Hybrid quantum-classical algorithms for solving quantum chemistry in Hamiltonian-wavefunction space
HHybrid quantum-classical algorithms for solving quantum chemistry inHamiltonian-wavefunction space
Zhan-Hao Yuan, Tao Yin, ∗ and Dan-Bo Zhang
3, 4, † National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Yuntao Quantum Technologies, Shenzhen, 518000, China Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,GPETR Center for Quantum Precision Measurement and SPTE,South China Normal University, Guangzhou 510006, China Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China
Variational quantum eigensolver (VQE) typically optimizes variational parameters in a quantumcircuit to prepare eigenstates for a quantum system. Its applications to many problems may involvea group of Hamiltonians, e.g., Hamiltonian of a molecule is a function of nuclear configurations. Inthis paper, we incorporate derivatives of Hamiltonian into VQE and develop some hybrid quantum-classical algorithms, which explores both Hamiltonian and wavefunction spaces for optimization.Aiming for solving quantum chemistry problems more efficiently, we first propose mutual gradientdescent algorithm for geometry optimization by updating parameters of Hamiltonian and wavefunc-tion alternatively, which shows a rapid convergence towards equilibrium structures of molecules. Wethen establish differential equations that governs how optimized variational parameters of wavefunc-tion change with intrinsic parameters of the Hamiltonian, which can speed up calculation of energypotential surface. Our studies suggest a direction of hybrid quantum-classical algorithm for solvingquantum systems more efficiently by considering spaces of both Hamiltonian and wavefunction.
I. INTRODUCTION
Variational quantum eigensolver opens a promisingparadigm for solving eigenstates of Hamiltonians on near-term quantum processors with hybrid quantum-classicaloptimization [1–6]. It has received intensive studies sinceit provides a practical avenue to exploit the power ofquantum computing for many fundamental problems,ranging from quantum chemistry [7–12], quantum many-body systems [13–15], and many other applications [16–18]. The original VQE is designed to solve the groundstate for a single Hamiltonian, and variants of VQE havebeen developed for solving excited states [19–23], finite-temperature quantum systems [13, 24–28], and so on.Many practical problems may involve a group ofHamiltonians characterized by intrinsic parameters thatdescribes the system, for instance, electronic Hamilto-nian for a molecule is a function of nuclear configura-tions. The energy dependence of nuclear configurations,namely potential energy curve/surface (PEC/PES) withbond lengths and angles, account for many properties ofchemical reaction such as transition states, reaction rate,etc [29]. While it requires VQE for solving moleculesunder many different configurations and the computa-tional cost can be large, it is possible to explore relationsbetween Hamiltonians to give more efficient algorithm.For instance, collective optimization has been developedto update variational parameters of wavefunctions fordifferent Hamiltonians jointly [30], guided by the con-tinuousness of Hamiltonian. As such a simple strategy ∗ [email protected] † [email protected] works, it is natural to exploit more concrete relations,such as first order derivative of Hamiltonian with bondlengths or angles. While calculations of energy deriva-tives have been proposed in Ref. [31–33], it is still lack ofexploiting Hamiltonian derivatives for variational solvingquantum chemistry in an enlarged space of Hamiltoniansand wavefunctions, where both variational parameters ofwavefunction and intrinsic parameters of Hamiltonianscan be searched and optimized.In this paper, we incorporate Hamiltonian derivativein the framework of VQE, and develop hybrid quantum-classical algorithms for solving quantum chemistry inthe space of both Hamiltonians and trial wavefunctions.We first propose mutual gradient descent algorithm forgeometry optimization, which aims to find equilibriumstructure of molecules more efficiently. As optimizedvariational parameters of wavefunctions are expected tochange continuously with varying Hamiltonians, we alsoestablish differential equations that reveal the relation.With a set of differential equations and an initial condi-tion, we can calculate the energy potential surface with-out using VQE for every configurations, including bothground state and excited states. By numeral simulations,we demonstrate the algorithms for some representativemolecules. Our work suggests Hamiltonian derivativeas an important ingredient in VQE for solving quantumchemistry or other physical systems involving a group ofHamiltonians.The paper is organized as follows. In section. II, wefirst introduce VQE and propose two new algorithms ex-ploiting Hamiltonian derivatives. In section. III, we ap-ply those methods to molecules such as H , LiH and H numerically. Finally, we give a summary of this work insection. IV. a r X i v : . [ qu a n t - ph ] A ug II. VARIATIONAL QUANTUM EIGENSOLVERSWITH HAMILTONIAN DERIVATIVE
Variational quantum eigensolver aims to solve eigen-states for a given Hamiltonian. In quantum chemistry,the Hamiltonian depends on the nuclear configurationsof the molecule, as positions of atoms are fixed by theBorn-Oppenheimer approximation. Then, we can solvea molecule for a range of nuclear configurations, e.g., toget the energy potential surface. As Hamiltonian willchange continuously with the configuration parameters,we can expect accessing the Hamiltonian derivative withnuclear configuration can be helpful for solving quantumchemistry problems more efficiently. We incorporate thisidea into VQE into two different ways. First, we pro-pose mutual gradient descent algorithm for geometry op-timization, which finds the lowest-energy molecule struc-ture for a molecule in an enlarged parameter space. Sec-ond, we establish differential equations that uncovers therelation between the optimized parameter of variationalwavefunctions and the Hamiltonians.
