Quantum Simulation of Many-Body Effects in the Li^-, Be, and B^+ Isoelectronic Systems using the Variational Quantum Eigensolver Algorithm
QQuantum Simulation of Many-Body Effects in the Li − , Be, and B + IsoelectronicSystems using the Variational Quantum Eigensolver Algorithm Sumeet, V. S. Prasannaa, , B. P. Das and B. K. Sahoo Qu & Co B.V., Palestrinastraat 12H, 1071 LE Amsterdam, The Netherlands Centre for Quantum Engineering, Research and Education,TCG CREST, Salt Lake, Kolkata 700091, India Department of Physics, Tokyo Institute of Technology,2-12-1-H86 Ookayama, Meguro-ku, Tokyo 152-8550, Japan Atomic, Molecular and Optical Physics Division,Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India (Dated: January 15, 2021)The emerging field of quantum simulation of many-body systems is widely recognized as a killerapplication of quantum computing. A crucial step towards realizing its full potential requires un-derstanding the role of electron correlation effects that lie at the heart of such calculations. In thispilot study, we investigate the trends in the electron correlation effects in the ground state ener-gies of atomic systems using the classical-quantum hybrid variational quantum eigensolver (VQE)algorithm. To facilitate a comparative analysis, we consider three isoelectronic systems, namelyLi − , Be, and B + , that is, an anion, a neutral atom and a cation, respectively. We choose theunitary coupled-cluster (UCC) ans¨atz for our computations. We conduct a rigorous analysis ofthe factors that could affect the accuracy with which electron correlation effects can be captured,namely mappings, backend simulators, and basis sets. The obtained results are also compared withthe those calculated by using the full configuration interaction method, coupled-cluster theory andUCC method, on a classical computer. A noteworthy part of the study involves a careful analysisto find the number of shots required for calculations with Qiskit’s QASM simulator backend, whichmimics an ideal quantum computer, at which one can obtain results in very good agreement withthose from a classical computer. I. INTRODUCTION
Quantum information science and technology, also her-alded as the second quantum revolution, has witnessed ameteoric rise in recent times, thus opening new pathwaysto tackle the notoriously challenging electronic structureproblem using quantum computers and simulators [1–8]. The interest in quantum chemistry using quantumcomputers/ simulators stems from the potential speed-up that a quantum computer promises to offer [9, 10] incalculating properties such as energies. This is in starkcontrast to a steep cost incurred on a classical computer(ClC) [11]. An overview of the developments in this fieldcan be found in Ref. [1]. Among the algorithms thatcalculate the energy of a many-body system, approachessuch as the quantum phase estimation algorithm [10, 12]may produce energy estimates with high accuracy, butdemand long coherence times [13–15]. An alternativethat promises to alleviate this problem, especially in thenoisy-intermediate scale quantum (NISQ) era that we arenow in, is the Variational Quantum Eigensolver (VQE)algorithm [16, 17]. This algorithm is a quantum-classicalhybrid approach that uses the variational principle toobtain the ground state energies of many-body systems.It has been experimentally realized in platforms such asphotonic processors [17], superconducting qubits [18],ion traps [19], etc.Accurate quantum many-body calculations are cen-tered around capturing electron correlation, which ariseas a result of electron-electron interactions in atomic and molecular systems. It is, therefore, necessary that aquantum computation/ simulation, specifically the VQEalgorithm in our work, appropriately capture these ef-fects. Although works exist in literature that performmany-body calculations in the framework of a quantumsimulation, not much attention is paid to the correlationeffects. In this work, we endeavour to understand thedegree to which these many-body effects are accountedfor, in a typical VQE calculation. This requires one tocarefully choose appropriate physical systems, as well astune the various parts of a VQE computation.The energies of a whole host of molecular systems, suchas H O [20], H [13, 21, 22] (also Ref. [23] for an ex-cited state treatment using an extended version of VQE),HeH + [17, 24], LiH, BeH [21], and H [25], have been cal-culated in literature. However, atoms have received littleattention, in spite of finding many applications, for exam-ple, see Refs. [26–32]. Adopting the viewpoint that atomsare merely subsets of molecules could be misleading, inthat the correlation effects and trends in a molecule andits constituent atoms can be quite dissimilar. Atomicsystems have shown to display their own unique featuresin this regard. For instance, it is easy to explain theresults of atomic calculations from their electronic con-figurations and the trends can be more or less generalizedto other systems. Moreover, atomic systems are betterplatforms