Abstract
Establishing a predictive ab initio method for solid systems is one of the fundamental goals in condensed matter physics and computational materials science. The central challenge is how to encode a highlycomplex quantummanybody wave function compactly. Here, we demonstrate that artificial neural networks, known for their overwhelming expressibility in the context of machine learning, are excellent tool for firstprinciples calculations of extended periodic materials. We show that the groundstate energies in real solids in one, two, and threedimensional systems are simulated precisely, reaching their chemical accuracy. The highlight of our work is that the quasiparticle band spectra, which are both essential and peculiar to solidstate systems, can be efficiently extracted with a computational technique designed to exploit the lowlying energy structure from neural networks. This work opens up a path to elucidate the intriguing and complex manybody phenomena in solidstate systems.
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Introduction
Artificial neural networks (ANNs) are a class of expressive mathematical models originally designed to imitate the high computing power of the human brain. Driven by the outstanding success over existing data processing methods in the field of machine intelligence^{1,2,3}, ANNs have been used in a wide range of applications, from physical science^{4,5,6,7,8}, medical diagnosis, to astronomical observations. Remarkable among numerous factors underlying their performance is their ability to perform efficient feature extraction from highdimensional data.
As universal approximators, ANNs have a rich expressive power, which can also be exemplified by encoding complicated quantum correlations^{9}. Carleo and Troyer^{10} showed that ANNs, employed as a quantum manybody wavefunction ansatz, can solve strongly correlated lattice systems at stateoftheart level. Such quantumstate ansatze, often referred to as neural quantum states (NQS), capture quantum entanglement that even scales extensively^{11}. The use of such a powerful nonlinear parametrization has been keenly investigated in the quantum physics community: both equilibrium^{12,13} and outofequilibrium^{14,15,16,17} properties, extension of the network structure^{18,19,20}, and quantum tomography^{21,22,23,24}. Meanwhile, we point out that the application of ANNs to fermionic systems is much less explored, despite their practical significance, such as the modeling of real materials and the experimental realizability in quantum simulators^{25,26}. The proof of concept for small molecular systems was first presented by Choo et al.^{27} which applied the ANNs to solve the manybody Schrödinger equation governed by the secondquantized Hamiltonian for molecular orbits. Few implementations have been further performed to simulate the electronic structures using ANNs^{28,29,30,31}. Thus, a crucial question remains to be answered: are ANNs powerful enough to represent the electronic structures of real solid materials? This is related to one of the fundamental problems in condensedmatter physics and computational materials science; namely, establishing a predictive ab initio method for solids or surfaces. In particular, it must be demonstrated that the ANNs are capable of investigating the thermodynamic limit.
We stress that no current firstprinciples method can take into account both weak and strong electron correlations compactly and sufficiently. For instance, it is well known that the accuracy of the de facto standard method, density functional theory (DFT), is semiquantitative and it is very difficult to improve significantly^{32,33}. Manybodywavefunctionbased methodologies are, in contrast, systematically improvable. Such techniques, mainly based on coupledcluster (CC) theory (or manybody perturbation theory)^{34}, have been successful for the electronic states of molecules. This has encouraged the application of quantum chemical methods to solidstate physics^{35,36}. However, methods such as CC specialize in describing weak electronic correlations, and only work well for electronic states where the meanfield approximation is valid.
Methods for dealing with strongly correlated electrons, called multireference theory, also exists in quantum chemistry^{37}; but these assume that the number of strongly correlated electrons is small. Such a condition usually holds in the case of molecules, because the number of strongly correlated electrons is often localized and limited. In contrast, there can be a large number of moderately or strongly correlated electrons in solidstate systems, owing to their high symmetry and dense structure. Based on its success in spin systems, it is natural to expect that the NQS have the potential to compactly describe a variety of electron correlations appearing in firstprinciples calculations of solids with a moderate computational cost (See Fig. 1 for a schematic diagram of the hierarchy of quantum chemical methods^{38,39,40,41,42}).