A. Variational quantum eigensolver
We consider Hamiltonians of a molecule are H ( λ )with intrinsic parameter λ . The variational wavefunc-tion ansatz is denoted as | ψ ( θ ) (cid:105) , where θ ∈ R K . We alsoadopt a density matrix denotation ψ ( θ ) = | ψ ( θ ) (cid:105)(cid:104) ψ ( θ ) | .The Hamiltonian can be written as a summation of localHamiltonians, H ( λ ) = N (cid:88) i =1 c i ( λ ) L i , (1)where a local Hamiltonian L i can be written asproduct of Pauli matrices. We denote c ( λ ) =[ c ( λ ) , c ( λ ) , ..., c N ( λ )] T and L = [ L , L , ..., L N ] T . Thuswe can write H ( λ ) = c T ( λ ) L . As c ( λ ) shall be con-tinuous function of λ , derivative of H ( λ ) is defined as ∂ c T ( λ ) ∂λ L .For a single Hamiltonian with particular λ , the VQEworks as follows. An variational ansatz | ψ ( θ ) (cid:105) = U ( θ ) | ψ (cid:105) is used to parametrize a ground state. Theinitial state | ψ (cid:105) is usually choosen as a good classicalapproximation for the ground state of H . In quantumchemistry, for instance, | ψ (cid:105) can be chosen as a Hartree-Fock state. U ( θ ) is an unitary operator parameterizedwith θ , which can encode quantum correlation into theground state. The essential task is to find parameters θ that minimizes the energy E ( θ ; λ ) = (cid:104) ψ ( θ ) | H ( λ ) | ψ ( θ ) (cid:105) = Tr( ψ ( θ ) H ( λ )) . (2)In the process of optimization, the quantum processorprepares ψ ( θ ) and performs measurements to evaluate E ( θ ; λ ), which can be reduced into E ( θ ; λ ) = c T ( λ ) L ( θ ) . where L ( θ ) = Tr( ψ ( θ ) L ). Here, quantum average ofeach component of L corresponds to a joint measure-ment on multiple qubits that can be implemented on aquantum processor. The classical computer updates pa-rameters θ according to received data from the quantumprocessor, e.g., using gradient descent θ t = θ t − − η A ∂∂ θ E ( θ t − ; λ ) , (3)where η A is the step size. Calculating the gradientswith respect to a target cost function (here is E ( θ ; λ ))which can be obtained with the same quantum circuiton a quantum processor, using the shift rule [34, 35] ornumeral differential. The optimization for energy mini-mization reaches a zero gradient descent ∂ E ( θ ; λ ) ∂ θ ≡ c T ( λ ) ∂ L ( θ ) ∂ θ = 0 , (4)which we may call it as wavefunction-matching condition. B. Mutual gradient descent algorithm forgeometry optimization
Geometry optimization is an important task in com-putational chemistry, which is key to understand molec-ular structures and chemistry reactions. VQE has beenapplied for geometry optimization, which optimizationalong the path of energy potential surfaces [31]. In thiswork, we propose a more efficient hybrid algorithm by di-rectly minimizing E ( θ ; λ ) in the enlarged parameter space( θ , λ ), without referring to optimize VQE for each fixed λ . We may divide the procedure into wavefunction-matching and Hamiltonian matching processes. The for-mer is just VQE that optimizes θ by fixing λ . The Hamil-tonian matching can be formulated as follows. For a ψ ( θ ), to find a Hamiltonian in the group of Hamiltonians H ( λ ), we propose the Hamiltonian matching condition, ∂ E ( θ ; λ ) ∂λ ≡ ∂ c T ( λ ) ∂λ L ( θ ) = 0 . (5)To reach this condition, λ can be updated using gradientdescent as following, λ t = λ t − − η B ∂ E ( θ ; λ t − ) ∂λ . (6)It should be noted that the term ∂ c T ( λ ) ∂λ is given withclassical computers using opensource packages (Open-Fermion [36] or HiqFermion [37]). Then, once L ( θ ) hasbeen evaluated on a quantum process, the optimizationwith Eq. (6) can be run iteratively on a classical com-puter. This is in contrast to the optimization processof Eq. (3) where the quantum processor and the classi-cal computer should be used repeatedly. For this reason,finding a best matched Hamiltonian is a task withoutusing quantum resources.We propose mutual gradient descent for geometry op-timization, which start from an initial ψ ( θ ) for a givenHamiltonian H ( λ ), and optimize θ and λ alternatively.The algorithm is as follow (N, T are hyper-parameterscontrolling the iterative step): Algorithm 1
