## Abstract

To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) basis set with a rectangular partitioning strategy was recently introduced [McClean et al, New J. Phys. 22, 093015, 2020]. We propose and numerically scrutinize a more general DG basis set construction based on a Voronoi decomposition with respect to the nuclear coordinates. This allows the construction of DG basis sets for arbitrary molecular and crystalline configurations. We here employ the planewave dual basis set as primitive basis set in the supercell model; as a set of grid-based nascent delta functions, the planewave dual functions provide sufficient flexibility for the Voronoi partitioning. The presented implementation of this *DG-Voronoi* approach is in Python and solely based on PySCF. We numerically investigate the performance, at the mean-field and correlated level of theory for quasi-1D, quasi-2D and fully 3D systems, and exemplify the application to crystalline systems.

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## Acknowledgements

This work was partially supported by the Department of Energy under Grant No. DE-SC0017867, the National Science Foundation under grant number OMA-2016245 (L.L.), and by the Air Force Office of Scientific Research under award number FA9550-18-1-0095 (F.F.,X.W.,L.L.). L.L. is a Simons Investigator.

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## Appendices

### Appendix A: H\(_4\) Model

### 1.1 Condition number, grid spacing and egg-box effect

We provide additional computation of the “egg-box” effect for the DG-V procedure using the cc-pVDZ basis set. We confirm that the egg-box effect caused by the DG-V procedure is on the order of \(3\times 10^{-5}\) Hartree and is negligible. The size of the error is also comparable to that in the pbc (periodic boundary condition) module of PySCF Fig. 11.

We have furthermore computed the condition number of the overlap matrix as a function of the system parameter \(\alpha \) for the (aug-)cc-pVDZ basis set in Fig. 12 as well as for different basis sets in Fig. 13. We see that the adding diffuse basis functions increase in general the condition number, and that the condition number in all cases is significantly higher than one, which is the condition number of the DG-V overlap matrix.

### 1.2 Mean-field computations

Figures 14, 15 and 16 show the truncation sensitivity of the Hartree–Fock energy in the DG-V basis discretization for a number of Pople basis set with added diffuse basis functions. Picking the truncation threshold for each basis set based on the calculations presented in Figs. 14, 15 and 16, we can extend the results presented in Section 3.1 on the correlated calculations for the respective basis sets. The results are presented in Figs. 17, 18, 19.

We here also present numerical values showing the maximal and minimal improvement by means of the DG-V procedure, see Tables 4, 5. We highlight that the main limitation of this improvement is restricted by the choice of the active basis.

### Appendix B: Hydrogen chains

Given the more general implementation using the Voronoi decomposition, we numerically confirm that the performance for quasi one-dimensional systems presented in Ref. [5]. To that end, we investigate the same system of hydrogen chains of various lengths, aiming at empirical data for cost factors in quantum simulation of chemistry. This quasi one-dimensional system is well suited for examining different scaling behavior of the DG-V formalism as a function of the system size. As we systematically increase the number of atoms, we choose a consistent truncation procedure of the DG-V basis functions by means of the absolute size of the singular values. In the particular case of hydrogen chains, we observe that around 90% of singular values are enclosed in the interval [0, 1], and the largest singular value is approximately 10. Hence, in this special case, the truncation of the singular values with respect to the absolute size can be straightforwardly related to a truncation criterion with respect to relative size.

Aligning with the previous simulations in Ref. [5], we use the cc-pVDZ basis. Furthermore, as we aim at a cost-factor comparison for quantum simulation of chemistry, we need to construct a second quantized Hamiltonian with an orthonormal basis set. To that end we compare the performance of the DG-V basis with the Löwdin orthonormalized [23] cc-pVDZ atomic orbital basis.

We also present numerical evidence that the number of DG-V basis functions per Voronoi cell as function of the system size converges to a constant that depends on the truncation tolerance. The hydrogen chains H\(_n\) with \(n-1\) bond length of 1.7 (a.u.) are centered in a supercell of size \((n\cdot 3.6+ 10) \times 10 \times 10\) (a.u.). The underlying grid spacing was chosen to be 0.2 (a.u.). In the case of H\(_{30}\) this yields a super cell of size \(118 \times 10 \times 10\) with \(590 \times 50 \times 50\) grid points which corresponds to 1475000 planewave dual functions, i.e., approximately 49000 basis functions per atom. The large number of planewave dual functions is needed since cc-pVDZ is a basis set designed for all-electron calculations. Using this configuration, we find that, even for an aggressive truncation threshold, the result from the DG-V basis can reliably match the accuracy of the results of the active-space basis, while each DG-V basis function is far more compact; see Fig. 20 for a comparison of potential energy surfaces (PES) in DG-V discretizations of different tolerances with the corresponding PES in the active-space basis set. In fact, the energies obtained from the DG-V basis are slightly lower than those from the Gaussian basis set. This shows that the primitive basis set is more expressive by design than the active-space Gaussian cc-pVDZ basis against which the DG-V fit is performed. This allows even fairly loose DG-V fits to match the accuracy of the active-space basis.

