Efficient step-merged quantum imaginary time evolution algorithm for quantum chemistry
Niladri Gomes, Feng Zhang, Noah F. Berthusen, Cai-Zhuang Wang, Kai-Ming Ho, Peter P. Orth, Yongxin Yao
EEfficient step-merged quantum imaginary timeevolution algorithm for quantum chemistry
Niladri Gomes, † Feng Zhang, † Noah F. Berthusen, † , ‡ Cai-Zhuang Wang, † , P Kai-Ming Ho, † , P Peter P. Orth, † , P and Yongxin Yao ∗ , † , P † Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA ‡ Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa50011, USA. P Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
E-mail: [email protected]
Abstract
We develop a resource efficient step-mergedquantum imaginary time evolution approach(smQITE) to solve for the ground state ofa Hamiltonian on quantum computers. Thisheuristic method features a fixed shallow quan-tum circuit depth along the state evolution path.We use this algorithm to determine bindingenergy curves of a set of molecules, includingH , H , H , LiH, HF, H O and BeH , and findhighly accurate results. The required quantumresources of smQITE calculations can be furtherreduced by adopting the circuit form of the vari-ational quantum eigensolver (VQE) technique,such as the unitary coupled cluster ansatz. Wedemonstrate that smQITE achieves a similarcomputational accuracy as VQE at the samefixed-circuit ansatz, without requiring a gener-ally complicated high-dimensional non-convexoptimization. Finally, smQITE calculations arecarried out on Rigetti quantum processing units(QPUs), demonstrating that the approach isreadily applicable on current noisy intermediate-scale quantum (NISQ) devices. One of the most promising near-term applica-tions of quantum computing is to solve the elec- tronic structure of molecules and condensed mat-ter systems.
This is because the number ofbinary bits required to store a general many-body state of a fermionic Hamiltonian growsexponentially with the dimension of the single-particle basis in classical computers, while quan-tum computers offer a natural representation ofmany-body states using qubits whose requirednumber only scales linearly with the size of thesingle-particle basis. A many-body wave func-tion can thus be efficiently stored in memoryusing qubits. The pioneering proposal of quan-tum phase estimation algorithm (PEA) needs O (1 /(cid:15) ) controlled- U operators and O (log(1 /(cid:15) ))ancillary qubits to reach an accuracy (cid:15) , where U is the time-evolution operator of a given systemHamiltonian. This represents a very stringentrequirement for the quantum resources in termsof number of qubits, gate fidelity and coherencetime, which is beyond the current or near-termNISQ computing technology. While the numberof ancillary qubits can be significantly reducedby adopting the recursive PEA, the generalcondition of deep quantum circuits in the PEAand the adiabatic state preparation (ASP) re-mains prohibitive for practical calculations onNISQ devices.A large class of algorithms adapted to NISQhardware have been developed in recent years,to exploit the new technology in Hamiltoniansimulations, or a wider set of optimization prob-1 a r X i v : . [ phy s i c s . c o m p - ph ] S e p ems. The variational quantum eigensolver(VQE) represents a most promising approach toaddress open quantum chemistry problems us-ing NISQ technologies.
Within VQE, thestate wavefunction is parameterized by a varia-tional ansatz. The cost function, which is usu-ally the expectation value of the system Hamilto-nian with respect to the variational ansatz, canbe efficiently calculated on NISQ devices withrelatively shallow circuits. The variational pa-rameters are adjusted to extremize the cost func-tion using classical computers. The effectivenessof VQE is determined by the variational wave-function form and the high-dimensional clas-sical optimization. The unitary coupled clus-ter ansatz with single and double excitations(UCCSD) represents a commonly used varia-tional form, motivated by the success of theCCSD method in classical quantum chemistrycalculations for systems free of multi-referencecharacters.
Many efforts have been devotedto improve the variational ansatz regarding thecomputational accuracy and variational circuitcomplexity.
For examples, the hardware-efficient ansatz prepares the variational state bya sequence of native two-qubit entangling gatesalternating with single qubit Euler rotationsto an initial state such as Hartree-Fock (HF)state. The k -UpCCGSD ansatz is composedof k products of generalized unitary paired dou-ble excitations and a complete set of generalizedsingle excitations, which can be systematicallyimproved toward exact answers. The quantumapproximate optimization algorithm (QAOA)provides an alternative way to construct a vari-ational ansatz in the form of applying the sys-tem Hamiltonian and mixing Hamiltonian to areference state. The variational wavefunctionform has also been proposed to be dynamicallyoptimized, which provides a compact system-dependent ansatz with systematically improv-able accuracies.
While the variational wavefunction form inVQE can be optimized to some extent, the num-ber of variational parameters is deemed to growwith the system size under study. The cost func-tion of VQE is generally non-convex in the high-dimensional parameter space, which renders theclassical optimization problem susceptible to lo- cal minima and very challenging. Recently, aquantum imaginary time evolution algorithm(QITE) has been proposed as an alternative ap-proach to determine eigenstates of an Hamilto-nian on quantum computers without the compli-cation of high-dimensional optimization. Theidea originates from the classical imaginary timeevolution algorithm, which is a sophisticatedway to obtain Hamiltonian eigenstates usingclassical computers.
Within the QITE algo-rithm, the non-unitary imaginary time evolutionoperator is replaced by a unitary operator whichpreserves the induced variation in the quantumstate. The unitary operator is uniquely deter-mined by solving a system of linear equationsand can be conveniently applied on quantumcomputers. The QITE method has been demon-strated by solving a set of finite spin modelson quantum simulators, including a two-siteIsing model and H dimer on real quantum de-vices. As the current and near term NISQ hardwaresuffers short coherence time, gate infidelity, andother noises, the direct application of QITEon real devices is limited by the rather deepquantum circuits, in particular for systems withlong-range correlations. The circuit depth growslinearly with the QITE steps, similar to the cir-cuit to study the quantum dynamics followingTrotter decomposition for the time-evolution op-erator.