Mutual Gradient Descent (MGD)
Input: ψ ( θ ), λ , N , T Output: λ t function MGD ( ψ ( θ ) , λ , N, T ) λ t ← λ θ t ← θ repeat for i = 0 → N − do λ t = λ t − − η B ∂ E ( θ ; λ t − ) ∂λ end for for i = 0 → T − do θ t = θ t − − η A ∂∂ θ E ( θ t − ; λ ) end for until Convergence end functionC. Differential equations for calculating energypotential surface
We turn to solve the energy potential surface, whichlies at the heart of quantum computational chemistry.The energy potential surface describes the dependenceof ground or low-energy excited states with bond lengthsand angles for a molecule. For VQE, it can be expectedthat optimized θ ∗ vary continuously with intrinsic pa-rameters λ of the Hamiltonian, namely θ ∗ ( λ ) is a func-tion of λ . We can reveal explicitly this function, whichcan very useful for calculating energy potential surface.For this, we set λ (cid:48) = λ + δλ and θ (cid:48) = θ + δ θ for thewavefuntion matching condition in Eq. (4), and then ex-pand c T ( λ (cid:48) ) ∂ L ( θ (cid:48) ) ∂ θ (cid:48) = 0 as (omitting second order terms)( c T ( λ ) + ∂ c T ( λ ) ∂λ δλ )( ∂ L ( θ ) ∂ θ + ∂ L ( θ ) ∂ θ δ θ ) = 0 . (7)Then, one can get a differential equation, ∂ c T ( λ ) ∂λ ∂ L ( θ ) ∂ θ + c T ( λ ) ∂ L ( θ ) ∂ θ d θ ( λ ) dλ = 0 . (8)The system of differential equations of Eq. (8) containsa number of N × K equations. We can use a simple case N = K = 1 to illuminate the meaning of this differentialequation. Eq. 8 is simplified as c ( λ ) T L (cid:48)(cid:48) ( θ ) dθ = − c (cid:48) ( λ ) T L (cid:48) ( θ ) dλ. (9)A more inspiring way is to write Eq. (9) as, ∂ θ ∂ θ E ( θ ; λ ) dθ ( λ ) = − ∂ θ ∂ λ E ( θ ; λ ) dλ (10) We may take κ θλ = ∂ θ ∂ λ E ( θ ; λ ) and κ θθ = ∂ θ ∂ θ E ( θ ; λ )as two elastic coefficients. The numeric algorithm can bedone as follows, θ ∗ i +1 = θ ∗ i − ( λ i +1 − λ i ) κ θλ κ θθ . (11)Note that κ θλ and κ θθ can be evaluated with a quantumcomputer (by numeral differential or analytic differentialusing the shift rule). Calculating each θ ∗ i may inevitablybring some errors. A slightly modification can be madeby using gradient descent in Eq. (3) to make sure anoptimized θ ∗ i is indeed obtained, which is expected totake very few steps.Let’s consider cases with higher dimensional parame-ters. For case of θ = ( θ , θ ), apply Eq. (8) to Eq. (10)similarly, we can write the differential equation as follow. ∂ θ ∂ λ E ( θ ; λ ) dλ + ∂ θ ∂ θ E ( θ ; λ ) dθ + ∂ θ ∂ θ E ( θ ; λ ) dθ = 0 ∂ θ ∂ λ E ( θ ; λ ) dλ + ∂ θ ∂ θ E ( θ ; λ ) dθ + ∂ θ ∂ θ E ( θ ; λ ) dθ = 0(12)Apparently, the discussion above can be promoted tocases with any dimension, so a matrix form of the dif-ferential equation can be represented as A d θ = − b dλ, (13)where A = ∂ θ ∂ θ E ∂ θ ∂ θ E · · · ∂ θ ∂ θ n E ∂ θ ∂ θ E ∂ θ ∂ θ E · · · ∂ θ ∂ θ n E ... ... . . . ...∂ θ n ∂ θ E ∂ θ n ∂ θ E · · · ∂ θ n ∂ θ n E , b = ∂ θ ∂ λ E ∂ θ ∂ λ E ...∂ θ n ∂ λ E (14)one can neatly write the numeral solver as, θ ∗ i +1 = θ ∗ i − ( λ i +1 − λ i ) A − b . (15)It can be viewed as a numeral solution for the differen-tial equation with a hybrid quantum-classical algorithm.More advanced numeral methods may be applied for im-proving the precision.The different equations can be readily for solving theenergy potential surface with an initial condition, whichcan be obtained by using VQE for a molecule at a fixedconfiguration. This avoids to apply VQE for every con-figurations. Moreover, the different equations are in prin-ciple applicable for all eigenstates, which is determinedby the initial condition. Thus, different equations can beeasily incorporated into VQE for calculating EPS con-sisting of excited states [20, 23]. III. NUMERICAL RESULTS