than molecules to test scaling with number ofqubits, which in turn is decided by the choice of single-particle basis. Therefore, it is worthwhile to carry outatomic calculations in the framework of quantum simu-lation. a r X i v : . [ phy s i c s . a t o m - ph ] J a n FIG. 1: An overview of the work carried out in this paper,including listing the systems considered for analysis. Theground state energy, E ( θ ), which can be written as a sum ofthe Hartree-Fock part ( E HF ) and the correlation part ( E corr )is parametrized by the set θ , and is constructed from theHartree-Fock wave function, Φ , the atomic Hamiltonian, H a ,and the variational form, U ( θ ). Using the variational quan-tum eigensolver (VQE) algorithm, we study the correlationeffects that are captured by different combinations of map-ping, basis sets, and backend simulator, shown in the bracketsnext to the respective headings. We will now discuss our choice of atomic systems forthis pilot study. On physical grounds, many-body effectsare expected to behave differently in ions and neutralatoms of isoelectronic systems. Among them, electroncorrelation effects in the negative ions are vastly differ-ent [33, 34] owing to the short-range potentials that bindthe outer valence electron in these ions [35]. Negativeions find several applications, and details on this aspectcan be found in Refs. [35–37]. Also, atomic calculationsfrom literature have shown that electron correlation ef-fects in the alkaline earth-metal atoms are very strongdue to strong repulsion between the outer two valenceelectrons in these atoms [38–40]. For these two reasonsand keeping in mind the steep cost of simulation in theNISQ era, we consider here isoelectronic lithium anion(Li − ), neutral beryllium (Be), and boron cation (B + )as representative systems to investigate roles of electroncorrelation effects in the determination of their groundstate energies. We also stress that the study undertakenin this work is general in nature, and should be appli-cable to other heavier atomic systems in higher qualitybasis sets, when such simulations become feasible. It isalso worth adding that the systems that have been cho-sen in this work find many applications. For example,Group III A ions have been known to hold great promisefor atomic clocks [41]. Specifically, B + , holds promise,since the transition of interest has an extremely long life- time in its excited state. Moreover, because the B + ion’s mass is closer to that of Be + , there would be ef-ficient state exchange for quantum logic detection [42].Light systems such as Be can serve as excellent systemsin probing roles of Coulomb interactions [43, 44], as wellas obtaining nuclear charge radii from measurements ofisotope shifts [45]. Systems such as Li − may find appli-cations in plasma diagnostics [46].In view of the points discussed above, we reiterate thegoal of the present work: to study the electron correlationeffects in Li − , Be, and B + , using the VQE algorithm, andcompare with results from a traditional quantum chem-istry computation. A VQE calculation depends uponseveral factors, including the crucial aspect of choosinga variational form. In this work, we choose the unitarycoupled-cluster (UCC) ans¨atz. It is the unitary version ofthe well-known and physically motivated coupled-clustertheory, which is the gold standard of electronic struc-ture calculations in atomic and molecular physics [47],due to its ability to accurately capture correlation effects.The other important aspects that we need to consider arethe choice of mapping technique used to convert the sec-ond quantized fermionic operators to their spin counter-parts, backend simulator for running quantum circuits,and optimizer, besides the more intuitive and traditionalfeatures such as the choice of single-particle basis. Weexplore these facets in detail in this work. We focus ex-tensively on the required number of shots for obtainingreliable results using Qiskit’s QASM simulator backend.This investigation is especially important, as it providesestimates for expected error from a measurement-basedscheme. This sets the ground for future analyses withnoise models and error mitigation, which then would bemore realistically comparable to a calculation performedon a real quantum computer. The overall objective of thecurrent work is depicted pictorially in Fig. 1. We reem-phasize that this pilot study serves to pave way for futurecalculations and applications to heavier atomic systems,as well as to problems of a more general nature. II. THEORY AND METHODOLOGYA. General many-body formalism
