Variational Quantum Simulation for Periodic Materials

Nobuyuki Yoshioka,1, ∗ Yuya O. Nakagawa,2 Yu-ya Ohnishi,3 and Wataru Mizukami4, 5, 6, †

1Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako-shi, Saitama 351-0198, Japan2QunaSys Inc., Aqua Hakusan Building 9F, 1-13-7 Hakusan, Bunkyo, Tokyo 113-0001, Japan

3Materials Informatics Initiative, RD Technology & Digital Transformation Center,JSR Corporation, 100 Kawajiri-cho, Yokkaichi, Mie, 510-8552, Japan

4Center for Quantum Information and Quantum Biology,Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.

5JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan6Graduate School of Engineering Science, Osaka University,

1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.(Dated: September 18, 2020)

We present a quantum-classical hybrid algorithm that simulates electronic structures of periodic systems suchas ground states and quasiparticle band structures. By extending the unitary coupled cluster (UCC) theory todescribe crystals in arbitrary dimensions, for a hydrogen chain, we numerically demonstrate that the UCC ansatzimplemented on a quantum circuit can be successfully optimized with a small deviation from the exact diag-onalization over the entire range of the potential energy curves. Furthermore, by using the quantum subspaceexpansion method, in which we truncate the Hilbert space within the linear response regime from the groundstate, the quasiparticle band structure is computed as charged excited states. Our work establishes a powerfulinterface between the rapidly developing quantum technology and modern material science.

Introduction.— Achieving decisive ab initio descriptions ofelectronic properties in solid systems is one of the most sig-nificant issues in modern material science. For weakly cor-related systems, the development of the density functionaltheory (DFT) [1–3] and GW approximation [4, 5] have re-alized increasingly accurate numerical simulations. Wave-function-based techniques have also been studied intensively:time-dependent Hartree-Fock theory [6], second-order Mller-Plesset perturbation theory (MP2) [7, 8] and coupled-cluster(CC) theory with single and double excitations (CCSD) [7,9, 10]. More recent reports include CCSD with perturba-tive triple excitations (CCSD(T)) [11] and full configuration-interaction (FCI) quantum Monte Carlo method for periodicsolids [12]. Meanwhile, it must be noted that periodic sys-tems contrast sharply with molecular systems, in that one mustsimulate the thermodynamic limit. In general, a large numberof particles, or the Brillouin zone sampling, are required toachieve convergence to the thermodynamic limit. The growthof computational resources requirements rapidly exceeds su-percomputing capacity, which severely limits the explorationof realistic materials. Therefore, algorithms with both favor-able scaling and high accuracy beyond the current schemesare indispensable.

The surging development in quantum technology may of-fer a path to achieving this goal. The variational quantumeigensolver (VQE) algorithm and its variants, for instance,enable the simulation of eigenstates of a given Hamiltonianon noisy intermediate-scale quantum (NISQ) devices [13].Although the rigorous computational speed-up by the VQE-based calculations over classical algorithms still remains elu-sive (a situation related to the present lack of quantum er-ror correction), many studies have focused on its demonstra-tion in actual quantum devices [14–18] and its extension tosolve a various classes of problems [19–25]. From a quan-tum chemistry perspective, an intriguing question is whether

the NISQ devices become capable of implementing classi-cally intractable wave function ansatz such as the unitary cou-pled cluster (UCC) ansatz [26–31], a variational parametriza-tion of CC wave functions based on unitary transformation.Although classical computers suffer from an exponential in-crease in the computational cost, quantum computers natu-rally simulate such ansatze with only a polynomial numberof quantum gates. For efficient implementation on the NISQdevices, more hardware-friendly and/or sophisticated ansatzehave been proposed [32–34]. However, to the best of ourknowledge, existing VQE algorithms and their demonstra-tions have been mainly performed for small molecules, andperiodic systems have not been successfully simulated.

In the present work, we propose and demonstrate a VQE-based framework enables simulations of solid materials at theab-initio level. Our work is the first to show that electronicground states of periodic systems such as the hydrogen chaincan be computed accurately even in the strongly correlatedregime where the classical gold-standard methods such asCCSD(T) break down. Furthermore, we present a method tocalculate the quasiparticle band structure from the VQE quan-tum state. Our approach is to describe quasiparticle excita-tions by linear-response-based calculations, i.e., the quantumsubspace expansion (QSE) [35]. The present work establishesa powerful interface between two major fields, namely therapidly developing quantum computing technology and mod-ern material science.