In this work, we demonstrate that neuralnetworkbased manybody wave functions can readily simulate the essense of firstprinciples calculations for extended periodic materials: the groundstate and excitedstate properties. The secondquantized fermionic Hamiltonian is transformed into a spin representation, such that the problematic sign structure of fermions, which usually imposes severe limits on the numerical accuracy, is naturally encoded. Employing the variational Monte Carlo (VMC)based stochastic optimization, we show that the thermodynamic limit of a onedimensional system can be simulated within chemical accuracy. For real solids in both two and three dimensions, the static electronic correlation in the minimal active space is compactly represented by the NQS. Our work’s main contribution is that multiple excited states, forming quasiparticle band spectra, are computed by constructing an effective Hamiltonian in the truncated Hilbert space. To the best of our knowledge we offer the first demonstration that the NQS can be applied to simulate lowlying eigenstates in the identicalquantumnumber sector.
Results
Secondquantization representation of solid systems
To alleviate the notorious difficulty of simulating the manybody problem of solid systems, we employ a linear combination of the singleparticle basis. Namely, we construct crystalline orbitals (COs) using the solution of the crystalline Hartree–Fock (HF) equation^{43,44}. The secondquantization form of the manybody fermionic Hamiltonian is
where c_{pk} (\({c}_{p{\bf{k}}}^{\dagger }\)) denotes the annihilation (creation) operator of an electron on the pth CO with crystal momentum k. Here, the anticommutation relation \(\{{c}_{p{{\bf{k}}}_{p}},{c}_{q{{\bf{k}}}_{q}}^{\dagger }\}={\delta }_{pq}{\delta }_{{{\bf{k}}}_{p}{{\bf{k}}}_{q}}\) is imposed, and onebody (twobody) integrals are given as \({t}_{pq}^{{\bf{k}}}\) (\({v}_{pqrs}^{{{\bf{k}}}_{p}{{\bf{k}}}_{q}{{\bf{k}}}_{r}{{\bf{k}}}_{s}}\)). For simplicity, hereafter we denote the suffix as μ ≔ (pk). While the general framework of the crystalline HF equation is common with that for molecular systems, it must be noted that the contribution from the reciprocal lattice vector G = 0 requires extra numerical care owing to the divergence of the exchange integrals. In this work, we employ the crystalline Gaussianbased atomic functions as the singleparticle basis. The Gaussian density fitting technique is applied to efficiently compute the twobody integrals^{45}.
The summation in the first term of Eq. (1) is taken over a uniform grid, which is typically obtained by shifting the k’s obeying the Monkhorst–Pack rule^{46}. Note that the number N_{k} of sampled kpoints can be arbitrary. The primed summation in the second term satisfies the conservation of crystal momentum, which follows from translational invariance:
where \({\mathcal{G}}\) is the set of reciprocal lattice vectors. With the number of COs at each kpoint denoted as N, the total number of terms in Eq. (1) is given as \({\mathcal{O}}({N}^{4}{N}_{k}^{3})\).
To solve the fermionic manybody Hamiltonian (1), we must explicitly impose the antisymmetric sign structure in the quantum state. Here, we map the Hamiltonian into the spin1/2 representation such that the sign structure is encoded in the operators rather than the quantum states, as Choo et al.^{27} considered in their application of the NQS to small molecules. The Jordan–Wigner (JW) transformation^{47} defines the relation of fermionic and spin operators as \({c}_{\mu }^{(\dagger )}={(1)}^{\mu 1}{\prod }_{\nu \,{<}\,\mu }{\sigma }_{\nu }^{z}{\sigma }_{\mu }^{+()}\), where \({\sigma }_{\mu }^{+()}\) is the raising (lowering) operator of the μth spin. Such a mapping yields a nonlocal spin Hamiltonian
where P_{Q} ∈ ⨂_{μ}{I, X, Y, Z} is a product of Pauli matrices for a corresponding Pauli string Q.
Let us make two remarks on the application of JW transformation. First, the use of the fermiontospin transformation for stochastic variational calculations was initially considered in the context of nearterm quantum computers^{48}, including the application to real solids^{49,50,51}, while the spintofermion mapping has been long applied in condensedmatter and statistical physics community, e.g., to solve exactly soluble quantum spin models. Second, the JW transformation merely generates the spin operator representation of the Hamiltonian (1) and does not alter the computational basis. The evaluation of physical observables in the Monte Carlo approach by the occupationnumber basis of the fermionic representation is identical to that by the spin computational basis of the spin representation. This is not the case when we apply other transformations developed in quantum information, such as the Bravyi–Kitaev transformation^{52}.