The ground state energy, E , of an atomic system canbe determined by evaluating the expression E = (cid:104) Ψ | H a | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) , (1)where | Ψ (cid:105) is the ground state wave function of the atom.The atomic Hamiltonian, H a , in the second quantizedform, can be cast as H a = N (cid:88) pq h pq a † p a q + 12 N (cid:88) pqrs h pqrs a † p a † q a r a s . (2) FIG. 2: A diagrammatic overview of the Variational Quantum Eigensolver algorithm applied to electronic structure problem.A traditional quantum chemistry program performs the Hartree-Fock (HF) calculations and it also generates the one- and two-electron integrals. The Hamiltonian, H a , as well as the trial wave function, Ψ , is mapped to its qubit form by an appropriatemapping, and recast as circuits. This is mathematically shown for the Hamiltonian in the ‘Mapping’ step of the flowchart,where α is a single collapsed index of two indices for one- and four indices for the two-electron integrals. Similarly, P α is ashort hand notation for the corresponding second quantized operators. ⊗ Nj =1 σ j,α refers to the tensor product of a string ofPauli operators. We choose the unitary coupled-cluster variational form as our ans¨atz. The expectation values of each of theresulting terms are now evaluated in the quantum module, and are added up with a classical adder. The guess parameters arethen updated by the classical optimizer until a global minimum is reached. Here, h pq and h pqrs denote the amplitudes of the associ-ated one-body and two-body operators, respectively, andare basically the integrals involving the Hamiltonian andthe single particle wave functions, while N represents thenumber of electrons in the system.Since it is not possible to solve the Schr¨odinger equa-tion for a many-electron system, | Ψ (cid:105) is determined byemploying an appropriate approximate many-body for-malism. The simplest of such approaches is the Hartree-Fock (HF) method, whose governing equations can bevariationally derived. In this approach, the wave func-tion, | Φ (cid:105) , is a Slater determinant. However, since theHF theory does not take into account electron correla-tion effects, one needs to adopt post-HF methods. Em-ploying the full configuration interaction (FCI) methodfor a many-electron system within a given single-particlebasis gives the ‘exact’ atomic wave function within thatbasis. In this approach, the wave function of the sys-tem of interest is expressed as a linear combination ofall possible determinants that can be generated by ex-citing orbitals from the HF wave function. However, itis not feasible to perform FCI calculations on even the lighter systems with a reasonably high quality basis set,because of extremely steep computational cost. Trun-cated configuration interaction (CI) method is usuallyemployed in such scenarios. However, at a given levelof truncation, coupled-cluster (CC) theory can accountfor electron correlation effects more rigorously than theCI method. Moreover, truncated CC method satisfiessize consistency and size extensivity, which are desirableproperties of a many-body theory, in contrast to the CImethod, owing to the former expressing the atomic wavefunction in an exponential form as (e.g. see Ref. [48]) | Ψ (cid:105) = e T | Φ (cid:105) , (3)where for an N-electron system, T = T + T + ... + T N is the excitation operator, which generates particle-hole excitations. Once the amplitudes associated withthe T operators are obtained, the energy of the systemis calculated by E = (cid:104) Φ | ( H a e T ) c | Φ (cid:105) , (4)where the subscript, ‘c’, means that only the connectedterms between H a and T are retained. For practical rea- FIG. 3: Plot showing the variation in percentage fraction er-ror taken with respect to full configuration interaction (FCI)method, with the number of shots chosen up to 512, for Bein the STO-3G basis and with the Jordan-Wigner mapping. sons, the CC method is truncated at a given level ofparticle-hole excitation. When we truncate T at the sec-ond term, the approach is called the CCSD (coupled-cluster singles and doubles) method. Due to the expo-nential structure of the operator, the CCSD method isstill sufficient to predict properties with reasonable accu-racy for most closed-shell systems, as compared to othermany-body methods.As the above equation shows, the expression for en-ergy involves an effective non-hermitian Hamiltonian,( H a e T ) c . In the framework of quantum computation/simulation, it is desirable to work with unitary operators.For this purpose, we take recourse to the unitary versionof the CC (UCC) theory [49]. In the UCC framework,the wave function is given by | Ψ (cid:105) = e T − T † | Φ (cid:105) . (5)One can immediately see from the above equation thatthe UCC operator involves not only the excitation opera-tor T but also the de-excitation operator T † . The energyexpression follows, and is given by E = (cid:104) Φ | e T † − T H a e T − T † | Φ (cid:105) . (6)Clearly, unlike in the traditional version of the CCmethod, e T † − T H a e T − T † does not terminate naturally.There is no efficient method to evaluate the UCC am-plitude equations and evaluate the expression for energyon a classical computer without resorting to any approx-imation. However, as we shall see later, this issue is cir-cumvented on a quantum computer/simulator. Here too,we truncate T and T † at the level of singles and doublesexcitations (UCCSD method).It is also