Second quantized ab-initio crystal Hamiltonian.— Ab ini-tio fermionic Hamiltonian with periodic boundary conditions

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is given in the second quantization representation as

H =∑pq

∑k

tkpq c†pkcqk

+∑pqrs

′∑kpkqkrks

vkpkqkrkspqrs c†pkp

c†qkqcrkr csks , (1)

where cpk (c†pk) is the annihilation (creation) operator of thep-th Bloch or crystalline orbital (CO) with crystal momentumk. The complex coefficients tkpq and vkpkqkrks

pqrs are one- andtwo-body integrals between COs. Note that, because of trans-lational symmetry, the two-body term must obey the conser-vation law written as

kp + kq − kr − ks = G, (2)

where G is a reciprocal lattice vector of the unit cell. Such arequirement is indicated by the primed summation in Eq. (1).

In the present work, we determine COs from the crystalHartree–Fock theory with the Gaussian-based atomic orbitals(AOs), for which we employ minimal basis sets (i.e., STO-3G) [36, 37]. Two remarks concerning the computation ofintegrals are in order. First, the divergence corrections for ex-change integrals are computed separately. Because the G = 0contribution in the exchange integrals simply shifts the bandstructure according to their particle number, we initially ne-glect the divergent term and add the corresponding correc-tion after computing the correlation energy [38]. Second, theGaussian density fitting technique [39] is used for the two-body coefficients vkpkqkrks

pqrs to accelerate the integral calcula-tion.

To solve the Schrodinger equation defined by the Hamil-tonian (1) on a quantum computer, we map fermionic oper-ators into spin-1/2 operators. A widely-known technique isthe Jordan–Wigner transformation [40], which naturally en-codes the fermionic anticommutation relation as the parity ofthe particle number. Although we adopt the Jordan–Wignertransformation in the present work, one may also consult onother techniques with improved non-locality [41], which maybecome crucial for suppressing Pauli measurement error innoisy devices. Here, the fermion-qubit mapping algorithm isno different in the case of crystalline systems than in molec-ular systems. However, the number of qubits required in aperiodic system could be considerably larger than that in anisolated system; if the number of k-point samples is set toNk,the number of qubits required is increased by Nk [See Fig. 1for a graphical description.].

Variational quantum eigensolver and unitary coupled clus-ter theory for solids.— Once the qubit representation of thecrystal Hamiltonian is prepared, the ground state wave func-tion and its energy can be calculated on a quantum computerusing the VQE algorithm. That is, one constructs a quantumcircuit U(θ) where θ denotes the circuit parameters, and a trialwave function |ψ(θ)〉 = U(θ) |0〉, where |0〉 is an input quan-tum state. The ground state is approximated by the variational

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FIG. 1. Encoding crystalline orbitals into variational quantum cir-cuits. The example is for a linear hydrogen chain where each unitcell contains two atoms. The number of the qubits increases linearlywith respect to the number of k-points sampled from the Brillouinzone.

ansatz |ψ(θ∗)〉 whose parameters are taken to minimize theenergy function as follows,

θ∗ = arg minθ

E(θ), (3)

E(θ) := 〈ψ(θ)|H|ψ(θ)〉. (4)

Various quantum circuits (i.e., ansatze) have been proposedfor the VQE to describe many-body wave functions accu-rately and compactly. In this study, we choose the unitarycoupled cluster singles and doubles (UCCSD) ansatz [26–31]with one-step Trotter expansion, or the disentangled UCCSDansatz [42]:

|ψ〉 =

∏pq

∏kpkq

eAkpkqpq

∏pqrs

∏kpkqkrks

eAkpkqkrkspqrs

|0〉 ,(5)

where Akpkqpq = a

kpkqpq c†pkp

cqkq− c.c. and A

kpkqkrkspqrs =

akpkqkrkspqrs c†pkp

c†qkqcrkr

csks−c.c. are cluster operators for sin-

gle and double excitations, respectively. The Hartree–Fockstate is chosen as the input state |0〉. The coefficients a arethe variational parameters and correspond to θ. Note that thevariational parameters a of the UCCSD ansatz are in gen-eral complex values for a periodic system, while those witha standard time-independent molecular Hamiltonian are realbecause its matrix elements can be chosen to exclude imagi-nary values. One of the strengths of the UCC in general is thatit enables the symmetry, such as the number of particles or thetotal spin angular momentum, to be easily introduced into theansatz (i.e., quantum circuit) because the UCC is based on thefermionic representation. In addition, the translational sym-metry can be straightforwardly implemented into the UCC by

