Development and application of the transcorrelated method
! UNDER CONSTRUCTION !
Accurate methods are mandatory to correctly describe strongly correlated systems, but have an unfavorable computational scaling with the size of the system. An exact calculation called full configuration interaction (FCI), even scales exponentially with the system size. In addition to this hierarchy of methods, there is a hierarchy of basis set expansions. In electronic structure theory, the Schrödinger equation is usually expanded in one-electron basis functions (plane waves for extended systems/basis sets for molecular problems), which makes it tractable to solve, but fails to capture the electron-cusp condition[1], a form of electron correlation. For a quantitatively accurate description of the physics/chemistry of a system it is crucial to capture this form of correlation. This necessitates using many basis functions, which results in more computational effort for highly accurate methods.
Explicitly correlated methods[2] can reduce the need for large basis set expansions by directly incorporating the electronic cusp condition in the wavefunction Ansatz. In the transcorrelated (TC) approach, a correlated Ansatz -- exactly incorporating the cusp condition -- is applied and used to perform a similarity transformation of the electronic Hamiltonian describing the ab initio chemical system. The benefit of the TC method is that it yields highly accurate results already with a small number of basis functions. The challenge, however, is that the TC approach renders the Hamiltonian non-Hermitian and introduces 3-body terms. Thus, most computational methods need to be modified to deal with these intricacies, which I was able to do during my Ph.D. for the full configuration interaction QMC (FCIQMC) method and on quantum computing hardware during my postdoc.
We investigate the optimization of flexible tailored real-space Jastrow factors for use in the transcorrelated (TC) method in combination with highly accurate quantum chemistry methods, such as initiator full configuration interaction quantum Monte Carlo (FCIQMC). Jastrow factors obtained by minimizing the variance of the TC reference energy are found to yield better, more consistent results than those obtained by minimizing the variational energy. We compute all-electron atomization energies for the challenging first-row molecules C2, CN, N2, and O2 and find that the TC method yields chemically accurate results using only the cc-pVTZ basis set, roughly matching the accuracy of non-TC calculations with the much larger cc-pV5Z basis set. We also investigate an approximation in which pure three-body excitations are neglected from the TC-FCIQMC dynamics, saving storage and computational costs, and show that it affects relative energies negligibly. Our results demonstrate that the combination of tailored real-space Jastrow factors with the multi-configurational TC-FCIQMC method provides a route to obtaining chemical accuracy using modest basis sets, obviating the need for basis-set extrapolation and composite techniques.
@article{Haupt2023,doi={10.1063/5.0147877},url={https://doi.org/10.1063/5.0147877},year={2023},month=jun,publisher={{AIP} Publishing},volume={158},number={22},author={Haupt, J. Philip and Hosseini, Seyed Mohammadreza and R{\'{\i}}os, Pablo L{\'{o}}pez and Dobrautz, Werner and Cohen, Aron and Alavi, Ali},title={Optimizing Jastrow factors for the transcorrelated method},journal={The Journal of Chemical Physics},dimensions={true},project_tc={true}}
Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more “compact” ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength (U/t=4), enabling the use of shallower quantum circuits for wave-function Ansätzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.
@article{PhysRevResearch.5.023174,title={Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method},author={Sokolov, Igor O. and Dobrautz, Werner and Luo, Hongjun and Alavi, Ali and Tavernelli, Ivano},journal={Phys. Rev. Res.},volume={5},issue={2},pages={023174},numpages={19},year={2023},month=jun,publisher={American Physical Society},doi={10.1103/PhysRevResearch.5.023174},url={https://link.aps.org/doi/10.1103/PhysRevResearch.5.023174},dimensions={true},project_quantum={true},project_tc={true},project_hubbard={true}}
Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields, including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and limited connectivity) hamper the straightforward implementation of most quantum algorithms and call for more noise-resilient solutions. In quantum chemistry, the limited number of available qubits and gate operations is particularly restrictive since, for each molecular orbital, one needs, in general, two qubits. In this study, we propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach, which transfers – without any approximation – correlation from the wavefunction directly into the Hamiltonian, thus reducing the number of resources needed to achieve accurate results with noisy, near-term quantum devices. In particular, we show that the exact transcorrelated approach not only allows for more shallow circuits but also improves the convergence towards the so-called basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, therefore, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using just 4 and 6 qubits, respectively. Conventional methods require at least ten times more qubits for the same accuracy.
@article{2303.02007,author={Dobrautz, Werner and Sokolov, Igor O. and Liao, Ke and Ríos, Pablo López and Rahm, Martin and Alavi, Ali and Tavernelli, Ivano},title={Ab Initio Transcorrelated Method enabling accurate Quantum Chemistry on near-term Quantum Hardware},year={2023},eprint={arXiv:2303.02007},journal={arXiv},project_tc={true},project_quantum={true}}
In this work, we investigate the performance of a recently proposed transcorrelated (TC) approach based on a single-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)] for systems involving more than two electrons. The benefit of such an approach relies on its simplicity as efficient numerical–analytical schemes can be set up to compute the two- and three-body integrals occurring in the effective TC Hamiltonian. To obtain accurate ground state energies within a given basis set, the present TC scheme is coupled to the recently proposed TC–full configuration interaction quantum Monte Carlo method [Cohen et al., J. Chem. Phys. 151, 061101 (2019)]. We report ground state total energies on the Li–Ne series, together with their first cations, computed with increasingly large basis sets and compare to more elaborate correlation factors involving electron–electron–nucleus coordinates. Numerical results on the Li–Ne ionization potentials show that the use of the single-parameter correlation factor brings on average only a slightly lower accuracy (1.2 mH) in a triple-zeta quality basis set with respect to a more sophisticated correlation factor. However, already using a quadruple-zeta quality basis set yields results within chemical accuracy to complete basis set limit results when using this novel single-parameter correlation factor. Calculations on the H2O, CH2, and FH molecules show that a similar precision can be obtained within a triple-zeta quality basis set for the atomization energies of molecular systems.
