Orbital optimized unitary coupled cluster theory for quantum computer

We propose an orbital optimized method for unitary coupled cluster theory (OO-UCC) within the variational quantum eigensolver (VQE) framework for quantum computers. OO-UCC variationally determines the coupled cluster amplitudes and also molecular orbital coefficients. Owing to its fully variational nature, first-order properties are readily available. This feature allows the optimization of molecular structures in VQE without solving any additional equations. Furthermore, the method requires smaller active space and shallower quantum circuit than UCC to achieve the same accuracy. We present numerical examples of OO-UCC using quantum simulators, which include the geometry optimization of the water molecule.


I. INTRODUCTION
Coupled cluster theory (CC) is one of the most representative electron correlation methods in quantum chemistry. 1,2It has some vital features to describe molecular electronic structures reliably.CC is size-extensive and can be improved systematically by increasing excitation level.It converges to full configuration interactions (FCI) faster than truncated configuration interactions (CI) or Møller-Plesset perturbation theory.Furthermore, its energy is invariant to unitary transformations among the occupied/virtual orbitals.It is well known that, for an electronic state where the mean-field approximation works well, the CC models considering up to triple excitations can provide the chemical accuracy (e.g., 1 kcal/mol), if sufficiently large basis functions are employed.In particular, it has been established that CC singles and doubles with perturbative triples (i.e., CCSD (T)), which incorporates three-body interactions perturbatively, is a highly accurate method.It is often called the golden standard of molecular electronic structure theory. 3n the framework of the traditional CC (TCC), the wave-function parameters are determined by solving projected amplitude equations and not variationally.Owing to its non-variational properties, the validity of TCC is strongly dependent on the reference Hartree-Fock wave function.Indeed, CCSD and CCSD(T) often breakdown in the systems where static electron correlations are strong, for example, multiple-chemical-bond breaking systems.
a) Electronic mail: wataru.mizukami.857@qiqb.otri.osaka-u.ac.jp b) Electronic mail: mitarai@qc.ee.es.osaka-u.ac.jp c) Electronic mail: Yuuya Oonishi@jsr.co.jp5][6] Nonetheless, VCC has a factorial-scaling in the computational-cost.Therefore, it is only applicable to a tiny system where FCI can be performed.However, it has recently been shown that a variant of VCCunitary coupled cluster (UCC) 7-14 -can be solved at a polynomial-scaling cost using a quantum computer. 15A UCC wave function can be prepared on a quantum computer using the Trotter approximation with polynomial number of quantum gates.Although the gate count for the accurate UCC can be much larger than what today's quantum devices are capable of, 16 UCC can be a good starting point for analyzing the power of the quantum computer in the field of quantum chemistry.
The method enabling UCC on a quantum computer is called variational quantum eigensolver (VQE), which is a kind of quantum-classical hybrid algorithm. 15n the VQE, a wave function is prepared through a parametrized quantum circuit corresponding to a wave function ansatz (e.g., UCC).Then, we measure its energy for given circuit parameters.The parameters of the circuit are iteratively tuned by a non-linear optimizer running on a classical computer to minimize the energy.][25][26][27][28][29][30][31][32] Although VQE allows the determination of UCC parameters based on the Rayleigh-Ritz variational procedure, the obtained UCC wave functions are not fully variational.UCC and its variants employ a Hartree-Fock determinant as a reference wave function; the orbitals are fixed and not altered during a UCC calcula-tion.However, it is well-known that the Hartree-Fock orbitals are not optimal orbitals for a correlated wave function.One method for obtaining such optimal orbitals is to optimize orbitals in such a way that the gradients of the energy with respect to orbital rotation parameters vanish.][36][37][38] This article concerns the orbital-optimization technique to UCC in the context of VQE.At this moment, the size of the orbital space that can be handled by a quantum computer is severely limited because of the number of the available qubits.Therefore, active space approximation is indispensable when we wish to use a quantum computer for quantum chemical problems.Improvement of the active space can be achieved by optimizing molecular orbitals with the VQE, which leads to the reduced number of qubits.Furthermore, the orbital-optimized VQE (OO-VQE) is a fully variational method, and the molecular gradients of OO-VQE (e.g., forces) can be calculated without solving response equations.The idea of using the orbital-optimization techniques for quantum computers has been already reported by Reiher et al. for the phase estimation algorithm (PEA) 39 and by Takeshita et al. for the VQE. 40To the best of our knowledge, this study is the first to implement OO-VQE using a quantum circuit simulator; we propose an orbital optimized unitary coupled cluster doubles (OO-UCCD) as a wave-function model for OO-VQE.
The remainder of this paper is organized as follows: First, Sec.II following chapter describes theory of orbital optimized UCC (OO-UCC) based on VQE.Sec.III provides a brief description of the implementation of OO-UCC using a quantum circuit simulator.Sec.III also discusses simple numerical experiments to exhibit its usefulness.Finally, Sec.IV concludes the paper.

