i sub-algebras,with h corresponding to the CAS closest to the wave func-tion of interest, then in the h i problem we partitioned (inanalogy to Eq.(33) set of parameters θ ( h i ) into sub-set θ CP ( h i ) that refers to the common pool of the amplitudesdetermined in preceding steps (say, for h j ( j = , . . . , i − ) )and sub-set θ X ( h i ) that is uniquely determined in the h i minimization step, i.e,min θ X ( h i ) (cid:104) Ψ ( θ X ( h i ) , θ CP ( h i )) | A eff ( h i ) | Ψ ( θ X ( h i ) , θ CP ( h i )) (cid:105) ( i = . . . , M ) . (58)Each computational block coupled into a flow correspondsto a minimization procedure that optimizes parameters θ X ( h i ) using quantum algorithms such as the VQE ap-proach. At the end of the iterative cycle, once all amplitudesare converged, in contrast to the SR-CC flows, the energycan be calculated by diagonalization the h problem, i.e., A eff ( h ) e σ int ( h ) | Φ (cid:105) = Ee σ int ( h ) | Φ (cid:105) . (59)The DUCC flow is composed of classical computing stepswhere approximate second-quantized form of the A eff ( h i ) operators are calculated and quantum computing steps,where cluster amplitudes are determined using the VQEalgorithm. The discussed formalism introduces a broadclass of control parameters, which define each computa-tional step’s dimensionality. These are the numbers of oc-cupied/unoccupied active orbitals defining h i sub-algebras x R i and y S i .In the second strategy for defining DUCC flows, in anal-ogy to the secondary DUCC steps and amplitudes univer-sality strategy, we assume that the σ int ( h ) operator can beapproximated by σ int ( h i ) , ( i = , . . . , M ) , i.e., σ int ( h ) (cid:39) M ∑ i = σ int ( h i ) + X ( h , h , . . . , h M ) (60)where the X ( h , h , . . . , h M ) operator (or X for short) elimi-nates possible overcounting of the ”shared” amplitudes. Itenables to re-express σ int ( h ) as σ int ( h ) = σ int ( h j ) + R ( h j ) ( j = , . . . , M ) , (61)where R ( h j ) = ( j ) M ∑ i = σ int ( h i ) + X (62) and ( j ) ∑ Mi = designates the sum where the j -th element isneglected. Consequently, we get e σ int ( h ) | Φ (cid:105) = e σ int ( h j )+ R ( h j ) | Φ (cid:105) ( j = , . . . , M ) . (63)Using Trotter formula we can approximate right hand sideof (63) for a given j as e σ int ( h ) | Φ (cid:105) (cid:39) ( e R ( h j ) / N e σ int ( h j ) / N ) N | Φ (cid:105) . (64)Introducing auxiliary operator G ( N ) j G ( N ) j = ( e R ( h j ) / N e σ int ( h j ) / N ) N − e R ( h j ) / N ( j = , . . . , M ) , (65)the ”internal” wave function (63) can be expressed as e σ int ( h ) | Φ (cid:105) (cid:39) G ( N ) j e σ int ( h j ) / N | Φ (cid:105) ( j = , . . . , M ) . (66)One should remember that G ( N ) j is a complicated functionof all σ int ( h i ) ( i = , . . . , M ) and the above expression doesnot decouple σ int ( h j ) from the G ( N ) j term. Instead, this ex-pression may help define the practical way for determiningcomputational blocks for flow equations without makingany assumption about the universality of the amplitudes.To see this, let us introduce expansion (66) to Eq.(49) (with H eff ( h ) replaced by the A operator), pre-multiply both sidesby [ G ( N ) j ] − , and project onto ( P + Q int ( h j )) sub-space,which leads to non-linear eigenvalue problems ( P + Q int ( h j ))[ G ( N ) j ] − AG ( N ) j e σ int ( h j ) / N | Φ (cid:105) (cid:39) Ee σ int ( h j ) / N | Φ (cid:105) ( j = , . . . , M ) . (67)The above equations become exact in the N → ∞ limit. We will utilize these equations as a computational blocks for0the DUCC flow. To make a practical use of Eqs. (67) letus linearize them by defining the downfolded Hamiltonian Γ ( N ) j , Γ ( N ) j = ( P + Q int ( h j ))[ G ( N ) j ] − AG ( N ) j ( P + Q int ( h j )) asa function of σ int ( h i ) ( i = , . . . , M ) from the previous flow cycle (we will symbolically designate this fact by using spe-cial notation for Γ ( N ) j effective Hamiltonian, i.e., Γ ( N ) j ( P ) Hamiltonian). Now, in full analogy to (58), this representa-tion can be employed to optimize σ int ( h j ) amplitudes:min θ X ( h j ) (cid:104) Ψ ( θ X ( h j ) , θ CP ( h j )) | Γ ( N ) j ( P ) | Ψ ( θ X ( h j ) , θ CP ( h j )) (cid:105) ( j = , . . . , M ) . (68)At the cost of additional similarity transformations (or theirapproximate variants) induced by the G ( N ) j operators insmaller active sub-spaces, the above procedure eliminatesthe need for invoking amplitude universality assumption.Similar results can be obtained by using the Zassenhausformula. Moreover, the present formalisms, in analogy tothe SR-CC flows, can be extended to the time-domain.An essential feature of the DUCC flow equation is asso-ciated with the fact that each computational block (58) orequivalently Hamiltonian A eff ( h i ) can be encoded using amuch smaller number of qubits compared to the full sizeof the global problem. In fact, the maximum size of thequbit register ( QR ( m max ) ) required in DUCC quantum flowis associated with the maximum size of the sub-system andnot with the size of the entire quantum system of interest QR ( m max ) (cid:28) QR ( g ( N ) ) , (69)where QR ( g ( N ) ) is the total number of qubits required todescribe whole system. This observation significantly sim-plifies the qubit encoding of the effective Hamiltonians in-cluded in quantum DUCC flows, especially in formulationsbased on the utilization of localized molecular basis set asdiscussed in Section II.C (for early quantum algorithmsexploiting locality of interactions see Ref. ). B. Extraction of the analytical form of effectiveinter-electron interactions from DUCC flow equations
Once the stationary flow equations (58) reaches the con-vergence, which means that the energy of the systems can becalculated by diagonalizing any downfolded DUCC Hamil-tonian H eff ( h ) included in the flow equations in the cor-responding active space. In particular, this is true for theactive space, which is closest to the problem of interest andcorresponds to the sub-algebra h . For ground-state type calculations, one expects the h to contain valence naturalorbitals.In standard formulations of downfolding methods it isassumed (see Refs. )) that downfolded Hamiltonians aredominated by one- and two-body effects, i.e., using the lan-guage of second quantization H eff ( h ) can be approximatedas (for simplicity, let us assume that only virtual orbitalsare downfolded) H eff ( h ) (cid:39) ∑ PQ χ ( h ) PQ a † P a Q + ∑ P , Q , R , S χ ( h ) PQRS a † P a † Q a S a R , (70)where P , Q , R , S indices, χ ( h ) PQ , and χ ( h ) PQRS representactive spinorbitals and effective one- and two-body in-teractions, respectively (non-antisymmetrized matrix el-ements χ ( h ) PQRS are employed in (70)). Once the set of { χ ( h ) PQ , χ ( h ) PQRS } is known (at the end of flow procedure)this information can be further used to derive an analyticalform of effective inter-electron interactions. This can beaccomplished by fitting the general form of one-body u and two-body g interactions defined as functions of to-be-optimized parameters γ / δ as well as r , r , r = | r − r | , ∇ , ∇ , etc. operators: u = u ( γ , r , ∇ , . . . ) , (71) g = g ( δ , r , r , r , ∇ , ∇ , . . . ) , (72)These effective interactions replace standard one- andtwo-body interactions in non-relativistic quantum chem-istry and are defined to minimize the discrepancies with { χ ( h ) PQ , χ ( h ) PQRS } for a given discrete molecular spinor-bital set, i.e., min γ { ∑ PQ | u PQ ( γ ) − χ PQ ( h ) |} , (73)min δ { ∑ PQRS | g PQRS ( δ ) − χ PQRS ( h ) |} , (74)where u PQ ( γ ) = (cid:90) dx φ P ( x ) ∗ u ( γ , r , ∇ , . . . ) φ Q ( x ) , (75) g PQRS ( δ ) = (cid:90) dx dx φ P ( x ) ∗ φ Q ( x ) ∗ g ( δ , r , r , r , ∇ , ∇ , . . . ) φ R ( x ) φ S ( x ) . (76)We believe that the utilization of efficient non-linear opti-mizers or machine learning techniques can provide an effec- tive form of the interactions u and g defined in small-sizeactive spaces. These effective interactions can be utilized in1low-order methodologies, including Hartree-Fock (HF) anddensity functional theories (DFT). In the latter case func-tions u and g can be utilized to develop/verify new formsof exchange-correlations functionals. The access to theanalytical form of the inter-electron interactions can alsoenable affordable and reliable ab-initio dynamics driven bylow-order methods. IV. CONCLUSIONS