evident from the above discussions that theone-body and two-body integrals are the main ingredi-ents from a classical computer to carry out many-body calculations on a quantum simulator. These integrals areobtained using the HF single particle orbitals by employ-ing the PySCF program [50]. In this program, Gaussiantype orbitals (GTOs) [51], specifically contracted ver-sions of the minimal STO-3G and STO-6G basis [52],and Pople’s 3-21G basis and 6-31G basis [53], are em-ployed. Since the number of qubits required for the com-putations is equal to the number of spin-orbitals (whichis in turn decided by the choice of single-particle basisset), the qubit requirement for Li − , Be, and B + in STO-3G, STO-6G, 3-21G, and 6-31G basis sets is 10 for thefirst two and 18 for the remaining two basis sets. Wehave also carried out FCI and CCSD calculations usingPySCF [50], while the UCCSD computations were per-formed using the OpenFermion-PySCF [54] program. B. Mapping the Hamiltonian and wave functionfrom fermionic to qubit representation
To compute atomic energies in the framework of quan-tum simulation, one needs to map the second quantizedversion of the operators given on the right hand side ofEq. (6) into terms that contain a sequence of unitaryoperations. These structures are appropriately recast asgates in a circuit. This mapping is achieved by a tak-ing the fermionic creation and annihilation operators tospin operators. We use three such mapping techniques,namely the Jordan-Wigner (JW), parity (PAR), and theBravyi-Kitaev (BK) transformations. A comprehensivediscussion on all the three transformations can be foundin Ref. [55]. We additionally note that in the PAR map-ping scenario, we can use the two-qubit reduction thatresults from Z symmetry, thereby reducing the numberof required qubits for a given system by two. Furtherdetails on this can be found in Ref. [56]. C. The VQE algorithm
The VQE algorithm is a classical-quantum hybrid ap-proach, which is based on the well-known variationalprinciple. The energy functional is defined as in Eq. (1),with the unknown | Ψ (cid:105) replaced by a parametrized trialwave function, | Ψ ( θ ) (cid:105) = U ( θ ) | Φ (cid:105) , (7)with a set of arbitrary parameters, denoted compactlyas θ . We seek to find that set of parameters that takesus to the energy minimum. The energy thus obtainedis guaranteed to be an upper bound to the true groundstate energy. Mathematically, we express it as E ( θ ) = (cid:104) Ψ ( θ ) | H a | Ψ ( θ ) (cid:105)(cid:104) Ψ ( θ ) | Ψ ( θ ) (cid:105) (8)= (cid:104) Φ | U † ( θ ) H a U ( θ ) | Φ (cid:105) ≥ E . (9) FIG. 4: Analysis of energy versus the number of shots, with bigger step size than for Fig. 3 and up to 30000 shots. The fullconfiguration interaction (FCI), coupled-cluster method (CCSD), and unitary coupled-cluster method (UCCSD) results fromclassical computation are given for comparison. Each data point represents the mean of 20 runs for a given number of shots,and is accompanied by an error bar that quotes the maximum and the minimum obtained values within those 20 computations.
In the above equation, the unitary, U ( θ ), decides the vari-ational form (ans¨atz) to be used. E ( θ ) is minimized tofind the ground state energy, that is, E ≡ δE ( θ ) δθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ . (10)We will now briefly describe the VQE algorithm. State preparation:
The VQE procedure first requiresus to solve the HF equations and generate the relevantone- and two-electron integrals. The next step is statepreparation, where we choose a relevant ans¨atz. If thechoice for U ( θ ) takes the form of the UCC wave functionwith the t-amplitudes playing the role of the parameters, θ , we call such an ans¨atz as the UCC variational form. Itis this ans¨atz that we employ in this work. Given the ex-ponential form of the ans¨atz, one needs to appropriatelydecompose e ( T − T † ) into smaller operators that can be ef-ficiently implemented as quantum gates. Trotterization(for example, see Ref. [57]) is one such tool, where anexponential of a sum of operators can be recast as prod-uct of exponents of individual operators, and the level ofapproximation is decided by the Trotter number. Mapping:
Next, we map the Hamiltonian as well as thewave function to their spin operator form, by performingeither JW, PAR, or the BK transformation. The result-ing strings of tensor products of Pauli matrices (whichdepends on mapping and ans¨atz) can be implemented ascircuits, and this constitutes the quantum module of thealgorithm.
Backend simulator:
The computation is now carriedout with a suitable choice of a simulator (either stat-evector or qiskit’s QASM backend), and the resulting ex-pectation values are classically added. The statevectorsimulator executes the set of circuits associated with asystem without measurements or shots, given an inputstate vector. On the other hand, the QASM simulatormimics an ideal quantum computer, in that it gives prob-abilistic outcomes as counts for each of the states, aftermultiple shots.