@article{Dobrautz2022,doi={10.1063/5.0088981},url={https://doi.org/10.1063/5.0088981},year={2022},month=jun,publisher={{AIP} Publishing},volume={156},number={23},pages={234108},author={Dobrautz, Werner and Cohen, Aron J. and Alavi, Ali and Giner, Emmanuel},title={Performance of a one-parameter correlation factor for transcorrelation: Study on a series of second row atomic and molecular systems},journal={The Journal of Chemical Physics},dimensions={true},project_tc={true}}
2019
PhD Thesis
Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems
Full Configuration Interaction Quantum Monte Carlo (FCIQMC) is a prominent method to calculate the exact solution of the Schrödinger equation in a finite antisymmetric basis and gives access to physical observables through an efficient stochastic sampling of the wavefunction that describes a quantum mechanical system. Although system-agnostic (black-box-like) and numerically exact, its effectiveness depends crucially on the compactness of the wavefunction: a property that gradually decreases as correlation effects become stronger. In this work, we present two -conceptually distinct- approaches to extend the applicability of FCIQMC towards larger and more strongly correlated systems. In the first part, we investigate a spin-adapted formulation of the FCIQMC algorithm, based on the Unitary Group Approach. Exploiting the inherent symmetries of the nonrelativistic molecular Hamiltonian results in a dramatic reduction of the effective Hilbert space size of the problem. The use of a spin-pure basis explicitly resolves the different spin-sectors, even when degenerate, and the absence of spin-contamination ensures the sampled wavefunction is an eigenfunction of the total spin operator. Moreover, targeting specific many-body states with conserved total spin allows an accurate description of chemical processes governed by the intricate interplay of them. We apply the above methodology to obtain results, not otherwise attainable with conventional approaches, for the spin-gap of the high-spin cobalt atom ground- and low-spin excited state and the electron affinity of scandium within chemical accuracy to experiment. Furthermore we establish the ordering of the scandium anion bound states, which has until now not been experimentally determined. In the second part, we investigate a methodology to explicitly incorporate electron correlation into the initial Ansatz of the ground state wavefunction. Such an Ansatz induces a compact description of the wavefunction, which ameliorates the sampling of the configuration space of a system with FCIQMC. Within this approach, we investigate the two-dimensional Hubbard model near half-filling in the intermediate interaction regime, where such an Ansatz can be exactly incorporated by a nonunitary similarity transformation of the Hamiltonian based on a Gutzwiller correlator. This transformation generates novel three-body interactions, tractable due to the stochastic nature of FCIQMC, and leads to a non-Hermitian effective Hamiltonian with extremely compact right eigenvectors. The latter fact allows application of FCIQMC to larger lattice sizes, well beyond the reach of the method applied to the original Hubbard Hamiltonian.
@phdthesis{dobrautz-phd,title={Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems},author={Dobrautz, Werner},school={University of Stuttgart},year={2019},month=mar,url={http://dx.doi.org/10.18419/opus-10593},doi={10.18419/opus-10593},project_devel={true},project_fciqmc={true},project_guga={true},project_tc={true},project_thesis={true},owner={dobrautz},timestamp={2019.04.16}}
Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half filling in the intermediate interaction strength regime (U/t=4). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity-transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the full configuration interaction quantum Monte Carlo (FCIQMC) method and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimization of the correlator. The FCIQMC method, as a projective technique, is well suited for such non-Hermitian problems, and its stochastic nature can handle the three-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes up to 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off-half-filling regime, with no severe sign problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.
@article{PhysRevB.99.075119,title={Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations},author={Dobrautz, Werner and Luo, Hongjun and Alavi, Ali},journal={Phys. Rev. B},volume={99},issue={7},pages={075119},numpages={20},year={2019},month=feb,publisher={American Physical Society},doi={10.1103/PhysRevB.99.075119},url={https://link.aps.org/doi/10.1103/PhysRevB.99.075119},dimensions={true},project_tc={true},project_hubbard={true}}
By expressing the electronic wavefunction in an explicitly correlated (Jastrow-factorized) form, a similarity-transformed effective Hamiltonian can be derived. The effective Hamiltonian is non-Hermitian and contains three-body interactions. The resulting ground-state eigenvalue problem can be solved projectively using a stochastic configuration-interaction formalism. Our approach permits the use of highly flexible Jastrow functions, which we show to be effective in achieving extremely high accuracy, even with small basis sets. Results are presented for the total energies and ionization potentials of the first-row atoms, achieving accuracy within a mH of the basis-set limit, using modest basis sets and computational effort.
@article{Cohen2019,doi={10.1063/1.5116024},url={https://doi.org/10.1063/1.5116024},year={2019},month=aug,publisher={{AIP} Publishing},volume={151},number={6},author={Cohen, Aron J. and Luo, Hongjun and Guther, Kai and Dobrautz, Werner and Tew, David P. and Alavi, Ali},title={Similarity transformation of the electronic Schr\"{o}dinger equation via Jastrow factorization},journal={The Journal of Chemical Physics},dimensions={true},project_tc={true}}
