Abstract
For many-body methods such as MCSCF and CASSCF, in which the number of one-electron orbitals is optimized and independent of the basis set used, there are no problems with using plane-wave basis sets. However, for methods currently used in quantum computing such as select configuration interaction (CI) and coupled cluster (CC) methods, it is necessary to have a virtual space that is able to capture a significant amount of electron-electron correlation in the system. The virtual orbitals in a pseudopotential plane-wave Hartree–Fock calculation, because of Coulomb repulsion, are often scattering states that interact very weakly with the filled orbitals. As a result, very little correlation energy is captured from them. The use of virtual spaces derived from the one-electron operators has also been tried, and while some correlations are captured, the amount is quite low. To overcome these limitations, we have been developing new classes of algorithms to define virtual spaces by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, which are able to capture a significant amount of correlation. The focus in this manuscript is on using these derived basis sets to target full CI (FCI) quality results for H2 on near-term quantum computers. However, the initial results for this approach were promising. We were able to obtain good agreement with FCI/cc-pVTZ results for this system with just 4 virtual orbitals, using both FCI and quantum simulations. The quality of the results using COVOs suggests that it may be possible to use them in other many-body approaches, including coupled cluster and Møller–Plesset perturbation theories, and open up the door to many-body calculations for pseudopotential plane-wave basis set methods.
Highlights
Results using the quantum phase estimation (QPE) algorithm in the Microsoft QDK package (Svore et al, 2018; Low et al, 2019) in which the number of orbitals was reduced to 4 and 6 orbitals using the double unitary CC (DUCC) method are shown. These results showed that the DUCC QDK QPE method produces total energies that are within a few milli-Hartrees of the 20 orbital full CI (FCI) result with only 4 or 6 orbitals
Because one-qubit gates tend to be executed in a short timescale and are fairly insensitive to noise, we can use the number of CNOTs present in the circuit as indicative of the complexity in its implementation, which we provide in Figure 6, showing that the ansatzes generated from ADAPT-variational quantum eigensolver (VQE) are much more affordable than those obtained by ordinary unitary coupled cluster singles and doubles (UCCSD) VQE simulations
We have developed a new approach for defining virtual spaces with a pseudopotential plane-wave code for use in many-body methods described by second-quantized Hamiltonians
Summary
Quantum chemistry is one of the first and most successful scientific applications of digital computers (Mulliken et al, 1941; Mulliken and Rieke, 1941; Boys, 1950; Parr and Mulliken, 1950; Hall, 1951; Roothaan, 1951; Boys et al, 1956; Nesbet, 1960; Allen and Karo, 1962; Nesbet, 1963; Pople et al, 1965; Kohn and Sham, 1965; Reeves, 1966; Pulay, 1969). We present a downfolding method to define virtual orbitals for expanding the second-quantized Hamiltonian given in Eq 4 The differences in our approach compared to this previous work is that the variational space used by COVOs is significantly bigger because plane-wave basis sets are used instead of LCAO Gaussian basis sets and that a second-order Hylleraas functional (Hylleraas, 1928; Hylleraas, 1929; Hylleraas, 1930; Hylleraas, 1964; Koga, 1992) was used to describe the correlation in the OVOS procedure rather than a small CI Hamiltonian.
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