In this section, we will apply above methods in differentsystems including molecular hydrogen (H and H ) and (a) (b) FIG. 1. Landscape of H in parameter space. (a) Left: The black curve is corresponding to VQE, optimizing θ with respect todifferent bond length. The blue curve shows the process of Hamiltonian matching which optimize λ with fixed θ . The polylineswith arrows in red, yellow and purple demonstarte the process of mutual gradient descent (MGD) with different strategies(bycontrolling hyper parameters N, T ) which converge to equilibrium point. Right: The polylines with arrows in grey shows theresult of potential energy curve of H with STO-3G basis sets from a randomly initialized point. The black one is optimized bya few steps of gradient descent to fix the error. (b) Left: PEC calculated with different methods. Grey curves are exact diagonalresults without fixing the electronic neutral condition. Colored curves are ground states and excited states that calculated withdifferential equations using STO-3G basis and UCC ansatz. Right: First component of θ varies with bond length Lithium Hydride. With those representative molecules,the results show that mutual gradient descent algorithmcan search for equilibrium point in PEC in an enlargedparameter space and converge with only a few steps.Moreover, by solving the differential equation, we cancalculate the PEC/PES with same accuracy as normalVQE does. This method can also avoid parameters tobe trapped in local minima once the initial point is inglobal minima. Moreover, it is also compatible with ex-isting VQE for calculating excited states.
A. Molecular Hydrogen H We first use H to illustrate basic properties and advan-tages of two algorithms for geometrical optimization andEPS calculations. We adopt both two and four qubitseffective Hamiltonians obtained with OpenFermion pack-age [36] beforehand.We adopt the unitary coupled cluster(UCC) ansatzwith an unitary operator U ( θ ) = e − iθσ x σ y acting onthe Hartree-Fock reference state | (cid:105) [38]. Bond lengthsranging from 0.2 a.u. to 2.85 a.u. was discretized uni-formly into 50 points. Both gradient descent and scien-tific computing packages such as SciPy are available whenusing gradient-based methods to optimize the parameter θ . Fig. 1 shows different processes, including traditionalVQE and mutual gradient descent algorithm, of findingthe balance point.Traditional VQE requires to obtain the whole poten-tial energy curve and then locate the minimum energypoint while mutual gradient descent could converge tobalance point with much fewer steps. To further under-stand the route of mutual gradient descent, we show theHamiltonian matching results in the landscape. The blueline is the process of finding best matched bond lengthwith different fixed θ which corresponds to find parentHamiltonian with respect to a specific wavefunction. What impressed us is mutual gradient descent is follow-ing a path between Hamiltonian matching step(blue line)and wavefunction matching step(VQE). The predisposi-tion of the curve is due to the strategy we choose whichcan further control the portion of consumption betweenquantum computing resource and classical computing re-source (see Fig.1(a)). For comparison, more complicatedvariational ansatz (4 qubits) and basis (6-31g) are usedin MGD. We use results from previous works of equilib-rium geometry to compare our methods to traditionalones (see Table. I) [39, 40].We also apply differential equations which attempt toexplicitly establish the dependence of optimized θ on λ . From our numerical results, we notice that potentialsurface curve(surface) can be obtained by solving thoseequations indeed. However, we also find that the devia-tion of the result becomes larger along the line of inte-gration, which is due to finite step sizes (see Fig.1(b)).Those deviations can be further remedied with a few stepof gradient descents. Excited states.