In contrast, the VQE calculationswith an ansatz such as UCCSD features a varia-tional circuit of fixed depth. In this paper, wedevelop a resource-efficient “step-merged” QITE(smQITE) algorithm, which performs approxi-mate QITE calculations at fixed quantum circuitdepth. The smQITE method builds on the nu-merical observations that the accumulated uni-tary operators in the QITE calculation can oftenbe effectively combined. We will first presentthe smQITE formalism, followed by demonstra-tions that the smQITE method can producehigh-quality results beyond chemical accuracyon a set of molecules. We demonstrate that thecircuit depth of smQITE calculations can befurther reduced significantly by adopting com-pact wavefunction representations, such as theUCCSD variational form among others, which ef-fectively reduce the circuit depth down to that of2CCSD-VQE. It is shown that smQITE methodcan reach the accuracy of VQE with the sameUCCSD ansatz, in much fewer steps withoutresorting to high-dimensional optimizations. Fi-nally, we demonstrate the smQITE calculationsfor H dimer on a real quantum device, witha binding energy curve in reasonable accuracy.We argue that, supported by numerical evidence,a combination of smQITE with VQE offers away to address the highly complicated optimiza-tion problem of VQE when simulating largemolecules. To be self-contained, we first review the quan-tum imaginary time evolution algorithm pro-posed by Motta et al, and point out the limi-tations for practical implementations on NISQdevices. The presentation of the step-mergedQITE (smQITE) formalism then follows, whichaims to dramatically reduce the circuit depthof QITE calculations on quantum computers,hence is better adapted for the current and near-term quantum devices. Consider an N q -qubit system with Hamiltonianˆ H = (cid:80) M − m =0 ˆ h [ m ], which includes a sum of M weighted Pauli terms. The Pauli term ˆ h [ m ] is ageneral product of Pauli operators. The qubitHamiltonian can naturally describe spin- mod-els, or fermionic systems by mapping fermionicoperators to qubit operators. Starting froman initial state | Ψ (cid:105) , the imaginary time evolu-tion leads the system to the lowest eigenstate | Ψ f (cid:105) which has finite overlap with | Ψ (cid:105) in thelong time limit, | Ψ f (cid:105) = lim β →∞ e − β ˆ H | Ψ (cid:105) . (1)The imaginary time evolution can be carried out through Trotter decomposition e − β ˆ H = ( e − ∆ τ ˆ h [0] e − ∆ τ ˆ h [1] · · · ) N + O (∆ τ ) , (2)with the Trotter step size ∆ τ = βN . Literally,the above evolution operator e − β ˆ H consists of M × N steps, yielding an error of leading orderproportional to ∆ τ . For the convenience ofdiscussions later, we label the Trotter step by( n, m ) ≡ nM + m , with 0 ≤ n < N and 0 ≤ m < M . The associated intermediate state islabelled as Ψ ( n,m ) . After one additional Trotterevolution step, we have (cid:12)(cid:12) Ψ ( n,m )+1 (cid:11) = c − ( n,m ) e − ∆ τ ˆ h [ m ] (cid:12)(cid:12) Ψ ( n,m ) (cid:11) , (3)The wavefunction norm is given by c ( n,m ) = (cid:104) Ψ ( n,m ) | e − τ ˆ h [ m ] | Ψ ( n,m ) (cid:105) = 1 − τ (cid:104) Ψ ( n,m ) | ˆ h [ m ] | Ψ ( n,m ) (cid:105) + O (∆ τ ) , (4)where to leading order in ∆ τ the deviation of thenorm from unity is determined by the expecta-tion value of the Hamiltonian in the intermediatestate.The main idea of QITE algorithm is to replacethe non-unitary imaginary time Trotter evolu-tion operator in Eq. (3) by a unitary operatorwhich transforms Ψ ( n,m ) to a state closest toΨ ( n,m )+1 , (cid:12)(cid:12) Ψ ( n,m )+1 (cid:11) ≈ e − i ∆ τ ˆ A ( n,m ) (cid:12)(cid:12) Ψ ( n,m ) (cid:11) . (5)Here, ˆ A ( n,m ) is a Hermitian operator that canbe expanded in a complete Pauli basis set of adomain of D qubits around the support of ˆ h [ m ]:ˆ A ( n,m ) ≡ (cid:88) I a ( n,m ) I ˆ σ I . (6)Here, I = i i ...i D is a composite index run-ning through all the D qubits. The domain D includes at least all sites m , where ˆ h [ m ] actsnon-trivially. Generally, the domain size D canbe larger than the support of a qubit opera-tor due to correlation effects. The Pauli termˆ σ I = ˆ σ i ˆ σ i ... ˆ σ i D is a product of Pauli operators.3 σ i ∈ { I, X, Y, Z } is a Pauli operator associatedwith the i th qubit. Without loss of generality, a ( n,m ) is a set of real parameters of dimension4 D corresponding to rotation angles in the qubitHilbert space.In order to determine the operator ˆ A ( n,m ) , wedefine the change of the state wavefunction aftera Trotter imaginary time evolution step as (cid:12)(cid:12)(cid:12) ∆ ( n,m )0 (cid:69) = (cid:12)(cid:12) Ψ ( n,m )+1 (cid:11) − (cid:12)(cid:12) Ψ ( n,m ) (cid:11) ∆ τ (7) ≈ c − ( n,m ) − τ − c − ( n,m ) ˆ h [ m ] (cid:12)(cid:12) Ψ ( n,m ) (cid:11) , where the Trotter exponential operator in Eq.3is expanded to the first order of ∆ τ . Similarly,for the unitary evolution we define the variationof the state as. (cid:12)(cid:12)(cid:12) ∆ ( n,m )1 (cid:69) = e − i ∆ τ ˆ A ( n,m ) (cid:12)(cid:12) Ψ ( n.m ) (cid:11) − (cid:12)(cid:12) Ψ ( n,m ) (cid:11) ∆ τ ≈ − i ˆ A ( n,m ) (cid:12)(cid:12) Ψ ( n,m ) (cid:11) . (8)The objective function to be minimized is de-fined as f [ a ] = (cid:104) ∆ ( n,m )0 − ∆ ( n,m )1 | ∆ ( n,m )0 − ∆ ( n,m )1 (cid:105) (9)= f + (cid:88) I b I a ( n,m ) I + (cid:88) IJ a ( n,m ) I S IJ a ( n,m ) J with f = (cid:104) ∆ ( n,m )0 | ∆ ( n,m )0 (cid:105) , (10) b I = − i (cid:104) ∆ ( n,m )0 | ˆ σ I | Ψ ( n,m ) (cid:105) + c.c, (11) ≈ ic − ( n,m ) (cid:104) Ψ ( n,m ) | ˆ h [ m ]ˆ σ I | Ψ ( n,m ) (cid:105) + c.c., and S IJ = (cid:104) Ψ ( n,m ) | ˆ σ † I ˆ σ J | Ψ ( n,m ) (cid:105) . (12)The minimization of the function f [ a ] with re-spect to a ( n,m ) leads to a system of linear equa-tions (cid:0) S + S T (cid:1) a ( n,m ) = − b , (13)which is solved to determine the optimal expan-sion coefficients a ( n,m ) for the operator ˆ A ( n,m ) .Since f does not enter the above linear equa-tion, no explicit evaluation is needed. Quantumcomputers are employed to facilitate the setup of the linear equation (13) by determining the S -matrix and b -vector. As the quantum computa-tion only involves direct measurements of Pauliterms with respect to the state wavefunction,it is straightforwardly implemented on quan-tum devices. The number of linear equationsin Eq. (13) is 4 D , which scales exponentiallywith the number of qubits D in the relevantqubit domain. With increasing system size, thisrapidly becomes the bottleneck of the algorithm.We will discuss alternative ways to lift this con-straint in section 3.3. A key factor in determining the required quan-tum resources of the QITE approach is thepreparation of state Ψ ( n,m ) at Trotter step( n, m ), which will be repeated for all the mea-surements. The state Ψ ( n,m ) is constructed as (cid:12)(cid:12) Ψ ( n,m ) (cid:11) = m (cid:89) µ (cid:48) =0 e − i ∆ τ ˆ A ( n,µ (cid:48) ) × n − (cid:89) ν =0 M − (cid:89) µ =0 e − i ∆ τ ˆ A ( ν,µ ) | Ψ (cid:105) , (14)where the exponential operators are ordered ac-cording to the Trotter evolution path, as alsoillustrated in Fig. 1. Clearly, the depth of thestate preparation circuit grows linearly with theTrotter steps, which limits the system size andmaximal Trotter steps that the QITE algorithmcan perform in NISQ devices. In contrast, thevariational quantum algorithms, such as varia-tional quantum eigensolver with unitary coupledcluster ansatz, have an advantage of a varia-tional quantum circuit at fixed depth. Althoughsome approximate ways have been discussed inreferences, the linear growth of the quan-tum circuit depth with increasing Trotter stepshas not been addressed.Here, we propose a step-merged QITE(smQITE) approach to control the circuit depthat an effective single (or few) Trotter step level.The key idea is to combine Trotter evolutionunitaries along the state evolution path, whichact on a common set of qubits. The algorithmis schematically depicted in Fig. 1. While this4euristic approach does not become exact inthe limit N → ∞ , we show below that it leadsto results for ground state energies that arecomparable to VQE. This is remarkable as,unlike VQE, the smQITE approach does notrequire performing a difficult optimization ina high-dimensional feature space. We furtherdiscuss a systematic way to improve the accu-racy of smQITE at the cost of using deepercircuits. Finally, the smQITE method can alsobe combined with VQE, as it yields an efficientansatz for the ground state that can be furtheroptimized variationally. ℎ 0 ℎ[1] ℎ 2 ⋯ ℎ[𝑀 − 1] ⋯ℎ 𝜇 መ𝐴 (0,0) መ𝐴 (0,1) ⋯ መ𝐴 (0,𝑚) ⋯ መ𝐴 (0,𝑀−1) 𝜈 = 0 መ𝐴 (1,0) መ𝐴 (1,1) ⋯ መ𝐴 (1,𝑚) ⋯ መ𝐴 (1,𝑀−1) 𝜈 = 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ At Trotterstep (𝑛, 𝑚) መ𝐴 (𝑛,0) መ𝐴 (𝑛,1) ⋯ መ𝐴 (𝑛,𝑚) መ𝒜 (𝑛,𝑚)(0) መ𝒜 (𝑛,𝑚)(1) ⋯ መ𝒜 𝑛,𝑚𝑚 ⋯ መ𝒜 (𝑛,𝑚)(𝑀−1) መ𝒜 (𝑛,𝑚) ⋮𝜈 = 𝑛 Figure 1:
Schematic illustration of com-bining Trotter unitaries in smQITE algo-rithm.