Choice of optimizer:
We finally discuss the role of anoptimizer in the VQE algorithm. Once an expectationvalue is evaluated in the quantum module, we pass theenergy to an optimizer, which runs on a classical com-puter. This module uses an optimization algorithm, andminimizes the energy, obtained from the previous stepof the VQE algorithm, with respect to the parameters.Once the new parameters are obtained thus, they are fedback as inputs to the quantum circuit from the previousstep. This process is repeated until the energy is min-imized. In this work, we use a gradient-free approach,the COBYLA (Constrained Optimization BY Linear Ap-proximation) optimizer [58], which is commonly used inliterature [25, 59, 60]. The convergence for COBYLA op-timizer is slower than the gradient based methods as itrequires higher number of function evaluations to reachthe optimum value. However, stability comes as a no-table feature for this algorithm along with lesser numberof parameters to be tuned for performing optimization[61].The above mentioned structure of the algorithm is en-capsulated in Fig. 2. In this work, we use the qiskit 0.15.0
STO-3GMapping Method Li − Be B + ClC HF − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − + and Li − computed using the variationalquantum eigensolver (VQE) algorithm in STO-3G basis and adopting the unitary coupled-cluster (UCCSD) ans¨atz, withdifferent combinations of simulators and different fermion to qubit mapping techniques. Here, ‘ClC’ refers to the calculationsdone on a classical computer. The Jordan-Wigner transformation is abbreviated as ‘JW’, the Bravyi-Kitaev mapping as ‘BK’and the acronym for the parity transformation is ‘PAR’. The use of the UCCSD ans¨atz on QASM simulator is denoted by‘UQ’ and that on statevector simulator is written as ‘US’. Next to each numerical value of ground state energy for each of thecombinations mentioned above, a percentage fraction difference is shown computed with respect to the classical computationmethod, namely full configuration interaction. We add that the energies are specified up to ∼ µ Hartrees. Here, ‘Corr’ standsfor the correlation energy obtained using a particular method. package for quantum simulation [62].
III. RESULTS AND DISCUSSION
We present and analyze here the results for the groundstate energies obtained from the quantum simulation ofLi − , Be and B + , using the VQE algorithm. We show thedependence of the calculated energies on combinationsof different mappings and simulators, within a basis set.For the larger 3-21G and the 6-31G bases, we only pro-vide results obtained with the statevector simulator. Wealso provide the HF, CCSD, UCCSD, and FCI results,obtained with a ClC, for comparison. Explicitly givingthe HF energy allows us to visually check for the corre-lation effects captured by a VQE calculation for a givencombination of basis, mapping, and backend. In all ofour calculations, we set all the initial guess parametersfor the optimizer to zero. Also, we fix the Trotter numberto one and choose the COBYLA optimizer.We verified the errors that may arise with Trotter num-ber of one. For the Be atom in the STO-3G basis andwith JW mapping, we find that up to a Trotter step of 50, the error is at most ∼ ∼
14 H. For B + withthe same basis and mapping, the error can go as high as0.1 mH in ∼
24 H, and for Li − , the error does not exceed0.2 mH in ∼ ≈ a. b.c. d. FIG. 5: Graphical illustration of our results for the ground state energies of the Li − ion. The plots serve to compare for a givenatom and with the unitary coupled-cluster singles and doubles (UCCSD) variational form, the impact of different combinationsof fermion to qubit mapping techniques and simulators. The abbreviations used for the mappings in the plots are: Jordan-Wigner (JW), parity (PAR) and Bravyi-Kitaev (BK). The optimizer has been fixed to COBYLA for all the calculations. Theanalysis is carried out using the STO-3G, STO-6G, 3-21G and 6-31G basis sets, as shown in Sub-figures 5a., 5b., 5c. and 5d.,respectively. The dark blue bars indicate the energies obtained on a QASM simulator, while the bars in light blue specifythe energies computed using a statevector simulator. The calculated energies are compared with full configuration interaction(FCI) (dot-dash line), and also with CCSD (dotted line), and UCCSD (dashed line) methods. Each of the plots also show theHartree-Fock (HF) energy as a black solid line, that allows to visualize the correlation