3

limiting Akpkqpq and Akpkqkrks

pqrs to the operators satisfying thecrystal momentum conservation law (Eq. (2)). Nonetheless, inthis work, for simplicity, we limit the variational parametersto real numbers and do not impose the momentum conserva-tion condition. Our strategy is to compensate for the loss ofexpressive power due to the absence of complex variables byremoving the restriction of the translational symmetry. Wetentatively refer to this variant of the UCCSD ansatz as thebroken-translational-symmetry UCCSD ansatz with real vari-ables (bUCCSD-Real) here.

Computing quasiparticle bands from the VQE wavefunction.— Of course, the ground state energy is not the onlyimportant and interesting property for solids. The force oneach nucleus, for example, is often essential in practical calcu-lations, and can be obtained from the energy derivatives. En-ergy derivatives for periodic systems can be calculated in thesame manner as those for molecules. The analytical energyderivative calculation methods for the VQE quantum statehave already been established and applied [24, 43]. Anotherexample is the band structure, which is a property peculiar tosolids. It is a common and indispensable concept or tool foranalyzing the electronic structure of a crystal. Band calcula-tions for quantum many-body systems are often performed bysimulating quasiparticle excitations, assuming that the theo-retical framework defined in the one-body picture still holds.In this context, various classical algorithms, including the GWapproximation, have already been proposed to find the quasi-particle bands [4, 5]. In the present study, we also employ asimilar assumption and extend the QSE method to calculatequasiparticle bands from the VQE quantum state.

The aim of the QSE method is to compute a subset ofthe entire eigenspectrum in a subspace defined from a ref-erence quantum state |ψ〉. Concretely, we first prepare a setof many-body basis |Φi〉 = Ri |ψ〉, where |ψ〉 is the VQEquantum state and Ri = c†pkp

c†qkq. . . crkr

csks. . . is an ex-

citation operator with a multi-index i specifying a string ofannihilation and creation operators. Then, we diagonalizea subspace Hamiltonian Hsub defined in a truncated Hilbertspace spanned by {|Φi〉}. The non-orthogonality of suchmany-body bases requires us to solve the following general-ized eigenvalue problems:

HsubC = SsubCE, (6)

where Ssub is a metric of the subspace given by the overlapbetween bases, C are eigenvectors, and the diagonal matrixE yields eigenenergies. The matrix elements of the subspaceHamiltonian Hsub and the metric Ssub are given by

Hsubij = 〈Φi|H|Φj〉 = 〈ψ(θ)|R†i HRj |ψ(θ)〉, (7)

Ssubij = 〈Φi|Φj〉 = 〈ψ(θ)|R†i Rj |ψ(θ)〉. (8)

These quantities are evaluated as the expectation value of non-Hermitian operators R†i HRj and R†i Rj , which can be realizedby measuring real and imaginary parts separately for instance.

Thus far, the QSE method has been used with particle-number conserving excitation operators, which corresponds

to the so-called multi-reference configuration interaction(MRCI) method in the classical algorithms of quantum chem-istry [44, 45]. The QSE method differs from the MRCImethod in that it uses quantum measurements to evaluatethe matrix elements. Here, we propose a QSE method forcalculating quasiparticle bands using many-body bases cre-ated by ionization or electron-attachment operators, whichremove or add one particle from the VQE quantum state|ψ〉. The valence band energies are obtained at individ-ual k by performing the QSE using ionization operatorsRIPlk = clk where l runs over the occupied orbitals. In con-trast, the conduction bands are obtained by using electron-attachment operators REAbk = c†bk where b runs over un-occupied orbitals. Our method is closely related to a vari-ant of the equation-of-motion coupled cluster (EOM-CC),namely, ionization-potential/electron-attached EOM-CC (IP-EOM-CC, EA-EOM-CC) [10].