At Trotter step ( n, m ) with imaginarytime evolution operator e − ∆ τ ˆ h [ m ] , a new unitaryoperator e − i ∆ τ ˆ A ( n,m ) is appended to the quantumcircuit. The operator ˆ A ( n,m ) is defined in a qubitdomain D m around the support of ˆ h [ m ]. A set ofPauli terms in the qubit Hamiltonian can sharea common qubit domain, as indicated by thedotted ellipse. As the accumulated operators { ˆ A ( ν,µ ) } at Trotter step ( n, m ) share the samequbit domain if they have the same column index µ , they can be combined to ˆ A ( µ )( n,m ) = (cid:80) ν A ( ν,µ ) .By defining a union of the Pauli basis set inall qubit domains D = D ∪ . . . ∪ D M , the op-erators { ˆ A ( µ )( n,m ) } can be further combined to asingle operator ˆ A ( n,m ) = (cid:80) M − µ =0 ˆ A ( µ )( n,m ) .More specifically, by commuting terms with a common index µ next to each other in Eq. (14),we can rewrite the state evolution in this equa-tion as (cid:12)(cid:12) Ψ ( n,m ) (cid:11) = e − i ∆ τ (cid:80) M − µ =0 ˆ A ( µ )( n,m ) | Ψ (cid:105) + O (∆ τ ) . (15)Here, we have defined ˆ A ( µ )( n,m ) = (cid:80) nν =0 A ( ν,µ ) for µ ≤ m . For µ > m , the summation stops at ν = n −
1. This expression combines the oper-ators ˆ A ( ν,µ ) with a common index µ that sharethe same Pauli basis in the qubit domain D µ around the support of h [ µ ]. By commuting theexponential terms to bring terms with a com-mon µ next to each other, we have generateda number of terms that are all of the order of∆ τ . We discuss the issue of the Trotter errorin more detail below.Further grouping is possible if different qubitdomains D µ of different ˆ h [ µ ] overlap and someˆ A ( µ ) -operators can be further combined. With-out loss of generality, we define an extendedPauli basis set { σ I } as the union of all the Paulibasis sets in the different qubit domains D µ ofthe Hamiltonian terms { ˆ h [ µ ] } . This allows usto maximally combine the operators ˆ A ( n,m ) ≡ (cid:80) M − µ =0 ˆ A ( µ )( n,m ) and represent it in the extendedPauli basis set, as illustrated in Fig. 1. ThesmQITE wavefunction at Trotter step ( n, m ) isthen given by (cid:12)(cid:12) Ψ ( n,m ) (cid:11) = e − i ∆ τ ˆ A ( n,m ) | Ψ (cid:105) , (16)which corresponds to a single effective Trotterstep. Note that in the original QITE paper, ˆ h [ m ] is defined according to operational local-ity and can be a sum of Pauli terms sharing acommon qubit domain. Therefore, an effectivecombination of ˆ A ( ν,µ ) over index µ at a commonqubit domain has been performed, albeit at eachindividual Trotter step ν .In the case of ab initio molecular Hamiltonianswhere long-range one-body and two-body opera-tors are present, it is often the case that the setof { ˆ h [ µ ] } Hamiltonian terms share a commondomain of qubits, that often spans the full sys-tem. It is thus natural to consider the evolutionunder the full ˆˆ H : (cid:12)(cid:12) Ψ ( n +1) (cid:11) = c − ( n ) e − ∆ τ ˆ H (cid:12)(cid:12) Ψ ( n ) (cid:11) , (17)5ather than Eq. (3). Note that we do not in-troduce the domain index µ as there is only asingle domain spanning the full system. Thestate evolution Eq. (15) thus reads as (cid:12)(cid:12) Ψ ( n ) (cid:11) = e − i ∆ τ ˆ A ( n ) | Ψ (cid:105) , (18)with ˆ A ( n ) = (cid:80) n − ν =0 ˆ A ( ν ) = (cid:80) I (cid:16)(cid:80) n − ν =0 a ( ν ) I (cid:17) ˆ σ I ,where we have combined Trotter unitaries withdifferent step index ν . The step of combinedTrotter evolution of the state wavefunctionacross the whole set of { ˆ h [ µ ] } does not changethe quantum circuit depth. However, it poten-tially saves time for systems with largely over-lapping qubit domains { D µ } , such as molecules,due to the prevalence of nonlocal one-body andtwo-body operators. Furthermore, it introducea new perspective that a compact representationof the ˆ A n operator can be obtained through vari-ational wavefunction forms of VQE, which willbe detailed in section 3.3. It has been discussedrecently that the Pauli operator ordering inthe Trotterized circuits of the VQE-UCCSD ap-proach can introduce significant errors in energyevaluations beyond chemical accuracy. As aTrotterized form is also adopted in the smQITEmethod, similar operator ordering effects couldexist. Nevertheless, we will demonstrate thatdecent numerical results from smQITE calcula-tions can already be obtained without exploitingoptimum Pauli operator orderings. For the pur-pose of reproducibility of numerical results, allour calculations, including explicit ordered listof Pauli operators, are publicly accessible in theonline repository. As the number of Trotter steps N increases,the smQITE approach maintains a favorablefixed circuit depth. This is in stark contrast tothe linear growth of the depth with N found inQITE. But the gain in quantum resource effi-ciency is obtained at a price. In the worst casescenario where none of the operators { ˆ A ( ν,µ ) } commute with each other and all leading Trot-ter errors are of the same sign and add up, theabove step merging procedure introduces a con-stant error. The smQITE approach thus losesthe mathematical rigor of QITE and does notbecome exact in the limit of small Trotter step size ∆ τ = β/N N →∞ −−−→