effects. Setting the same scale for allfour sub-figures enables us to visually compare the trends. larger intervals, and all the way up to 30000 shots. Also,we explicitly plot the energy versus the number of shotshere. This analysis is rigorous, with the inclusion of max-imum, minimum, and mean values for the energies ob-tained for a given number of shots, as a result of repeat-ing the runs twenty times for each data point. We alsoshow the values of energy calculated on a ClC from HF,FCI, CCSD and UCCSD, so as to have a visual feel ofcorrelation effects. It is noticeable that at 100 shots, themean energy is above the Hartree Fock value and there-fore hardly satisfying the variational principle. One canalso see that at lower number of shots, the error bar (thedifference between the maximum and minimum values)is so large that its extent is greater than the difference between HF and the FCI energies, that is, the amount ofelectron correlation. As the number of shots increase, thecurve approaches and appears to converge to the UCCSDvalue that one obtains with a ClC, with a very small er-ror bar. It is worth noting here that had we increasedthe shots further, the curve would have, albeit gradually,yielded lower values. The inference that the curve wouldcontinue to monotonically decrease is based on a simplefit to the mean energy values. However, it is importantto see that it is non-trivial to find a rigorous fit due tothe statistical nature of each data point,and for our pur-poses, not necessary. The plot also shows that the errorbars reduce with increasing shots, with the only excep-tion being that for 25000 shots, where the error bar is a. b.c. d. FIG. 6: Plot showing the values of the ground state energies of the Be atom calculated using the variational quantum eigensolver(VQE) algorithm and with the unitary coupled-cluster (UCCSD) ans¨atz in different bases, with various fermion to qubit mappingtechniques and on different simulators. The notations and abbreviations are: JW: Jordan-Wigner, PAR: parity, BK: Bravyi-Kitaev, FCI: full configuration interaction, CCSD: coupled-cluster method, and HF: Hartree-Fock. The sub-figures 6a., 6b., 6c.and 6d. provide these results in the STO-3G, STO-6G, 3-21G and 6-31G bases, respectively. lesser than expected. This may be due to the fact that20 repetitions need not necessarily build the statistics al-ways, and a few more repetitions may be required forsuch cases. Based on these results, we performed compu-tations with the QASM backend for the rest of the basissets and mappings, as well as for the other atoms, set-ting the number of shots to 20000. The rationale is that20000 shots finds a golden mean between computationalcost and accuracy ( ∼ ∼ − and B + , whose results are presented inFigs. 5a and 7a, respectively. In comparison, the cor-relation effects from FCI are about 40, 50, and 60 mHfor Li − , Be, and B + , respectively. Therefore, we can in-fer that quantum simulation with statevector simulatoraccounts for electron correlations very accurately in theSTO-3G basis. This is perhaps not surprising, as a stat-evector simulator does not rely upon statistics built fromrepeated measurements in order to extract energy. We a. b.c. d. FIG. 7: Figure presenting our results for the ground state energies of the B + ion. The nomenclature is as follows: JW: Jordan-Wigner, PAR: parity, BK: Bravyi-Kitaev, FCI: full configuration interaction, CCSD: coupled-cluster method, UCCSD: unitarycoupled-cluster method, and HF: Hartree-Fock. The STO-3G, STO-6G, 3-21G and 6-31G basis sets given in sub-figures 7a.,7b., 7c. and 7d., respectively. also present our results from a QASM simulator. Theyare all in good agreement with the UCC results from aClC, and not FCI as expected, due to our choice of thenumber of shots (20000 of them) as seen earlier. A pecu-liar observation in the ClC part of the results is that forall the considered basis sets, the CCSD method agreesbetter with FCI than UCCSD. In principle, UCCSD isexpected to capture more many-body effects than CCSD,with the caveat that the energy expression for the formerdoes not naturally terminate, thereby relying upon thechosen truncation scheme to achieve the desired results.We suspect that the observed deviation is associated withthe truncation scheme of the UCCSD approach.Figs. 6b., 5b. and 7b. show the same results but withthe STO-6G basis. The results are an improvement overthe earlier basis as evident by lowering of the