Numerical examples.— Now that the theoretical frameworkis readily provided, we are ready to demonstrate our algo-rithms in periodic systems. First, we compute the ground stateof the linear hydrogen chain, which is known for its rich phys-ical feature that is still not completely understood despite itssimplicity [46–53]. The outcome of the electronic interac-tion varies diversely along the atom separation; the systemexperiences a metal-insulator transition with a strongly corre-lated regime in between. As shown in Fig. 2, the bUCCSD-Real ansatz correctly captures such a complex behaviour. Itis evident from the potential energy curve shown in Fig. 2(a)that strong electronic correlation develops as atoms becomeseparated. Therefore, the classical gold-standard CCSD andCCSD(T) methods result in a large deviation from the ex-act diagonalization, or the FCI; see Fig. 2(b). In contrast,the bUCCSD-Real ansatz can describe the behaviour of hy-drogen atoms much more accurately, owing to the enhancedrepresentability of the variational ansatz. The bUCCSD-Realansatz can simulate the weakly correlated region as precise asthe CCSD method and suppresses the deviation in the stronglycorrelated regime. Considering the fact that the ansatz isnot designed to capture the whole Hilbert space with higher-order electronic excitations, we expect that the calculation canbe systematically improved by applying more powerful andsophisticated ansatze such as the ADAPT or cluster-Jastrowansatze [33, 34].

It should be noted that, although the result by the bUCCSD-Real ansatz is presented in the current work, it may be desir-able to employ an ansatz with complex variables. Extendingreal variables to complex variables results in effectively dou-bling the number of variables. Nonetheless, the disadvantagesof extending to complex variables are presumably compen-sated by using the momentum conservation law (Eq. (2)); thetranslational symmetry leads to a considerable reduction in thenumber of parameters, especially when many k-points mustbe considered, such as in the three-dimensional systems.

Next, we turn to the band-structure calculation of the hydro-gen dimer chain. Such two-leg ladder systems are of strong in-terest from both the theoretical and experimental aspects, be-

4

(a) (b)

FIG. 2. (a) Potential energy curves of the linear hydrogen chain computed at UCCSD-Real, CCSD, MP2, RHF, and FCI with the STO-3Gbasis sets. Each unit cell consists of two hydrogen atoms each, and three k-points are sampled from a uniform grid. (b) Absolute error fromthe FCI calculation. In the weakly correlated region, the bUCCSD-Real ansatz are slightly more accurate than CCSD, whose deviation crossesfrom positive to negative near 2.6 Bohr. The dotted line indicates the chemical accuracy (1.6× 10−3 Hartree).

cause synthesized compounds on ladder structures may showexotic phenomena such as the unconventional superconductiv-ity and spin-liquid behavior [54]. In particular, the half-filledHubbard model on a two-leg ladder is gapped by both chargeand spin excitations, as opposed to that on the linear chain.Such a state with spin singlets on each rung has been foundto evolve into a superfluid phase by additional spin exchangeinteraction between rungs [55]. Here, we take a large distancebetween hydrogen dimers so that the system is described bythe coherent spin singlet state. The quasiparticle spectrum ofthe system is obtained by the ionized/electron-attached QSEmethod introduced previously. As can be seen from Fig. 3,both the highest occupied and lowest unoccupied bands aresimulated precisely. In particular, the direct band gap esti-mated at crystal momentum kL = π/4 (L: unit cell length)is 1.5047 Hartree, which is consistent with the EOM-CCSDcalculation with an error less than 3× 10−4 Hartree.

Summary and outlook.— We have presented a frameworkfor simulating the electronic structures of solids using NISQdevices at the ab initio level. The numerical results demon-strate that our VQE-based algorithm simulates the hydrogenchain well not only in the weakly-correlated electronic struc-tures but also for the strongly correlated regimes. Further-more, we have shown that the quasiparticle band structurecan be computed by applying the QSE method, which diag-onalizes the Hamiltonian in a truncated space described bythe linear response, to charged excited states. The use ofionization/electron-attachment operators yields a substantialimprovement in the measurement cost that scales quadrati-cally with respect to the qubit count, as opposed to the quar-tic (or higher) scaling required in the standard QSE methodwhich employs particle-number-conserved excitation opera-tors.