0. The smQITE methodshould thus be regarded as a heuristic approachthat can still work well in the average case, aswe demonstrate for a number of examples below.Even in this worst case scenario where the Trot-ter error is uncontrolled, the energy obtainedfrom the smQITE ansatz is still a variationalupper bound, and the smQITE wavefunction (cid:12)(cid:12) Ψ ( n,m ) (cid:11) in Eq. (16) can be used as a startingpoint for further variational optimization us-ing VQE. It is worth noting that the operatorˆ A ( n,m ) in Eq. (6) is first determined variation-ally at each smQITE step, and subsequentlymerged into the preceding unitary operators. Inother words, the QITE procedure is followedinitially, but in order to avoid a further growthof the circuit depth the preparation of state (cid:12)(cid:12) Ψ ( n,m ) (cid:11) is approximately achieved by using thestep-merged unitary in Eq. (16). Therefore, theeffective single-step smQITE ansatz is generallydifferent form the a QITE ansatz with a sin-gle Trotter step. In fact, because the smQITEapproach coincides with QITE at the first Trot-ter step where no combination of unitaries hasbeen performed, smQITE can always achieve thesingle-step QITE result as an upper bound. Theerror in smQITE calculations should be equalor smaller than that by effectively reducing theTrotter decomposition in Eq. (2) from order N to order 1, which will also be demonstrated nu-merically in section 3.2.Finally, let us describe a way to detect theTrotter errors induced by the step merging pro-cess and a way to iteratively reduce it. Oneway to estimate this error is to compare theenergy of the state obtained from merging allTrotter steps ν into a single effective step,ˆ A ( µ )( n,m ) = (cid:80) nν =0 A ( ν,µ ) , versus merging them intotwo effective steps, ˆ A ( ν,µ )( n,m ) = (cid:80) n ( ν +1) ν (cid:48) = n ν A ( ν (cid:48) ,µ ) with ν = 0 ,
1. If the energy decreases whenusing more effective Trotter steps, this processcan be repeated until convergence. Obviously,this process approaches the original QITE limitif we increase the range of the index ν and hencerequires increasingly deep circuits to prepare thewavefunction (cid:12)(cid:12) Ψ ( n,m ) (cid:11) .6 Application of step-mergedQITE to quantum chem-istry
In this section, we show that highly accurateresults beyond chemical accuracy can be ob-tained for the smQITE calculations for a setof molecules. In particular, we prove numeri-cally that the high accuracy of smQITE methodcannot be obtained by instead using a singleTrotter step calculation, even when using an op-timum step size ∆ τ . We further propose a wayto effectively adopt the variational wavefunctionform of VQE into smQITE. For a number ofmolecules, we show that smQITE yields resultsof similar accuracy as VQE with the same fixedvariational ansatz, yet with much fewer stepsand shallower circuits. Finally, we report resultsof smQITE calculations performed on RigettiQPUs. Consider an ab initio nonrelativistic molecularelectron Hamiltonianˆ H = (cid:88) pq (cid:88) σ h pq ˆ c † pσ ˆ c qσ + 12 (cid:88) pqrs (cid:88) σσ (cid:48) h pqrs ˆ c † pσ ˆ c † rσ (cid:48) ˆ c sσ (cid:48) ˆ c qσ , (19)with the one-electron core part of the Hamilto-nian given by h pq = (cid:90) d r φ ∗ p ( r )( T + V ion ) φ q ( r ) , (20)and the two-electron Coulomb integral h pqrs = (cid:90) d r (cid:90) d r (cid:48) φ ∗ p ( r ) φ ∗ r ( r (cid:48) ) V ee φ s ( r (cid:48) ) φ q ( r ) . (21)Here p, q, r, s are composite indices for atomand orbital, and σ is spin index with valuesof α for spin-up and β for spin-down. T isthe kinetic energy operator, V ion is the ionicpotential operator and V ee the Coulomb inter-action operator. { φ ( r ) } is a set of basis orbital functions, which are obtained from the stan-dard STO-3G minimal basis set. In the follow-ing smQITE calculations of molecules, a quan-tum chemistry package PySCF is first used toget the restricted Hartree-Fock(HF) solution. The molecular Hamiltonian (Eq. 19) is thentransformed to the molecular orbital represen-tation for the convenience of preparation of theinitial HF state in quantum computer. Thequbit representation of the Hamiltonian is ob-tained by parity transformation, with two qubitsreduced by exploiting the conservation of to-tal number of electrons and Z -component ofthe total spin operator, e.g., the Z symme-try. The smQITE code is implemented usingmodules from Qiskit and Forest, and isavailable as a module in the open-source packagePyGQCE. The smQITE method is a generalHamiltonian eigensolver, with potential appli-cations beyond quantum chemistry problems,such as the impurity models. Figure 2 shows the evolution of the Hamiltonianexpectation value E as a function of β = n × ∆ τ for H dimer and H chain in panel (a) and (d),which quickly converges to the exact result fromthe initial value of the HF solution. In the mid-dle panels (b) and (e), we plot the energy E after a single QITE step upon the initial HFwavefunction with varying the Trotter step size∆ τ , which shows a polynomial behavior with aunique minimum E (1) min at an optimal step size∆ τ opt . Accidentally, E (1) min coincides with the ex-act energy for H , which is due to the simplestructure of the Hamiltonian. Generally, E (1) min will be higher than the exact result. For thecase of H , the energy is overestimated by 5 kcal/mol , which is beyond the chemical accu-racy of 1 kcal/mol . For comparison, ∆ τ opt isused as the fixed step size ∆ τ for the smQITEcalculations. Fig. 2(b) clearly shows that thesmQITE calculation of H can reach a muchhigher accuracy ( 10 − kcal/mol ) after a fewsteps. The calculations are performed on a wave-function simulator as implemented in Forest, .0 2.5 5.01.131.12 E ( H a r . ) (a) smQITEexact 0.0 0.5 1.01.131.12(b) H single-step QITE 0.0 2.5 5.00.00.51.0 C i r c u i t d e p t h E ( H a r . ) (d) 0.0 0.5 1.02.182.162.142.12(e) H C i r c u i t d e p t h Figure 2:
Energy convergence and fixed circuit depth of the smQITE method.