Ground states in the thermodynamic limit
In general, it is classically intractable to solve for the ground state of the manybody Hamiltonian defined in Eq. (1) or (3). Here we alternatively rely on a variational method that exemplifies the expressive power of neural networks. Namely, a neural network is used as a variational manybody wavefunction ansatz. It is optimized so that the expectation value of the energy, estimated via the Monte Carlo simulation, is minimized by approximating the imaginarytime evolution. Such a technique, called variational Monte Carlo (VMC), has been successfully applied to condensedmatter systems^{53,54,55,56} and quantum chemistry problems^{57,58}, leading to stateoftheart numerical analysis on strongly correlated phenomena. The choice of the variational ansatz plays a key role for the accuracy, which, as has been pointed out by Carleo and Troyer^{10}, can be significantly improved by using neural networks.
Let us briefly review the general protocol of VMC for simulating ground states in manybody spin systems using the quantumstate ansatz based on the restricted Boltzmann machine (RBM)^{59}. First, we introduce the quantum manybody wave function expressed as follows^{10},
where \({{{\Psi }}}_{\theta }^{{\rm{RBM}}}(\sigma )\) is the unnormalized amplitude for a spin configuration \(\sigma \in {\{1,+1\}}^{{N}_{v}}\) where N_{v} = NN_{k} is the total number of spin orbitals and \(Z=\sqrt{{\sum }_{\sigma } {{{\Psi }}}_{\theta }^{{\rm{RBM}}}(\sigma ){ }^{2}}\) is the normalization factor. We denote the set of complex variational parameters as θ = {W_{μν}, a_{μ}, b_{ν}}, where the interaction W_{μν} denotes the virtual coupling between the spin σ_{μ} and the auxilliary degrees of freedom, or the hidden spin h_{ν}. Onebody terms a_{μ} and b_{ν} are also introduced to enhance the expressive power of the RBM state. In the present work, we find that the it suffices to take the total number of the hidden spin as N_{h} = N_{v}, and therefore the number of the complex variational parameters is \(({N}_{v}^{2}+2{N}_{v})\) in total. The alltoall connectivity between σ and h allows the RBM state to capture complicated quantum correlations such as topological orders^{13,60}, spinliquid behaviours^{61,62,63}, and electronic structures in small molecular systems^{27,28}.
Using the RBM state (4) as the manybody variational ansatz, the groundstate problem is solved in the VMC framework. In particular, we rely on the stochastic reconfiguration technique^{64} to approximate the imaginarytime evolution as
where the parameter update at the kth step Δθ_{k} is given by the Monte Carlo simulation, and the initial state \(\left{{{\Psi }}}_{0}\right\rangle\) is taken as the HF state in our simulation. Detailed information on the implementation and optimization techniques is provided in “Methods”.
As a first demonstration, we provide the potential energy curve for a onedimensional system whose electronic correlation varies drastically as the geometry is changed. Concretely, we consider a linear hydrogen chain with homogeneous atom separation d_{H} in a minimal basis set (STO3G)^{65,66}. Figure 2a presents the result of the calculation using the RBM state as well as the secondorder Møller–Plesset perturbation theory (MP2)^{67}, the coupledcluster singles and doubles (CCSD)^{41,68}, and CCSD with perturbative triple excitations (CCSD(T))^{69}, which is considered as the goldstandard in modern quantum chemistry. While the weakly correlated regime at nearequilibrium is simulated quite well by all the conventional methods, we see that they start to collapse as the correlation grows at the intermediate d_{H} regime, not to mention the Mottinsulating large d_{H} regime. In sharp contrast, the RBM state precisely describes the electronic correlation and achieves chemical accuracy at any atom separation d_{H}. Here, two kpoints are sampled from each unit cell, which contains four hydrogen atoms so that the interactions between nearby sites are reflected explicitly on the model.
To further illustrate the RBM state’s power and reliability, we calculate the energy in the thermodynamic limit by extrapolating N_{k} → ∞ in a system with a single atom per unit cell. The numerical result at nearequilibrium (d_{H} = 2.0a_{B}) is shown in Fig. 2b. We confirm the excellent agreement with conventional methods by comparing the result with the FCI for N_{k} ≤ 8 and CCSD for 10 ≤ N_{k} ≤ 18. Clearly, the thermodynamic limit is simulated precisely as well as the finitesize system.