Our VQE-based framework is expected to provide an ap-proach for investigating otherwise intractable systems. In ad-

(a) (b)

FIG. 3. (a) Band structure of the hydrogen dimer chain computedat UCCSD-Real, CCSD, and RHF with the STO-3G basis sets. Theenergy is shifted so that the highest energy of the occupied band iszero. Two bands are well separated by a gap owing to the coher-ent spin singlet formation. (b) Absolute deviation of the electronaffinity (upper panel) and ionization potential (lower panel) from theequation-of-motion CCSD calculation. A unit cell considered in thecalculation consists of a pair of hydrogen atoms that are 1.2 Bohrapart from each other, and the distance between dimers is taken as 4Bohr. Two k-points are sampled from a uniform grid.

dition to the insulating low-dimensional materials calculatedin the present work, real solid surface systems and stronglycorrelated materials are core targets that should be investi-gated once the quantum computers become sufficiently ma-ture. To address target materials having a tangible impact notonly for scientific knowledge but also for industrial applica-tions, it would be necessary to develop a qubit reduction tech-nique that explicitly makes use of the symmetry.

5

Acknowledgements.— This work was supported by MEXTQuantum Leap Flagship Program (MEXT Q-LEAP) GrantNumber JPMXS0118067394. WM wishes to thank JSPSKAKENHI No. 18K14181 and JST PRESTO No. JP-MJPR191A. Some calculations were performed using thecomputational facilities in the Institute of Solid State Physicsat the University of Tokyo, RIKEN, and in Research Insti-tute for Information Technology (RIIT) at Kyushu University,Japan. NY is supported by the Japan Science and Technol-ogy Agency (JST) (via the Q-LEAP program). We thankDrs. Takeshi Sato and Kosuke Mitarai for fruitful discus-sions. Numerical calculations were performed using Open-Fermion [56], PySCF (v1.7.1) [38], and Qulacs [57].

Note added.— Shortly after completion of this work, webecame aware of two independent works [58, 59] that havebeen carried out in parallel.

∗ nobuyuki.yoshioka@riken.jp† wataru.mizukami.857@qiqb.otri.osaka-u.ac.jp

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1

Supplemental Materials for: ‘Variational Quantum Simulation for Periodic Materials‘”

NUMERICAL RESULTS FOR MOMENTUM CONSERVATIONS AND FIDELITIES

As explained in the main text, we adopt the broken-translational-symmetry UCCSD ansatz with real variables (bUCCSD-Real)for demonstration of the variational quantum simulation for periodic materials. The total crystal momentum can be different fromthe input state of the ansatz, i.e., the Hartree-Fock state, in this ansatz. We numerically check the total crystal momentum K ofthe optimized wave function |ψbUCCSD−R〉 for the linear hydrogen chain investigated in Fig. 2 of the main text. We computethe expectation value of the translation operator TL defined for the unit cell length L as

⟨ψbUCCSD−R

∣∣∣ TL ∣∣∣ψbUCCSD−R

⟩, and

this expectation value is identified with eiKL. The result is shown in Fig. S1(a). As is expected from the uniform geometryof the chain, the total crystal momentum K is apparently zero within the numerical error caused by slight imperfectness of theoptimization of the wavefunction in the VQE: the mean of real and imaginary parts ofKL/π for all data points are below 10−11.

In addition, we present infidelity between |ψbUCCSD−R〉 and the exact wavefunction computed by the full configuration-interaction (FCI) |ψFCI〉, namely 1−| 〈ψbUCCSD−R |ψFCI〉 |, in Fig. S1(b). The infidelity is small and varies with the hydrogen-hydrogen distance of the chain in a similar way to the deviation from the exact energy (Fig. 2(b)). This result also indicates thatour choice of the ansatz properly reproduces the exact wavefunction of the system.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Nearest H-H distance [Bohr]

1

0

1

2

3

Mom

entu

m K

L/

1e 11(a)

RealImag

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Nearest H-H distance [Bohr]

10 5

10 4

10 3

10 2In

fidel

ity

(b)

FIG. S1. (a) Crystal momentum of the optimized bUCCSD-Real wavefunction whose energy is shown in Fig. 2(a) of the main text. A unitcell with the length L consists of two hydrogen atoms each, and three k-points are sampled from a uniform grid. Crystal momentum K isdetermined by the equation eiKL =

⟨ψbUCCSD−R

∣∣∣ TL

∣∣∣ψbUCCSD−R

⟩, where TL is the translation operator of the length L. (b) Infidelity

between the optimized bUCCSD-Real wavefunction and the exact wavefunction obtained by the FCI calculation.