Upperpanels show the energy evolution as a function of merged QITE steps (a), the energy as a functionof single QITE step size (b), and the quantum circuit depth of QITE and smQITE calculations (c)for H dimer at bond length 0 . chain at bond length 0 . is far beyond the capability of the currentNISQ devices.which is equivalent to perfect measurements onfault-tolerant quantum computers. We estimatethe quantum circuit depth by counting the num-ber of two-qubit controlled-NOT (CNOT) gatesin the algorithms, which are shown in Fig. 2(c)and (f) for calculations of H and H , respec-tively. As expected, the smQITE circuit has afixed depth at 8 for H and 14208 for H . In con-trast, the QITE circuit grows linearly in depthas the QITE step proceeds.We further apply the smQITE method to aset of molecules to map out the full binding anddissociation energy curves, which give a morecomplete assessment of the computational accu-racy. The smQITE results are reported with theexact curves in Fig. 3 for molecules H (a), H (b),LiH(c) and HF(d). The associated error, definedas the energy difference between the smQITEand exact diagonalization (ED, or full configu-ration interaction, FCI) calculations, is plottedin the lower panels (e-h). In all the cases, thesmQITE calculations yield energies in much bet- ter agreement with the exact answers beyond thechemical accuracy. The Hartree-Fock bindingenergy curves have also been shown for refer-ence, which provides a measure for the electroncorrelation effects in the system. For polyatomicmolecules composed of atoms with open-shell,such as H, Li and F atom, the correlation en-ergy, defined as the energy difference betweenHartree-Fock and exact calculations, increasesas the molecule is uniformly stretched towardthe dissociation limit. The smQITE methodrecovers almost all the correlation energy.In the Hartree-Fock calculations for LiHmolecule, the STO-3G minimal basis set de-scribes 1 s -orbital for H and 1 s , 2 s , and 2 p -orbitals for Li. The Li 1 s orbital is kept in thecore, as it is fully occupied and deep in energylevel. The 2 p y and 2 p z -orbitals are discardedbecause they do not participate in bonding andremain empty due to the symmetry constraintsfor the geometry aligned along x -axis. There-fore, four qubits are needed to represent the8 E ( H a ) (a) H H LiH
HFsmQITEExact 1 298.698.598.498.398.2 (d) HF Å )10 E rr o r ( k c a l / m o l ) (e) 1 2R ( Å )10 (f) 2 4R ( Å )10 (g) 1 2R ( Å )10 (h) Figure 3:
Binding energy curves from smQITE calculations.
The binding energy curves ofH (a), H chain (b), LiH (c) and HF (d) molecules from smQITE calculations are plotted togetherwith exact and HF results. The error of smQITE calculations, E smQIT E − E Exact , as shown in panels(e-h), are well below the chemical accuracy threshold, as indicated by the horizontal dotted line inthe lower panels.LiH Hamiltonian with Z symmetry. In the caseof HF molecule, the minimal basis contains H1 s -orbital and F 1 s , 2 s , and 2 p -orbitals. Herewe keep all the orbitals in the calculations, ex-cept F 1 s and 2 s -orbitals, as they are muchdeeper in the core. Thus six qubits are used torepresent the Hamiltonian of HF molecule, likethe simulation of H . The detailed setup of thecalculations can be found in online repository. A limitation in the above smQITE calculationsis that the dimension of the system of linearequations (13) grows exponentially as 4 D withrespect to qubit domain size D determined bythe electron correlations. To simulate systemsof increasing size, some approximate treatmenthas been introduced in the reference. Specifi-cally, QITE calculations can be performed witha reduced qubit domain size D (cid:48) , by choosing asubset of Pauli terms of length L ≤ D (cid:48) to repre- sent ˆ A in Eq. (6). This approximation becomesequivalent to mean-field solution for D (cid:48) = 1and approaches to exact result with increasing D (cid:48) . Approximate QITE calculations have beendemonstrated to be quite effective for 1D short-range spin models up to 20 qubits, as well as for1D long-range Heisenberg Hamiltonian, albeitof much shorter 6 qubits.The ab initio molecular Hamiltonian usuallyhas a much more complex structure than thespin models aforementioned, due to the pres-ence of long-range one-body hopping and two-body interaction terms. Hence the qubit domainassociated with a Pauli term in the Hamilto-nian could be significantly larger. For example,the qubit representation of the electron Hamil-tonian of H molecule contains a Pauli termwhich acts on all the qubits, independent ofthe choice for encoding: Jordan-Wigner, parityor Bravyi-Kitaev transformation. As a re-sult, the qubit domain should include all thequbits in the calculations, as adopted in thesmQITE calculations reported before. Note that9 .0 1.5 2.075.074.874.674.474.2 E ( H a ) H O (a) 1 2 315.615.415.215.014.814.614.4 BeH (b) HFsmQITEVQEExact 1 23.02.52.01.51.0 H (c)1.0 1.5 2.0 R O H ( Å )10 E rr o r ( k c a l / m o l ) (d) 1 2 3 R Be H ( Å )10 (e) 1 2 R H H ( Å )10 (f) Figure 4:
Application of smQITE to molecules using simplified UCCSD ansatz.
Thebinding energy curve from smQITE calculations is shown for H O in panel (a), BeH in (b) and H chain in (c). The results from exact diagonalization, VQE with the same simplified UCCSD ansatz,and HF calculations are also shown for comparison. The errors of smQITE and VQE calculations,which measure the difference from the exact answers, are plotted in panels (d-f).the operator domain size is dependent of specificrepresentations. For fermionic systems, the lin-ear equation (13) can also be constructed usingfermionic representation of ˆ h [ m ] and ˆ A , whichwill be expanded in the basis of tensor productof fermionic operators { I, ˆ c pσ , ˆ c † pσ , ˆ c † pσ ˆ c pσ } . Inthis fermionic representation, the domain size ofˆ h [ m ] generally depends on the many-body stateit acts upon, and the length of Pauli term ofˆ h [ m ] in qubit representation becomes irrelevant.To extend the application of smQITE tomolecules of increasing size, we propose an al-ternative approach to reduce the computationalcomplexity. The dimension of the system of lin-ear equations (13) can be effectively reduced bychoosing an optimal subset of Pauli basis for therepresentation of Hermitian operator ˆ A in Eq. 6.Note that the smQITE approach produces awavefunction ansatz in Eq. 18, which resemblesthe variational wavefunction form of VQE, such as the UCCSD ansatz in qubit representation (cid:12)(cid:12)(cid:12) Ψ( (cid:126)θ ) (cid:69) = e ˆ T ( (cid:126)θ ) − ˆ T † ( (cid:126)θ ) | Ψ (cid:105) = e − i (cid:80) j θ j f j ( { ˆ σ } ) | Ψ (cid:105) . (22)Here f j ( { ˆ σ } ) is a weighted sum of Pauli termsassociated with the j th fermionic operator forthe single or double excitation. However, theUCCSD ansatz includes much fewer Pauli terms,which naturally provides an alternative compactPauli basis set, rather than a complete Paulibasis set of exponentially growing dimension4 D , for the representation of the Hermitian ˆ A operator in Eq. 18, and equivalently reduces thedimension of the system of linear equations (13).As each f j ( { ˆ σ } ) in Eq. 22 usually includesseveral Pauli terms (2 for single excitations and8 for double excitations), this translates to aquite significant overhead for the quantum cir-cuit. Indeed, it has been demonstrated thatreformulating the exponential ansatz (22) utiliz-ing directly the qubit evolution operators (Pauliterms) leads to a generally much shallower cir-10uit. However, the introduced overhead forsimulations is a screening process for selectingqubit operators, which inevitably renders theansatz system-dependent and lose the general-ity of the wavefunction form of the UCCSDansatz in Eq. 22. Here we take an alternativeapproach to simplify UCCSD ansatz preservingthe general wavefunction form without operator-screening. The proposal is to replace f j ( { ˆ σ } ) byone of the list of Pauli terms in f j . The advan-tage is that it preserves the general variationalwavefunction form and extremely easy to imple-ment based on an existing UCCSD code. Al-though the simplified UCCSD (sUCCSD) ansatzremains generally subject to static correlationerror as the UCCSD ansatz, it serves well ourpurpose here to demonstrate that adopting thecompact list of Pauli operators in the UCC-type exponential ansatz enables quite accuratesmQITE calculations of molecules with increas-ing size. In numerical examples to be discussedbelow, we do not find a significant effect on thespecific choice of Pauli term in f j ( { ˆ σ } ) basedon our preliminary tests. A systematic study onthe optimum choice of Pauli terms and the effecton the quantum circuit structure and numericalaccuracy is of interest and will be addressed infuture work. The details of our calculations canbe found in the open repository. We includeexplicitly lists of Pauli basis set ordered accord-ing to real calculations for reference, since it hasbeen demonstrated recently that different qubitoperator orders in VQE calculations with theTrotterized form of UCC ansatz could affect finalresults quite significantly. We note that thevariational ansatz-based quantum simulation ofimaginary time evolution (VQITE) recently pro-posed by McArdle, et al resembles our smQITEmethod with representations from VQE ansatzin some aspect. However, VQITE is derived us-ing McLachlans time-dependent variational prin-ciple and the guiding equations are completelydifferent.