Next, we provide the demonstration in both 2D and 3D real solids: graphene and the lithium hydride (LiH) crystal in the rocksalt structure. Here, we restrict the active space per each kpoint to its highest occupied CO and lowest unoccupied CO. The results for graphene [Fig. 3a] and the crystalline LiH [Fig. 3b] are both in remarkable agreement with the FCI or CCSD(T). Clearly, the RBM ansatz gives a quantitatively accurate description, which may allow crystal structure determinations of weakly to moderately correlated real solid systems.
Quasiparticle band structure from the oneparticle excitation
Interest beyond the groundstate electronic structures in solids is diverse: the response against electromagnetic fields, impurity effects, phononic dispersions, and so on. Here, we focus on the band structure, which is a peculiar yet fundamental property that characterizes solid systems. We stress that variational calculations for the lowest bandgap, which can be experimentally measured from photoemissions, are already few, not to mention the simulation of the band spectra based on stochastic methods^{70}. Furthermore, to the best of our knowledge, there is no NQS simulation of excited states in the identical sector of quantum numbers except the first excited state^{19}. This motivates us to perform the first attempt to calculate multiple lowlying states and deepen our understanding on the representability of the NQS beyond the wellstudied regimes.
In general, the calculation of band structures is based on the assumption that the system is weakly to moderately correlated. In other words, the meanfield approximation is qualitatively valid, so that oneparticle excitations dominate the lowlying spectrum. By employing such a picture in a quantum manybody context, we can also simulate the band structure via quasiparticle excitations. We take a similar approach here and compute the band structure from the singleparticle linearresponse behavior of the ground state.
Let us construct an appropriately truncated Hilbert space which captures the lowlying states in a stochastic manner. It is justified from the above argument that we consider a subspace spanned by a set of nonorthonormal bases \(\{{R}_{\alpha }\left{{{\Psi }}}_{{\rm{GS}}}\right\rangle \}\), where R_{α} denotes the αth singleparticle excitation operator. Here, the valence (conduction) bands are obtained from the ionization (electron attachment) operators \(\{{c}_{p{{\bf{k}}}_{p}}\}\) (\(\{{c}_{p{{\bf{k}}}_{p}}^{\dagger }\}\)), which allows us to compute the quasiparticle band with an additional computational cost of \({\mathcal{O}}({N}_{v}^{3})\). Although it is possible to include higherorder excitation operators, here we avoid them from the viewpoint of computational cost and size inconsitency. It can be shown that the diagonalization of the effective Hamiltonian given the nonorthonormal basis is done by the following generalized eigenvalue equation^{71},
where \(E={\rm{diag}}({E}_{1},...,{E}_{{N}_{v}})\) denote the eigenvalues and C is an array of eigenvectors. The matrix elements of the nonhermitian matrix \(\widetilde{H}\) and the metric \(\widetilde{S}\) are estimated via the Monte Carlo sampling as expectation values:
where the ground state is now replaced by the RBM ansatz \(\left{{{\Psi }}}_{{\theta }^{* }}^{{\rm{RBM}}}\right\rangle\), with the optimized variational parameter θ^{*}. In the field of quantum chemistry, this procedure is referred to as the internally contracted multireference configuration interaction^{72,73}.
To enhance the numerical reliability, we incorporate the effect of orbital relaxation by estimating the bandgap from the extended Koopmans’ theorem^{74,75,76}. The energies are shifted so that the first valence and conduction bands coincide with the energy difference ΔE^{IP} and ΔE^{EA} as
where \({E}_{{\mathrm{GS}}}^{n}\) is the energy of the RBM optimized in the particlenumber sector n (See “Methods”).
We provide a demonstration for the quasiparticle band structure of the polyacetylene [Fig. 4a] using the STO3G basis sets. The result is compared with a variant of the equationofmotion coupledcluster theories (EOMCC): ionizationpotential (electronattached) EOMCC (IPEOMCC, EAEOMCC), which considers up to 2hole and 1particle (2particle and 1hole) excitations^{41}. The agreement with EOMCCSD(T)(a)*^{77} is very good for the first valence and conduction bands, while it becomes slightly worse for higher excitations. As is shown in Fig. 4b, the first conduction band is simulated almost within chemical accuracy, which is partly due to the cancellation of the optimization errors induced by Eq. (9). Meanwhile, Fig. 4c indicates that errors in the higher excitations can be an order of magnitude larger in the worst case, which cannot be explained merely from the variational simulation error. Rather, it can be understood as a systematic error originating in the insufficiency of the truncated Hilbert space; there is a tradeoff between the computational cost and the accuracy. Systematic improvement can be expected from using higherorder excitation operators, e.g., twoelectron excitation operators \(\{{c}_{p{{\bf{k}}}_{p}}^{\dagger }{c}_{q{{\bf{k}}}_{q}}\}\) for the lowest energy state in the particlenumber sectors (N_{v} ± 1).