calculatedenergies, although the qubit number is the same for agiven system, since more functions are contracted in theSTO-6G case. Not too surprisingly, the trends are verysimilar to those in the STO-3G basis. We now proceed to examine the results obtained frombigger bases as shown as Figs. 6c. and d., 5c. and d. and7c. and d.. We reiterate that QASM results are not com-puted, in view of the requirement of a large number ofshots to obtain a reasonably accurate result. We observefrom the figures that the effect of electron correlationon FCI energy is about 30 mH, 40 mH, and 50 mH forLi − , Be, and B + , respectively, whereas the difference inthe correlation energies between FCI and quantum sim-ulation are about 10 mH for all the systems. This dis-crepancy is possibly due to the slow convergence of theCOBYLA optimizer. To check this, we choose the JWmapping and the STO-3G basis set for a representativecalculation, and increase the number of iterations to be-yond the default maximum threshold of 1000 iterations(which we employ to report our results in this work). Wefound that while the percentage fraction error with re-spect to the FCI result is ∼ − at 1000 iterations, itdecreases further to ∼ − at 2000 iterations. We ex-pect that with the 3-21G basis as well as the 6-31G basis,0 STO-6GMapping Method Li − Be B + ClC HF − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , Be and B + obtained using the variational quantumeigensolver (VQE) algorithm with the use of unitary coupled-cluster (UCCSD) ans¨atz in STO-6G basis. The results arepresented using the different methods of fermion to qubit mapping, various simulators and different techniques of classicalcomputation of ground state energy. The notations are as follows: JW: Jordan-Wigner, PAR: parity, BK: Bravyi-Kitaev, FCI:full configuration interaction, CCSD: coupled-cluster method, HF: Hartree-Fock, ClC: classical computer, Corr: correlationcontribution, US: UCCSD statevector, and UQ: UCCSD QASM. the results would improve slightly with larger number ofiterations, which comes with higher computational cost.Alternatively, one could employ an optimizer that con-verges faster, such as L-BFGS-B and conjugate gradient,which we find after a preliminary survey to have con-verged within a lesser number of iterations but not assmoothly as COBYLA. We note that for a given atom,between different maps, the change in correlation ener-gies are ∼ IV. CONCLUSION
We have carried out VQE calculations of the groundstate energies of three isoelectronic systems, the nega-tively charged Li − , neutral Be, and the positively chargedB + , with the aim of studying the correlation trends inthese systems, and finding the degree to which these ef-fects is captured with various components of a quantumsimulation. We employ the UCCSD variational formfor this purpose, and compare our results with CCSD,UCCSD, and FCI computations performed on classicalcomputers. We study the sensitivity of the correlation1 − − Be B + ClC HF − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , Be and B + in the 3-21Gbasis. The abbreviations used in the Table are: JW: Jordan-Wigner, PAR: parity, BK: Bravyi-Kitaev, FCI: full configurationinteraction, CCSD: coupled-cluster method, HF: Hartree-Fock, ClC: classical computer, Corr: correlation contribution, US:UCCSD statevector, and UQ: UCCSD QASM. − − Be B + ClC HF − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − , Be and B + in the 6-31G basis.The terminology used is: JW: Jordan-Wigner, PAR: parity, BK: Bravyi-Kitaev, FCI: full configuration interaction, CCSD:coupled-cluster method, HF: Hartree-Fock, ClC: classical computer, Corr: correlation contribution, US: UCCSD statevector,and UQ: UCCSD QASM. effects to basis sets, fermionic to qubit maps, as wellas choice of backend simulators. We do not find sig-nificant dependence of the results on mapping, and asexpected, the choice of higher quality basis, in general,does improve (and lower) the energies. We observe thatthe Trotter number does not impact the results signif-icantly either, and that the COBYLA optimizer couldimprove the results slightly with increased iterations. En route, we also find that 3-21G basis is perhaps not thebest choice for future calculations among the basis setsconsidered in this work, as it gives results somewhat com-parable to the STO-6G basis, but with more number ofqubits, and hence substantially higher cost. On the sim-ulator front, we find that the statevector simulator cap-tures correlation effects efficiently with the results almostbeing independent of mapping. The QASM results rely2heavily upon a large number of shots, thereby making thecomputations expensive in order to obtain accurate re-sults. We also observe that with a larger basis, the VQEresults move away from the FCI values, as expected. Acknowledgements