Differential equations can also be usedto calculate excited states. Combined with the idea ofweighted SSVQE introduced in Ref. [41], which utilizethe orthogonality between different eigenstates (Hartree-Fork reference states, in our cases), we use a cost functionas a weighted summation of energies, L w ( θ ) = k (cid:88) j =0 ω j (cid:104) ψ j | U † ( θ ) H U ( θ ) | ψ j (cid:105) (16)instead of Eq.(2) in differential equations. Eachstate in Eq.(16) required to be orthogonal with eachother( (cid:104) ψ i | ψ j (cid:105) = δ ji ). The weight vector w can be anyvalue in (0 ,
1) and should chosen w i < w j when i < j .After using Eq.(16) in differential equations, | ψ j (cid:105) will bethe j th excited states. We use H as examples and re-sults can be seen in Fig.1(b). Six orthogonal initial statesare chosen | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) . As (a)MGD optimization epoch (b) θ flows with differential equations (c)PEC of LiH FIG. 2. (a)Optimization process for finding equilibrium geometry with mutual gradient descent algorithm. Bond length isconverge after only a few steps of iterative calculation. (b)The variations of parameters θ and θ as λ changes calculatedwith differential equations. The red dots are VQE results and blue dots are results computed with differential equation. Foromitting higher order terms in those equations, there are deviations with respect to VQE results. The green crosses are resultsby adding a few steps of gradient descent which is almost the same comparing to VQE. (c)PEC calculated with differentmethods,ED (exact diagonal), DE (differential equations), VQE (variational quantum eigensolver).TABLE I. Equilibrium geometry of H and LiHMGD MGD CID/6-31G HF/6-31GH (a.u.) 0.744(STO-3G) 0.753(6-31G) 0.746 0.730LiH (a.u.) 1.520(STO-6G) 1.641(6-311G) 1.649 1.636 UCC operator keeps the particle number, those parame-terized wavefunctions spans a subspace of electronic neu-tral states of H , which correspond to four energy curvesin Fig. 1(b), as some of them are degenerate. B. Lithium Hydride
We now consider a more complicated molecule, LiH.We first use three-qubit effective Hamiltonians (six-qubitHamiltonian will be used later) that construct with STO-6G and 6-311G basis sets for illustration of geometry op-timization. Following Ref. [42], we use an UCC ansatzwith two parameters, U ( θ , θ ) = e − iθ σ x σ y e − iθ σ x σ y ,and use | (cid:105) as Hartree-Fock reference state, more de-tail of UCC implementation can be seen in Appendix.A.Bond lengths ranging from 0.2 a.u. to 3 a.u. are dis-cretized uniformly into 30 points.In this case, we found the convergence of stationarypoint only needs a few steps. This is reasonable since itmeans parametrized wavefunction with different param-eters are all in local minima with only slightly differentbond length. This may not a general argument but itseems work well in all simple cases we study. Six qubitsvariational ansatz and 6-311g basis are used in MGD,too. And results from previous works of equilibrium ge-ometry to compare our methods to traditional ones canbe seen in Table. I. The optimization process of differen-tial equations for three-qubit Hamiltonian are shown inFig. 2(a). We also use differential equations for calculating po-tential energy curve of LiH (see Fig. 2(c)). By cancelingthe error produced by finite step size with a few steps ofgradient descent, the result is as good as VQE. For moreaccurate results, we increase the complexity of our quan-tum circuit. We use a 6 qubits UCC ansatz for calculat-ing the potential energy curve with differential equationsand VQE. C. H molecule Molecules with many atoms are challenging in quan-tum chemistry, since the Hilbert space growths exponen-tially and the wavefunciton ansatz should be much morecomplicated. One protocol model can be multi-hydrogensystems such as hydrogen chain, which may be a bench-mark for computational methods and also exhibits inter-esting physical phenomenons [43, 44]. So we apply MGDto a H molecule in order to exhibit the potential of