This paper discussed the properties of SR-CC sub-systemflow equations stemming from the SES-CC formalism forRHF reference functions. It was shown that flow equationsdefine an alternative (to the canonical formulations) wayof introducing selected classes of higher-rank excitationsbased on system partitioning or choice of sub-system exci-tation sub-algebras corresponding to various active spaces.An essential feature of the SR-CC flow lies in the fact thatflow equations can be built upon g ( N ) ( x R , y S ) sub-algebraswith x R and y S chosen in a way that makes the flow tun-able to available computational resources. We also demon-strated that the idea of CC flow naturally extends to thetime-domain, offering a possibility of performing calcula-tions 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 oflocal CC formulations and the concept of ”locality” of corre-lation effects. As explained in section II.B, the SR-CC flowsbased on the utilization of local molecular orbitals providea rigorous way of defining sub-system through the effec-tive Hamiltonian corresponding to ( i , j ) -determined SES, g ( N ) ( i j ) . The ( i , j ) -pair density matrix can be further usedto calculate pair-natural orbitals and select leading excita-tion as postulated in the DLPNO-CCSD formulations. Webelieve that the SR-CC flows defined by larger active spacesalso provide a natural way of introducing higher-rank exci-tations, although maintaining linear-scaling of the resultinglocal CC formulation may not be possible. On the otherhand, following the SR-CC flow philosophy for localizedorbitals, although numerically more expensive, may help inre-establishing the desired level of accuracy in perturbative(non-iterative) energy corrections due to the higher-rankcluster excitations (for example, it is known that the neterrors of DLPNO perturbative triples (T) corrections maysignificantly surpass the discrepancies of DLPNO-CCSDcorrelations energies with respect to the canonical CCSDand (T) energies/correction).Due to the non-commutative nature of the general typeunitary CC formulations, the direct extension of SR-CCsub-system flows to DUCC-type flow is a rather challeng-ing endeavor. However, assuming the ”universality” ofcluster amplitudes or utilizing Trotter formula in downfold-ing procedures lead to computationally feasible algorithms.In the quantum computing variant, the flow represents asequence of coupled Hermitian eigenproblems, where diag-onalization is replaced by VQE type optimization to obtaina corresponding sub-set of amplitudes. In this formula-tion, the flow of quantum information corresponding toshared/external amplitudes (defined for a given sub-system) can be easily implemented at the level of the quantum cir-cuit. In analogy to the SR-CC flows, the DUCC-flows canbe tuned to the available quantum resources. As such, theDUCC flows offer an interesting possibility of decomposinglarge-dimensionality problems into a collection of reduceddimensionality computational blocks. We believe that theDUCC flow methods can significantly push the envelopeof system-size tractable in quantum simulations. It shouldalso be stressed that SR-CC/DUCC flow methods allow oneto enlarge the size of probed space systematically whileretaining the size-extensivity of the calculated in this wayenergies, which is especially important in applications toextended systems.An exciting feature of the SES-CC formalism and CCflows is their universal character irrespective of the partic-ular form of interactions defining correlated many-bodysystems. V. ACKNOWLEDGEMENT
This work was supported by the Quantum Science Center(QSC), a National Quantum Information Science ResearchCenter of the U.S. Department of Energy (DOE). Part ofthis work was supported by the “Embedding QC into Many-body Frameworks for Strongly Correlated Molecular andMaterials Systems” project, which is funded by the U.S.Department of Energy, Office of Science, Office of BasicEnergy Sciences (BES), the Division of Chemical Sciences,Geosciences, and Biosciences.
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