Furthermore, the evaluation of co-efficients in the VQITE equation of motion onquantum computers introduces additional over-head of an ancillary qubit and generally compli-cated controlled-unitary operators.
We demonstrate the smQITE calculationswith the above sUCCSD Pauli operator set on molecules H O, BeH and H , as shown in Fig. 4.The binding energy curves from exact diagonal-ization and VQE calculations with the samesUCCSD ansatz are also shown for compari-son. The HF results are given as a reference toestimate the dynamic and static correlation ef-fects. The smQITE calculation results generallystay in close agreement with VQE calculations,and they both reach chemical accuracy whenthe bond length near or smaller then the ener-getically optimum value, where the dynamicalcorrelation effect dominates. As the bond lengthincreases towards the dissociation limit wherethe static correlation takes over, the errors startto go beyond chemical accuracy, due to the singlereference nature of the sUCCSD ansatz. ThesmQITE and VQE binding curves are gener-ally very smooth, except one energy point ofH O at O-H bond length of 2.0˚A, which we at-tribute to a possible limitation of the sUCCSDvariational wavefunction form. We expect thatmore sophisticated variational forms, such as k -UpCCGSD or UCC with paired double exci-tations plus orbital optimization, may givebetter compact Pauli representation for smQITEcalculations, and improve the accuracy near dis-sociation limit.In QITE or smQITE calculations, the Trotterstep size ∆ τ can significantly affect the con-vergence speed of the Hamiltonian expectationvalue. Generally, ∆ τ can be gradually increasedfor molecules with increasing bond length forfaster convergence, where static correlation ef-fects become stronger. Take the smQITE calcu-lation of H in Fig. 3 as an example. It takes only3 smQITE steps to reach chemical accuracy with∆ τ = 0 . at bond length R = 0 . R = 2 . τ = 1 .
5, it takes only 4 steps toreach the chemical accuracy. Although the opti-mum ∆ τ is system-dependent and not known apriori, smQITE calculations with auto-tuned ∆ τ can be easily implemented. More precisely, it isfeasible to choose a large enough initial value for∆ τ to start the smQITE calculation. The energyat each smQITE step is monitored. If the energystarts to increase, ∆ τ will be scaled down by aconstant factor (e.g., 5) and the smQITE solu-11ion returns to the lowest energy point achievedin the previous steps. The smQITE calculationthen continues with the updated ∆ τ , which canbe further reduced accordingly. The smQITEcalculation terminates if ∆ τ is sufficiently small(e.g., ∆ τ < .e − ) or energy converges to thedesired accuracy. The smQITE calculations forH O, BeH and H in Fig. 4 are carried outwith the Trotter step size ∆ τ dynamically ad-justed as described above. All the calculationsconverge in energy of 0 . mHa within 80 steps.In contrast, the VQE calculations require fromseveral hundred up to two thousand steps toachieve similar convergence, if the sequentialleast squares programming (SLSQP) optimiza-tion method is used. Significantly more stepsare necessary if the sUCCSD ansatz is optimizedusing constrained optimization by linear approx-imation (COBYLA) method.Rigorously speaking, the VQE step, character-ized by the calculation of Hamiltonian expecta-tion value with respect to an updated wave func-tion, can take much less time than the smQITEstep, as many additional terms defining Eq. 13must be evaluated in the smQITE method. Con-sequently, the computational time of smQITEand VQE calculations is comparable. For ex-ample, it takes about 102 seconds for smQITEand 188 seconds for VQE calculation of the H chain at R = 1 . close to dissociation limit. In principle,VQE should always lead to an energy, which isthe same or lower than the smQITE result at theglobal minimum in its variational space, giventhat both approaches share the same variational wavefunction form. In fact, VQE can furtherimprove the smQITE energy if the smQITEsolution is used as the starting point for thevariational optimization. For example, the finalenergy can be further improved by more than 2mHa for BeH at bond length of 3.8˚A. This sug-gests that a combination of smQITE and VQEmay offer a way to overcome the challenge ofhigh-dimensional non-convex optimization prob-lem inherent in the VQE approach. Note thatthe convergence of VQE calculations can alsobe improved by utilizing the analytical gradientof the cost function. However, the evaluationof gradient on quantum computers introducesthe similar overhead of an ancillary qubit andcontrolled-unitary operators as in the VQITEmethod mentioned above. In the above Hartree-Fock calculations forH O molecule, the STO-3G minimal basis setdescribes 1 s -orbital for H and 1 s , 2 s , and 2 p -orbitals for O. The O 1 s and 2 s orbitals arekept in the core, as they are fully occupied anddeep in energy level. Therefore, eight qubits areneeded to represent the H O Hamiltonian with Z symmetry. In the case of BeH molecule,the minimal basis contains H 1 s -orbital and Be1 s , 2 s , and 2 p -orbitals. Here we keep Be 1 s or-bital in the core and remove Be 2 p z as it doesn’tparticipate in bonding for the molecule alignedin xy-plane. Therefore, eight qubits are usedto represent the Hamiltonian of BeH molecule,like the simulation of H . The number of Pauliterms in the Hamiltonian of H O, BeH and H is 252, 252, and 919, and the associated num-ber of variational parameters of the sUCCSDansatz, or equivalently the dimension of thePauli basis in smQITE calculations, is 54, 54,and 59, respectively. Remarkably, the smQITEcalculation with sUCCSD ansatz for moleculesperforms much better than the previously pro-posed sparse representation based on reducingthe qubit domain size of ˆ h [ m ] in the Hamilto-nian. For example, the smQITE calculationwith qubit domains reduced to D (cid:48) = 4, whichamounts to a much larger dimension of 3648for the Pauli basis set, yields an energy over 30mHa higher for H O molecule at R = 1˚A.12 .5 1.0 1.5 2.0 2.5R( Å )1.21.00.80.60.40.20.00.2 E ( H a ) HFExact smQITE-simulationsmQITE-device
Figure 5:
Demonstration of smQITE cal-culations of H molecule on Rigetti quan-tum device. The binding energy curve fromsmQITE calculations using wavefunction sim-ulator and Rigetti Aspen-4 device are shown,together with the results from ED (FCI) and HFfor reference. Inset: smQITE energy evolutionas a function of Trotter steps β = n × ∆ τ forH molecule at R = 2 . τ = 0 . Finally, we benchmark the smQITE calcula-tions on real quantum devices through the quan-tum cloud service provided by Rigetti. TheH molecule is chosen as an example for thedemonstration. The smQITE calculations witha compact Pauli basis from sUCCSD ansatz arecarried out to make an efficient use of quan-tum resources. As a result, the Pauli basis iscomposed of a single Pauli term X Y , whichis essentially the same of the UCCSD ansatzemployed in the literature for VQE calculationsof H or other similar two-orbital systems. Here X ( Y ) is the x(y)-component of a single qubitPauli operator. Figure 5 shows smQITE cal-culations for the total energy of H moleculeas a function of bond length using wavefunc-tion simulator and Rigetti Aspen-4 device. Thewavefunction simulation data overlap with the ED (FCI) results, because the sUCCSD ansatzis exact for this example. The smQITE calcula-tions on real device follow the exact curve quitewell, with errors on the order of 10 mHa. Theinset plots the energy evolution as a function ofTrotter step β = n × ∆ τ with fixed ∆ τ = 0 . molecule at R = 2 . . The fidelity of the two-qubit gate is about 95%. At each smQITE step,five different quantum circuits are constructedto measure the expectation values of eight Pauliterms, with some of them measured simulta-neously due to mutual commutation. Readouterror symmetrization and mitigation, as imple-mented in the Forest package, have been usedto reduce the effects of noise. The readout sym-metrization is performed by exhaustively flip-ping the qubits before the measurements (2 = 4ways for the two-qubit system), and subsequentflipping back the measurement outcomes. Asthe effect of symmetric measurement error isto scale the expectation value of the Pauli ob-servable by a noise-dependent factor, the errormitigation is to rescale the measured observ-able expectation value accordingly. The readoutsymmetrization comes at a price, which effec-tively introduce 4 × shots during themeasurement of Pauli terms for each circuit.13 Conclusion
In conclusion, the smQITE algorithm has beendeveloped as a resource-efficient version of QITE,which adapts better to the current and near-term NISQ hardware. Highly accurate resultshave been demonstrated for the smQITE calcu-lations of the binding and dissociation energycurves of a set of molecules. To simulate molec-ular Hamiltonian of increasing size, a compactrepresentation of the smQITE unitary evolutionoperators has been proposed by adopting a vari-ational wavefunction form in VQE calculations.It has been shown that the smQITE calculationsconverge much faster, and achieve the similaraccuracy as VQE with the same variational cir-cuit. Finally, we demonstrate smQITE calcula-tions on a Rigetti quantum device, where thebinding energy curve of H molecule has beenobtained with a reasonable accuracy. Numericalresults suggest that the inherent challenge in thenon-convex high-dimensional optimization prob-lem of VQE calculations can potentially be ad-dressed by a combination of smQITE and VQE,where the fast-converged smQITE solution canbe fed into VQE for further optimizations. Acknowledgements
This work was supported by the U.S. Depart-ment of Energy (DOE), Office of Science, BasicEnergy Sciences, Materials Science and Engi-neering Division. The research was performedat the Ames Laboratory, which is operated forthe U.S. DOE by Iowa State University underContract No. DE-AC02-07CH11358.
References (1) Feynman, R. P. Simulating physics withcomputers.
Int. J. Theor. Phys. , ,467–488.(2) Aspuru-Guzik, A.; Dutoi, A. D.;Love, P. J.; Head-Gordon, M. Simu-lated Quantum Computation of MolecularEnergies. Science , , 1704–1707. (3) McArdle, S.; Endo, S.; Aspuru-Guzik, A.;Benjamin, S. C.; Yuan, X. Quantum com-putational chemistry. Rev. Mod. Phys. , , 015003.(4) Cao, Y.; Romero, J.; Olson, J. P.; Deg-roote, M.; Johnson, P. D.; Kieferov´a, M.;Kivlichan, I. D.; Menke, T.; Peropadre, B.;Sawaya, N. P., et al. Quantum chemistryin the age of quantum computing. Chem.Rev. , , 10856–10915.(5) Babbush, R.; Wiebe, N.; McClean, J.; Mc-Clain, J.; Neven, H.; Chan, G. K.-L. Low-depth quantum simulation of materials. Phys. Rev. X , , 011044.(6) Bauer, B.; Wecker, D.; Millis, A. J.; Hast-ings, M. B.; Troyer, M. Hybrid quantum-classical approach to correlated materials. Phys. Rev. X , , 031045.(7) Yao, Y.; Zhang, F.; Wang, C.-Z.; Ho, K.-M.; Orth, P. P. Gutzwiller HybridQuantum-Classical Computing Approachfor Correlated Materials. arXiv:2003.04211 ,(8) Kitaev, A. Y. Quantum measurementsand the Abelian stabilizer problem. arXiv:quant-ph/9511026 ,(9) Abrams, D. S.; Lloyd, S. Quantum Al-gorithm Providing Exponential Speed In-crease for Finding Eigenvalues and Eigen-vectors. Phys. Rev. Lett. , , 5162–5165.(10) Farhi, E.; Goldstone, J.; Gutmann, S. Aquantum approximate optimization algo-rithm. arXiv:1411.4028 ,(11) Peruzzo, A.; McClean, J.; Shadbolt, P.;Yung, M.-H.; Zhou, X.-Q.; Love, P. J.;Aspuru-Guzik, A.; Obrien, J. L. A vari-ational eigenvalue solver on a photonicquantum processor. Nat. Commun. , , 4213.(12) Wecker, D.; Hastings, M. B.; Troyer, M.Progress towards practical quantum varia-tional algorithms. Phys. Rev. A , ,042303.1413) McClean, J. R.; Romero, J.; Babbush, R.;Aspuru-Guzik, A. The theory of variationalhybrid quantum-classical algorithms. NewJ. Phys. , , 023023.(14) OMalley, P. J.; Babbush, R.;Kivlichan, I. D.; Romero, J.; Mc-Clean, J. R.; Barends, R.; Kelly, J.;Roushan, P.; Tranter, A.; Ding, N., et al.Scalable quantum simulation of molecularenergies. Phys. Rev. X , , 031007.(15) Biamonte, J.; Wittek, P.; Pancotti, N.;Rebentrost, P.; Wiebe, N.; Lloyd, S. Quan-tum machine learning. Nature , ,195–202.(16) Kandala, A.; Mezzacapo, A.; Temme, K.;Takita, M.; Brink, M.; Chow, J. M.; Gam-betta, J. M. Hardware-efficient variationalquantum eigensolver for small moleculesand quantum magnets. Nature , ,242–246.