thisalgorithm.For simplicity, we constraint this molecule in a waythat only need 2 parameters to describe (see Fig. 3). Thismolecule model is parametrized with bond length d andbond angle α . A more complicated unitary coupled clus-ters ansatz is used here and a 6 qubits quantum circuitis established for simulating it’s eigenstates. The detailof UCC implementation is illustrate in Appendix.A.Bond lengths ranging from 0.3 a.u. to 3 a.u. arediscretized uniformly into 24 points and bond angles (a) (b) FIG. 3. (a) The process of finding equilibrium geometry with mutual gradient descent algorithm. The inset shows H moleculemodel with four hydrogen atoms located on a broken line with equal distance d , angle β = 60 ◦ , d and angle α are two degreesof freedom.(b) First component of θ varies with bond length d and bond angle α . are discretized uniformly into 25 points from π/
20 to25 π/
20. The only difference between H and previousexamples is that the degrees of freedom become largerwhen λ = ( d, α ). Since technically there is no difficult tocompute the gradients with higher dimensional parame-ters, mutual gradient descent algorithm work well in thiscase too (see Fig.3(a)). MGD can find equilibrium ge-ometry with a few steps in our molecule model. Theparameters are continuously moving along the intrinsicdegrees of freedom in parameter spaces (see Fig.3(b)). IV. SUMMARY
To summary, we have proposed hybrid quantum-classical algorithms incorporated Hamiltonian derivativeinformation for solving potential energy surface in quan-tum chemistry. Firstly, we have developed a methodcalled mutual gradient descent algorithm, which havebeen shown to be efficient while finding equilibrium ge-ometry under the context of VQE. MGD algorithm hassuccessfully found the equilibrium bond length and bondangle of molecules like H , LiH and H with only a fewsteps of iteration. We have developed differential equa-tions incorporating Hamiltonian derivatives to calculateenergy potential surface of molecules, which is also appli-cable for excited states. The paradigm of solving quan-tum chemistry in a combined Hamiltonian and wavefunc-tion space on a quantum computer may inspire morepractical quantum algorithms on near-term quantum de-vices. Lastly, we point out that while those quantumalgorithms have been demonstrated for quantum chem-istry, it is potential for applying them to solve quantummany-body problems with tunable parameters, such asquantum phase transitions. ACKNOWLEDGMENTS
This work was supported by the Key-Area Researchand Development Program of GuangDong Province(Grant No. 2019B030330001), the National Key Re-search and Development Program of China (Grant No.2016YFA0301800), the National Natural Science Founda-tion of China (Grants No. 91636218 and No. U1801661),the Key Project of Science and Technology of Guangzhou(Grant No. 201804020055).
Appendix A: Unitary coupled cluster ansatz
In quantum chemistry, unitary coupled cluster (UCC)ansatz is used widely for a parametrization of wavefunc-tion due to its representation power. The variationalwavefunction can be written as | ψ ( θ ) (cid:105) = e T − T † | R (cid:105) , (A1)where T = T + T with T = (cid:88) pq θ pq c † p c q T = (cid:88) pqrs θ pqrs c † p c † q c r c s . (A2) T and T represent single particle excitations and dou-ble particle excitations, respectively, and c and c † arefermionic operators. The ansatz we used here also calledunitary coupled cluster with single and double excita-tions (UCCSD).This expression can not be implemented on quan-tum devices directly. Wigner transformation (or Bravyi-Kitaev transformation) and trotterization is needed be-fore optimization. Trotter steps can be a single step ormore based on required accuracy, the expression for nTrotter step is | ψ ( θ ) (cid:105) = n (cid:89) k =1 (cid:89) pq e ˆ t kpq (cid:89) pqrs e ˆ t kpqrs | R (cid:105) . (A3) where ˆ t kpq = θ p,kq ( c † p c q − c † q c p ) and ˆ t kpqrs = θ pq,krs ( c † p c † q c r c s − c † s c † r c p c q ) [1] M.-H. 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