We thank Mr. Ramanuj Mitra and Dr. Amar Vuthafor their help with computational resources for the cal-culations reported in this work. We are also grateful toDr. Kenji Sugisaki for useful discussions. Sumeet thanksPhysical Research Laboratory (PRL), Ahmedabad for providing the visiting fellowship to carry out part ofthis work. Most of the computations were performed onthe VIKRAM-100 cluster of PRL. VSP acknowledgesthe Graham cluster at the SciNet HPC Consortium(Compute Canada). SciNet is funded by: the CanadaFoundation for Innovation; the Government of Ontario;Ontario Research Fund - Research Excellence; and theUniversity of Toronto. VSP and Sumeet both usedGoogle Colab (Bisong E. (2019) Google Colaboratory.In: Building Machine Learning and Deep Learning Mod-els on Google Cloud Platform. Apress, Berkeley, CA.https : // doi . org / . / − − − − [1] S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin,and X. Yuan, Rev. Mod. Phys. , 015003 (2020).[2] J. Preskill, Quantum , 79 (2018).[3] Y. Nam et al , Nature J. Phys. Q. Info. , 33 (2020).[4] T. Takeshita, N. C. Rubin, Z. Jiang, E. Lee, R. Babbush,and J. R. McClean, Phys. Rev. X , 011004 (2020).[5] G. H. Low and I. L. Chuang, Quantum , 163 (2019).[6] N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A.Aspuru-Guzik, and G. G. Guerreschi, Nature J. Phys. Q.Info. , 49 (2020).[7] D. Chivilikhin, A. Samarin, V. Ulyantsev, I. Iorsh, A. R.Oganov, and O. Kyriienko, arXiv:2007.04424v1 (2020).[8] J. P. Dowling and G. J. Milburn, Philos. Trans. R. Soc.Lond. A , 1655 (2003).[9] D. S. Abrams and S. Lloyd, Phys. Rev. Lett. , 2586(1997).[10] D. S. Abrams and S. Lloyd, Phys. Rev. Lett. , 5162(1999).[11] T. Saue et al , J. Chem. Phys. , 204104 (2020).[12] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. HeadGordon, Science , 1704 (2005).[13] B. P. Lanyon, et al , Nat. Chem. , 106 (2010).[14] M. Hamed et al , arXiv:1910.11696 (2019).[15] S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N.Wiebe, D. P. Tew, J. L. O’ Brien, and M. G. Thompson,Phys. Rev. Lett. , 100503 (2017).[16] M. H. Yung, J. Casanova, A. Mezzacapo, J. McClean, L.Lamata, A. Aspuru-Guzik, and E. Solano, Sci. Rep. ,3589 (2014).[17] A. Peruzzo, J. McClean, P. Shadbolt, M. H. Yung, X. Q.Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’ Brien,Nat. Comm. , 4213 (2014).[18] J. Clarke and F. K. Wilhelm, Nature , 1031 (2008).[19] J. I. Cirac and P. Zoller, Phys. Rev. Lett. , 4091 (1995).[20] H. Wang, S. Kais, A. Aspuru-Guzik, and M. R. Hoff-mann, Phys. Chem. Chem. Phys. , 5388 (2008).[21] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M.Brink, J. M. Chow, and J. M. Gambetta, Sci. Reps. ,3589 (2015).[22] M. H. Yung, J. Casanova, A. Mezzacapo, J. McClean, L.Lamata, A. Aspuru-Guzik, and E. Solano Nature ,242 (2017).[23] J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M.E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. deJong, and I. Siddiqi, Phys. Rev. X , 011021 (2018). [24] Y. Wang et al , Am. Chem. Soc. Nano , 7769 (2015).[25] J. Romero , R. Babbush , J. R. McClean , C. Hempel ,P. J. Love, and A. Aspuru-Guzik, Quantum Sci. Technol. , 014008 (2019).[26] F. Schmidt-Kaler, T. Pfau, P. Schmelcher and W. Schle-ich, New J. Phys. , 065014 (2010).[27] C. W. Chou, D. B. Hume, T. Rosenband, and D. J.Wineland, Science , 1630 (2010).[28] L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans,K. Jungmann, B. P. Das, and D. Mukherjee, Phys. Rev.A , 050501(R) (2008).[29] T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu,V. Vuletic, and M. D. Lukin, Nature , 241 (2014).[30] N. Fortson, Phys. Rev. Lett. , 2383 (1993).[31] B. K. Sahoo and B. P. Das, Phys. Rev. Lett. , 203001(2018).[32] C. Sur, K. V. P. Latha, B. K. Sahoo, R. K. Chaudhuri,B. P. Das and D. Mukherjee, Phys. Rev. Lett. , 193001(2006).[33] V. Kell¨o, M. Urban and A. J. Sadlej, Chem. Phys. Lett. , 383 (1996).[34] B. K. Sahoo, Phys. Rev. A , 022820 (2020).[35] T. Andersen, Phys. Reps. , 157 (2004).[36] H. S . W . Massey, Negative Ions , 3rd ed., CambridgeUniversity Press, London and New York (1976).[37] V. Dudinikov,
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