(17) Ryabinkin, I. G.; Yen, T.-C.; Genin, S. N.;Izmaylov, A. F. Qubit coupled clustermethod: a systematic approach to quan-tum chemistry on a quantum computer. J. Chem. Theory Comput. , , 6317–6326.(18) Lee, J.; Huggins, W. J.; Head-Gordon, M.;Whaley, K. B. Generalized unitary coupledcluster wave functions for quantum com-putation. J. Chem. Theory Comput. , , 311–324.(19) Grimsley, H. R.; Economou, S. E.;Barnes, E.; Mayhall, N. J. An adaptivevariational algorithm for exact molecularsimulations on a quantum computer. Nat.Commun. , , 1–9.(20) Tang, H. L.; Barnes, E.; Grimsley, H. R.;Mayhall, N. J.; Economou, S. E. qubit-ADAPT-VQE: An adaptive algorithm forconstructing hardware-efficient ansatze ona quantum processor. arXiv:1911.10205 ,(21) Scuseria, G. E.; Janssen, C. L.; Schae-fer Iii, H. F. An efficient reformulation of the closed-shell coupled cluster singleand double excitation (CCSD) equations. J. Chem. Phys. , , 7382–7387.(22) Bartlett, R. J.; Musia(cid:32)l, M. Coupled-clustertheory in quantum chemistry. Rev. Mod.Phys. , , 291.(23) Harsha, G.; Shiozaki, T.; Scuseria, G. E.On the difference between variational andunitary coupled cluster theories. J. Chem.Phys. , , 044107.(24) Sokolov, I. O.; Barkoutsos, P. K.; Olli-trault, P. J.; Greenberg, D.; Rice, J.; Pis-toia, M.; Tavernelli, I. Quantum orbital-optimized unitary coupled cluster meth-ods in the strongly correlated regime: Canquantum algorithms outperform their clas-sical equivalents? J. Chem. Phys. , , 124107.(25) Motta, M.; Sun, C.; Tan, A. T.;ORourke, M. J.; Ye, E.; Minnich, A. J.;Brand˜ao, F. G.; Chan, G. K.-L. Determin-ing eigenstates and thermal states on aquantum computer using quantum imagi-nary time evolution. Nat. Phys. , ,205–210.(26) Wick, G. C. Properties of Bethe-SalpeterWave Functions. Phys. Rev. , ,1124–1134.(27) Lehtovaara, L.; Toivanen, J.; Eloranta, J.Solution of time-independent Schr¨odingerequation by the imaginary time propaga-tion method. J. Comput. Phys. , ,148–157.(28) Yeter-Aydeniz, K.; Pooser, R. C.; Siop-sis, G. Practical Quantum Computation ofChemical and Nuclear Energy Levels Us-ing Quantum Imaginary Time Evolutionand Lanczos Algorithms. arXiv:1912.06226 ,(29) Lamm, H.; Lawrence, S. Simulation ofNonequilibrium Dynamics on a QuantumComputer. Phys. Rev. Lett. , ,170501.1530) Smith, A.; Kim, M.; Pollmann, F.;Knolle, J. Simulating quantum many-bodydynamics on a current digital quantumcomputer. npj Quantum Inf. , , 1–13.(31) Bravyi, S. B.; Kitaev, A. Y. Fermionicquantum computation. Ann. Phys. , , 210–226.(32) Tranter, A.; Sofia, S.; Seeley, J.;Kaicher, M.; McClean, J.; Babbush, R.;Coveney, P. V.; Mintert, F.; Wilhelm, F.;Love, P. J. The B ravyi–K itaev transfor-mation: Properties and applications. Int.J. Quantum Chem. , , 1431–1441.(33) Trotter, H. F. On the product of semi-groups of operators. Proc. Am. Math. Soc. , , 545–551.(34) Barkoutsos, P. K.; Gonthier, J. F.;Sokolov, I.; Moll, N.; Salis, G.; Fuhrer, A.;Ganzhorn, M.; Egger, D. J.; Troyer, M.;Mezzacapo, A., et al. Quantum algo-rithms for electronic structure calculations:Particle-hole Hamiltonian and optimizedwave-function expansions. Phys. Rev. A , , 022322.(35) Nishi, H.; Kosugi, T.; Matsushita, Y.-i. Implementation of quantum imaginary-time evolution method on NISQ devices:Nonlocal approximation. arXiv:2005.12715 ,(36) Grimsley, H. R.; Claudino, D.;Economou, S. E.; Barnes, E.; May-hall, N. J. Is the Trotterized UCCSDAnsatz Chemically Well-Defined? J.Chem. Theory Comput. , , 1–6.(37) Yao, Y.; Gomes, N.; Zhang, F.;Berthusen, N.; Wang, C.-Z.; Ho, K.-M.; Orth, P. Step-merged quantumimaginary time evolution (smQITE)calculations for quantum chemistry. https://10.6084/m9.figshare.12574154 ,(38) Sun, Q.; Berkelbach, T. C.; Blunt, N. S.;Booth, G. H.; Guo, S.; Li, Z.; Liu, J.; McClain, J. D.; Sayfutyarova, E. R.;Sharma, S., et al. PySCF: the Python-based simulations of chemistry framework. Wiley Interdiscip. Rev.: Comput. Mol. Sci. , , e1340.(39) Abraham, H.; Akhalwaya, I. Y.;Aleksandrowicz, G.; Alexander, T.;Alexandrowics, G.; Arbel, E.; As-faw, A.; Azaustre, C.; AzizNgoueya,;Barkoutsos, P.; Barron, G.; Bello, L.;Ben-Haim, Y.; Bevenius, D., et al. Qiskit:An Open-source Framework for QuantumComputing. 2019.(40) Smith, R. S.; Curtis, M. J.; Zeng, W. J. APractical Quantum Instruction Set Archi-tecture. 2016.(41) Karalekas, P. J.; Tezak, N. A.; Peter-son, E. C.; Ryan, C. A.; da Silva, M. P.;Smith, R. S. A quantum-classical cloudplatform optimized for variational hybridalgorithms. Quantum Sci. Technol. , , 024003.(42) Yao, Y. Python driver of Gutzwillerquantum-classical embedding sim-ulation framework (PyGQCE). http://doi.org/10.6084/m9.figshare.11987616 ,(43) Pople, J. A. Nobel lecture: Quantum chem-ical models. Rev. Mod. Phys. , ,1267.(44) Zhang, F.; Gomes, N.; Berthusen, N. F.;Orth, P. P.; Wang, C.-Z.; Ho, K.-M.; Yao, Y.-X. Shallow-circuit varia-tional quantum eigensolver based onsymmetry-inspired Hilbert space partition-ing for quantum chemical calculations. arXiv:2006.11213 [quant-ph] ,(45) McArdle, S.; Jones, T.; Endo, S.; Li, Y.;Benjamin, S. C.; Yuan, X. Variationalansatz-based quantum simulation of imag-inary time evolution. npj Quantum Inf. , , 1–6.1646) McLachlan, A. A variational solution ofthe time-dependent Schrodinger equation. Mol. Phys. , , 39–44.(47) Broeckhove, J.; Lathouwers, L.;Kesteloot, E.; Van Leuven, P. Onthe equivalence of time-dependent varia-tional principles. Chem. Phys. Lett. , , 547–550.(48) Li, Y.; Benjamin, S. C. Efficient variationalquantum simulator incorporating activeerror minimization. Phys. Rev. X , ,021050.(49) Stein, T.; Henderson, T. M.; Scuseria, G. E.Seniority zero pair coupled cluster doublestheory. J. Chem. Phys. , , 214113.(50) Sokolov, I. O.; Barkoutsos, P. K.; Olli-trault, P. J.; Greenberg, D.; Rice, J.; Pis-toia, M.; Tavernelli, I. Quantum orbital-optimized unitary coupled cluster meth-ods in the strongly correlated regime: Canquantum algorithms outperform their clas-sical equivalents? J. Chem. Phys. , , 124107.(51) Romero, J.; Babbush, R.; McClean, J. R.;Hempel, C.; Love, P. J.; Aspuru-Guzik, A.Strategies for quantum computing molecu-lar energies using the unitary coupled clus-ter ansatz. Quantum Sci. Technol. ,4