Conclusion
We have shown that a shallow neural network with a moderate number of variational parameters allows us to perform the essence of firstprinciples calculations in solid systems, i.e., the groundstate property and the quasiparticle band spectra. In the weakly to moderately correlated regions of the linear hydrogen chain, we have demonstrated that even the thermodynamic limit can be simulated using the RBM state. The representability of the RBM is also exhibited in the strongly correlated regions, where the standard approaches break down. We have furthermore shown that the electronic structures of real solids in both 2D and 3D can be described accurately. Furthermore, we have successfully obtained the quasiparticle band spectra of a polymer in the linearresponse regime. To the best of our knowledge, this is the first demonstration proving that NQS are capable of computing multiple excited states, in addition to precise groundstate simulations that reach their chemical accuracy.
Numerous future directions can be envisioned. We remark the following three points. First is the extension towards the complete basis limit. While we have here focused on relatively simple basis sets, the quantitative prediction and comparison with experiments would necessarily require larger basis sets. Working in the continuum space is a possibility, but the calculation would be much more involved than in molecular systems. Second is the systematic improvement of the calculations for excited states. It is intriguing to investigate the quantitative performance; whether higherorder subspace expansions can be efficiently implemented, how the accuracy is compared to other excitedstate calculation framework such as the equationofmotion and timedependent linear response^{78}, and so on. Third is the behavior of physical observables. One may want to know the optical/magnetoelectric/thermal responses, so that experimental results can be directly compared. If the system is either quasistatic or static, those properties can be evaluated as derivatives of the energy with respect to an external perturbation (e.g., electric field)^{79}.
The main bottleneck that prevents the simulation by the NQS in larger systems is the sampling efficiency. As mentioned by Choo et al. for the case of RBM^{27}, and as known before in the VMC community, accurate calculations for relatively weak electronic correlations in the HF basis requires increasingly larger number of Monte Carlo samplings, because the amplitudes for multielectron excitations are small. One may consider applying efficient sampling techniques, such as parallel tempering, heatbath configuration interaction^{80}, or even employ nonHF bases.
Methods
Stochastic imaginarytime evolution by variational Monte Carlo
Given an initial state \(\left{{{\Psi }}}_{0}\right\rangle\) whose overlap with the true ground state is nonzero (and desirably not exponentially small), the ground state \(\left{{{\Psi }}}_{{\rm{GS}}}\right\rangle\) can be simulated as
where H is the Hamiltonian of the system and η is a "learning rate" that determines the step of the imaginarytime evolution. The exact simulation of Eq. (10) for generic quantum manybody systems becomes exponentially inefficient as the system size grows. Hence, we approximate the quantum state by a variational ansatz \(\left{{{\Psi }}}_{\theta }\right\rangle\) and consider the update rule of the parameters θ such that Eq. (10) is realized approximately.
There are numerous variational principles that dictate the parameter updates. Here, we choose the stochastic reconfiguration method^{64,81}, which uses the FubiniStudy metric \({\mathcal{F}}\) to measure the difference between the exact and variational imaginarytime evolution. Given a set of variational parameter θ, the update δθ is determined as
where \({\mathcal{F}}[\left\psi \right\rangle ,\left\phi \right\rangle ]=\arccos (\sqrt{\left\langle \psi  \phi \right\rangle \left\langle \phi  \psi \right\rangle /\left\langle \psi  \psi \right\rangle \left\langle \phi  \phi \right\rangle })\) and elements of the generic force f_{i} and the geometric tensor g_{ij} are given as
where ∂_{i} is the derivative with respect to the ith element of the parameter θ_{i}. It is noteworthy that the geometric tensor g is the extension of the Fisher information to quantum states. The stochastic gradient method based on g, or the Fisher information, was independently developed in the machine learning community^{81}, and is frequently referred to as the natural gradient method.