II. THEORY Unitary coupled cluster
The molecular electronic Hamiltonian in a spin-free form is expressed as where h pq and h pqrs are one-and two-electron integrals, respectively.Êpq is a singlet excitation operator and is defined as Êpq = ĉ † p,α ĉq,α + ĉ † p,β ĉq,β , where ĉ † p,α and ĉp,β are creation and annihilation second quantized operators, respectively.p, q, r, s are the indices of general molecular spatial orbitals.
A wave function in the traditional coupled cluster ansatz is given as where T is an excitation operator T = T1 + T2 + T3 + • • • .In contrast, UCC uses an anti-hermite operator Â defined by the difference of the amplitude operator T of TCC and its hermitian conjugate, i.e., Â = T − T † .Therefore, a wave function of the UCC ansatz is expressed as, The Baker-Campbell-Hausdorff (BCH) expansion of the similarity transformed Hamiltonian of the traditional CC is terminated at the finite order, whereas that of UCC is not owing to de-excitation operators T † .The infinite BCH expansion makes the implementation of UCC on a classical computer unfeasible.

Orbital optimization
Optimizing orbitals is equivalent to minimizing a wave function with respect to orbital rotation parameters κ.The energy function of OO-UCC is given by where the orbital rotation operator is defined as κ = pq κ pq ( Êpq − Êqp ).When UCC parameters A are fixed, the second order expansion of the energy function becomes where Ê− pq = Êpq − Êqp .By taking the derivative with respect to κ, the following Newton-Raphson equation is obtained: whose elements are 7) H and g are often called electronic Hessian and gradients, respectively.One-particle and two-particle reduced density matrices (1RDM and 2RDM) are required to compute them in addition to the molecular Hamiltonian integrals h pq and h pqrs .They are readily available in VQE, because it measures 1RDM and 2RDM to compute electronic energy in a given quantum circuit.
Orbital Optimized Unitary Coupled Cluster Doubles The UCC singles and doubles (UCCSD) is given as where Ân = Tn − T † n consists of n-excitation operators Tn and their conjugates.Starting from the UCCSD ansatz Eq. 9, we consider the following wave-function model by separating the singles and doubles parts: The UCC operator Â is not commutable unlike TCC, because of the existence of de-excitation operators T † .Therefore, the decomposed UCCSD ansatz is different from the original ansatz.The singles part e Â1 in this model is identical to the orbital rotation unitary operator e κ appeared in Eq. 4. This implies that we can optimize its singles part e Â1 variationally using a classical computer via the well-established orbital-optimization technique.The singles only alters the Hartree-Fock determinant to another determinant | 0 = e A1 |0 .
Considering the Slater determinant | 0 as a reference wave function for UCC, we rewrite Eq. 10 and propose the orbital-optimised unitary coupled cluster doubles (OO-UCCD) model, given as The doubles part e Â2 in Eq. 11 is optimized by the VQE, while the reference determinant (i.e., the singles) is optimized by a classical computer using 1RDM and 2RDM from VQE.In practice, we repeatedly perform the VQE and the orbital optimization until convergence.In exchange for self-consistency, Eq. 11 requires less complicated quantum circuit than Eq. 10.