The results suggest that the overall accuracy of the energy is relatively insensitive even to rather aggressive singular value truncations in the DG-V decomposition procedure, at least for quasi one-dimensional systems.

We also confirm that the average number of DG-V basis functions per atom for fixed truncation thresholds in the DG-V procedure converges rather rapidly to a constant, see Fig. 21.

With the parameter settings above, we investigate the scaling of the number of nonzero two-electron integrals in Fig. 22. Analogous to the procedure in Ref. [5], we neglect two-electron repulsion integrals with an absolute value smaller than \(10^{-6}\). In particular, when the SVD-truncation tolerance is set to \(10^{-1}\), the DG-V basis set becomes advantageous when the system size is larger than 14 atoms. Comparing with results from Ref. [5], we find that the scaling of the number of nonzero two-electron integrals with respect to the number of atoms is lower in the present work using the Gaussian basis set, and the value of \(\alpha \) is reduced from 3.9 to 3.0. A more careful investigation reveals that the difference is affected by orthogonalization strategies of the GTOs. In this work, we apply the Löwdin orthonormalized Gaussian orbitals, while Ref. [5] uses the SVD-orthonormalized Gaussian orbitals (the number of nonzero two-electron integrals reported in Ref. [5] is reproduced in Fig. 23 using the PySCF based code in this work).

In a previous work [5], the number of nonzero two-electron repulsion integrals for hydrogen chains of increasing length was compared for discretizations through Gaussian type orbitals, DG-R and planewave dual basis, aiming at an empirical cost-factor scaling analysis for quantum simulations of chemistry. To that end, the Gaussian type orbitals were orthonormalized by means of a singular values decomposition, introducing a somewhat arbitrary gauge. We here extend this study by employing the Löwdin orthonormalization to the Gaussian type orbitals, and reporting the corresponding scaling for an active-space basis set of molecular orbitals. The considered active space of molecular orbitals (MO) consists of the \(N_\textrm{occ}\) occupied orbitals and the lowest \(3N_\textrm{occ}\) virtual orbitals. We confirm the scaling behavior of the number of nonzero two-electron repulsion integrals using an SVD orthonormalization reported in Ref. [5] for the considered system, see Fig. 23b. We furthermore observe that the scaling of the number of nonzero two-electron repulsion integrals for the molecular orbitals (MO) is close to \({\mathcal {O}}(N^4)\). Hence, although an active space introduces a beneficial offset, extrapolation of the curves in Fig. 23 shows that the use of molecular orbitals becomes more expensive than DG-V beyond 74 atoms. Comparing with the results obtained using the Löwdin orthonormalization Gaussians, see Fig. 23a, we observe that the scaling of the number of nonzero two-electron repulsion integrals in a Gaussian atomic orbital discretization is significantly reduced. Nonetheless, we observe a crossing with a DG-V curve between 12 and 14 atoms.

For certain types of quantum algorithms such as those based on the linear combination of unitaries approach [24, 25], the major part of the cost does not come directly from the bare number of nonzero two-electron repulsion integrals but from the \(\lambda \)-value, i.e., the \(\ell _1\)-norm of the two-body interaction tensor \(\lambda = \sum _{p,q,r,s}\vert v_{p,q,r,s}\vert \). We find that the Löwdin orthonormalization of the Gaussian type atomic orbital basis has a more significant effect to the scaling of the \(\lambda \)-value than on the number of nonzero two-electron repulsion integrals, see Fig. 24a. In particular, the \(\lambda \) values of DG-V and active basis are almost parallel. Note, however, that around 30 atoms, the \(\lambda \)-value for the molecular orbital basis (using an active space) crosses the Löwdin orthonormalized Gaussian basis set, i.e., in terms of the \(\lambda \)-value influenced cost factors in quantum simulations, the use of Löwdin orthonormalized Gaussians becomes more efficient compared to the use of molecular orbitals despite the use of a basis reduction through an active-space consideration.

Although the improvement in the \(\lambda \)-value scaling through the DG formalism is not as pronounced as it is for the number of nonzero two-electron repulsion integrals, the DG procedure gives rise to a two-body interaction tensor with a block-diagonal sparsity structure. We expect that this can still significantly reduce the cost of implementing quantum algorithms, such as those based on fermionic swap networks [4, 5].

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Faulstich, F.M., Wu, X. & Lin, L. Discontinuous Galerkin method with Voronoi partitioning for quantum simulation of chemistry.
*Res Math Sci* **9**, 68 (2022). https://doi.org/10.1007/s40687-022-00365-9

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DOI: https://doi.org/10.1007/s40687-022-00365-9

### Keywords

- Quantum chemistry
- Electronic structure
- Discontinuous Galerkin methods
- Voronoi decomposition
- Quantum computing