Note that both f and g can be estimated efficiently using Monte Carlo sampling. Indeed, any physical observable O can be estimated for a quantum state \(\left{{\Psi }}\right\rangle\) as
where \({O}_{{\rm{loc}}}(\sigma )={\sum }_{\sigma ^{\prime} }\frac{{{\Psi }}(\sigma ^{\prime} )}{{{\Psi }}(\sigma )}\left\langle \sigma  O \sigma ^{\prime} \right\rangle\) is introduced to enable the simulation of the expectation value from classical sampling over the probability distribution p(σ) = ∣Ψ(σ)∣^{2}/∑_{σ}∣Ψ(σ)∣^{2}. Using the Metropolis–Hastings algorithm with particlenumber conservation, we typically sample \({\mathcal{O}}(1{0}^{5})\) to \({\mathcal{O}}(1{0}^{7})\) spin configurations to estimate p(σ). Each configuration is drawn every 10–20 Monte Carlo steps so that the autocorrelation, and hence the sampling error, is sufficiently small when the optimization converges.
Three technical remarks are in order. First, we take the initial state \(\left{{{\Psi }}}_{0}\right\rangle (=\left{{{\Psi }}}_{{\theta }_{0}}^{{\rm{RBM}}}\right\rangle )\) as the HF state such that the overlap with the ground state is nonzero. Small noise is added to avoid the gradient vanishing problem, which arises when the parameters of the RBM state are tuned to express any computational basis exactly. Second, to stabilize the optimization, small number ϵ is uniformly added to the diagonal elements of g as g_{ii} → g_{ii} + ϵ. While large ϵ is beneficial in early iterations, it is necessary to decrease it, or otherwise one may result in undesirable local minima. Therefore, ϵ is initially set as \({\mathcal{O}}(1{0}^{2})\) and gradually decreased to \({\mathcal{O}}(1{0}^{3})\) after several hundred steps. Third, we find that it is crucial to adopt an appropriate scheduling of η to speed up the optimization and, more importantly, avoid local minima. In the present work, we exclusively employ the RMSProp method^{82}, which adaptively modifies η according to the magnitude of the gradient.
Energy corrections by the extended Koopmans’ theorem
In Fig. 5, we visualize the effect of the corrections to the energy bands by the extended Koopmans’ theorem, which are defined in Eq. (9) in the main text as
where \({E}_{{\mathrm{GS}}}^{n}\) is the energy of the RBM optimized in the particlenumber sector n. Here, panels (a) and (b) indicate the first conduction and valence bands, respectively. In both bands, we observe a systematic deviation, which we attribute to the lack of orbital relaxation effect caused by the removal or addition of a single electron. The order of the correction ΔE ~ 0.05 Ha is comparable to that of the electronic correlation (~0.1 Ha).
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
Codes written for and used in this study is available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Kenny Choo, Antonio Mezzacappo, and James Spencer for fruitful discussions. This work was supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant Number JPMXS0118067394 and JPMXS0120319794. N.Y. is supported by the Japan Science and Technology Agency (JST) (via the QLEAP program). W.M. wishes to thank Japan Society for the Promotion of Science (JSPS) KAKENHI No. 18K14181 and JST PRESTO No. JPMJPR191A. F.N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF1810358), Japan Science and Technology Agency (JST) (via the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134 and the JSPSRFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA23862014069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXiIAF1906. Numerical calculations were performed using OpenFermion^{83}, PySCF (v1.7.1)^{84}, and NetKet^{85}. Some calculations were performed using the supercomputer systems in RIKEN (HOKUSAI GreatWave), the Institute of Solid State Physics at the University of Tokyo, and in the Research Institute for Information Technology (RIIT) at Kyushu University, Japan.
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N.Y. and W.M. conceived the project and contributed equally to the numerical simulations. W.M. and F.N. supervised the research. All authors discussed the results and contributed to writing the paper.
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Yoshioka, N., Mizukami, W. & Nori, F. Solving quasiparticle band spectra of real solids using neuralnetwork quantum states. Commun Phys 4, 106 (2021). https://doi.org/10.1038/s42005021006090
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DOI: https://doi.org/10.1038/s42005021006090
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