Trotterization and Brueckner orbitals
The so-called Brueckner orbitals may coincide with the variational orbitals calculated from the proposed method when the Trotter approximation is used.Brueckner orbitals are optimal orbitals for a correlated wave function, where the singles' contribution (i.e., T1 or Â1 ) vanishes.It is known that, in the TCC framework, the variationally optimized orbitals are not the same as Brueckner orbitals.This is because of the difference between T1 and Â1 .This difference does not arise in UCC.Nonetheless, the OO-UCC method also does not satisfy the Brueckner condition because the anti-hermitian operators Ân are not always commutable with each other.
A UCC model can be modified such that the wave function simultaneously satisfies the variational and Brueckner conditions.This is done by embedding the Trotter Potential energy curves of the N2 molecule computed at OO-UCCD1, MP2, CISD, CCSD and FCI using the STO-3G basis set, where six orbitals and six electrons were correlated.Other orbitals were kept fixed during all the post-Hartree-Fock calculations; here, the external orbital rotation was omitted for OO-UCCD1.approximation a posteriori.This ansatz can be written as where Â2 = µ Â2,µ .We denote UCCSD and UCC doubles (UCCD) approximated by the n-step Trotter expansion as UCCSD n and UCCD n , respectively.Then, the ansatz of Eq. 12 corresponds to OO-UCCD

III. NUMERICAL EXAMPLES
In this section, we present some numerical results using the proposed methods.Computational details are as follows.We have implemented OO-UCCD in Python using Qulacs 42 , PySCF 43 , and OpenFermion 44 program packages.Qulacs is used to simulate quantum circuits.PySCF is used for orbital optimizations and for evaluating molecular Hamiltonian integrals.OpenFermion is employed for mapping molecular Hamiltonian into a quantum circuit based on the Jordan-Wigner transformation.
We first examine the potential energy curve (PEC) of N 2 .This system, involving triple bond breaking, is a Error of the OO-UCCD1 method along with potential energy curves of the LiH molecule using the STO-3G basis set.a) Deviations from FCI energies compared with UCCSD1 and the standard CCSD method.b) Deviations from UCCSD1 energies compared with UCCD1.
well-known benchmark for electron correlation methods, where the standard methods of many body perturbation theory such as MP2, CCSD, and CCSD(T) breakdown.We have computed the PEC at the OO-UCCD 1 /STO-3G level of theory by fixing the lowest four occupied orbitals and eight electrons.FIG. 1 shows the PEC along with those computed by MP2, CISD, CCSD, and FCI.It shows that OO-UCCD can treat a multiple-bondbreaking system appropriately where electrons of the breaking chemical bond are strongly correlated.In this system, the differences among UCCD, OO-UCCD, and UCCSD were small.They are at most 0.2 kcal/mol owing to the little orbital relaxation effect in the small basis set.The root mean square deviations (RMSD) of UCCD 1 and OO-UCCSD 1 with respect to UCCSD 1 PEC is 0.02 kcal/mol in the range of 1.0-2.1 Å.
Next, we investigate the PEC of LiH.FIG. 2 shows the errors of our method with respect to the PECs computed by FCI and UCCSD 1 with the STO-3G basis set.We did not employ the active space approximation.It can be seen that the energy difference between UCCSD 1 and OO-UCCD 1 is notably small in the entire range of the PEC.The energy deviation of OO-UCCD 1 from UCCSD 1 is 10 −7 mhartree at 1.3 Å Li-H distance, whereas at 2.1 Å it is 3 × 10 −5 mhatree.In contrast, the deviation of UCCD 1 is at least five orders of magnitude larger than OO-UCCD 1 .OO-UCCD is as accurate as UCCSD; likewise the standard OO-CCD calculations reproduce the CCSD results well.This indicates that UCCSD-level results can be obtained with a shallower quantum circuit using the orbital optimization technique at the cost of repeated VQE optimizations.Note that Tortterization introduces dependency on the operator ordering.Although Grimsley et al. have illustrated that the effect of the operator ordering could be significant, their results suggests that the impact of the ordering is not important on the systems below. 45he PEC of LiH with 6-311G basis set, given in FIG. 3, is another example.For comparison, we computed the PEC by OO-UCCD 1 and also by UCCSD 1 and FCI.Only four orbitals and two electrons were correlated in those VQE calculations, and the orbital optimization was performed for all the orbitals and electrons.This example shows how the orbital optimization effectively considers the electron correlations outside the active space.FIG. 3 illustrates that OO-UCCD has lower enery and is closer to FCI than UCCSD, indicating that OO-UCCD captures more electron correlation effects.
Finally, we report the geometry optimization of the water molecule with STO-3G basis set.For comparison, we carried out Hartree-Fock, MP2, CCSD, CCSD(T), and FCI calculations.The CCSD and CCSD(T) geometries were obtained by the ORCA program package using numerical gradients. 46,47All electrons were correlated in these calculations.FIG. 4 shows the error of the total energy from FCI for each optimised geometry, and RMSD of each optimized structure.It shows that OO-UCCD 1 is close to FCI in terms of both structure (i.e., 7 × 10 −5 Å) and energy (i.e., 0.1 mhartree).Fur- thermore, the comparison with CCSD and CCSD(T) suggests that OO-UCCD 1 's geometry is more accurate than that of CCSD and less than that of CCSD(T) when static correlation is small.This tendency is consistent with the findings of Kühn et al. for the total energy and reaction energy; 16 OO-UCCD 1 is slightly better than CCSD(T) in terms of energy in our case.

IV. SUMMARY
In this work, we have developed OO-UCCD.OO-UCCD treats singles contributions not on quantum computers but on classical computers.This reduce the number of gates and the depth in the UCCSD quantum circuit as noted by Takeshita and his coworkers, 40 while the conventional OO-CC simplifies energy expressions and amplitude equations.Moreover, all the wave function parameters are fully variationally determined in OO-UCC.This property makes the time-independent firstorder properties readily available.Therefore, geometry optimisation or ab initio molecular dynamics can be performed using VQE without solving orbital response equations.These aspects of OO-UCC seems useful, especially in the age of noisy intermediate-scale quantum computers (NISQ), 48,49 where the number of qubits and the coherence time are severely limited.The proposed method may be useful for solving quantum chemical problems once quantum computers become commonplace.

FIG. 1 .
FIG. 1.Potential energy curves of the N2 molecule computed at OO-UCCD1, MP2, CISD, CCSD and FCI using the STO-3G basis set, where six orbitals and six electrons were correlated.Other orbitals were kept fixed during all the post-Hartree-Fock calculations; here, the external orbital rotation was omitted for OO-UCCD1.

FIG. 3 .
FIG.3.Potential energy curves of the LiH molecule computed at OO-UCCD1, UCCSD1, and FCI using the 6-311G basis set.OO-UCCD1 and UCCSD1 calculations employed active space consisting of four orbitals and two electrons, while all the orbitals and electrons were correlated for FCI.

FIG. 4 .
FIG. 4. Structure and energy deviation of optimized the H2O molecule with OO-UCCD1 from Full CI.The STO-3G basis set was used.a) Comparison with MP2 and Hartree-Fock.b) Comparison with CCSD and CCSD(T).Bar graphs show energy differences from FCI energy in hartree.Line graphs illustrate RMSD errors of optimized geometries in angstrom.