Conventional methods to solve quantum many-body problems in physics and chemistry often struggle in the strongly correlated regime. Recent advances in quantum computing hardware have opened up new ways to tackle the strong correlation problem; however, existing hybrid quantum-classical algorithms typically start from conventional classical methods or take inspiration from them. In this thesis, we develop novel correlation methods on classical and quantum computers that are based the anti-symmetrized geminal power (AGP) wavefunction---a state equivalent to the number projected Bardeen--Cooper--Schrieffer (BCS) wavefunction. Our methods fall under the larger category of merging coupled cluster theory with symmetry--breaking and restoration. We showcase benchmark calculations for model Hamiltonians that exhibit strong correlation that are prototypical of those in molecules and condensed matter systems and demonstrate that our methods have the potential to address strong correlation in attractive and repulsive systems on an equal footing.
Conventional methods to solve quantum many-body problems in physics and chemistry often struggle in the strongly correlated regime. Recent advances in quantum computing hardware have opened up new ways to tackle the strong correlation problem; however, existing hybrid quantum-classical algorithms typically start from conventional classical methods or take inspiration from them. In this thesis, we develop novel correlation methods on classical and quantum computers that are based the anti-symmetrized geminal power (AGP) wavefunction---a state equivalent to the number projected Bardeen--Cooper--Schrieffer (BCS) wavefunction. Our methods fall under the larger category of merging coupled cluster theory with symmetry--breaking and restoration. We showcase benchmark calculations for model Hamiltonians that exhibit strong correlation that are prototypical of those in molecules and condensed matter systems and demonstrate that our methods have the potential to address strong correlation in attractive and repulsive systems on an equal footing.
Abstract Electronic structure methods typically benefit from symmetry breaking and restoration, specially in the strong correlation regime. The same goes for ansätze on a quantum computer. We develop a unitary coupled cluster method based on the antisymmetrized geminal power (AGP)—a state formally equivalent to the number-projected Bardeen–Cooper–Schrieffer wavefunction. We demonstrate our method for the single-band Fermi–Hubbard Hamiltonian in one and two dimensions. We also explore post-selection as a state preparation step to obtain correlated AGP and prove that it scales no worse than O ( M ) in the number of measurements, thereby making it a less expensive alternative to gauge integration to restore particle number symmetry.
The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree-Fock-Bogoliubov (HFB), and number-projected Bardeen-Cooper-Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin su(2) algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and ${\mathrm{J}}_{1}\text{\ensuremath{-}}{\mathrm{J}}_{2}$ Heisenberg models, both in one and two dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.
Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential Ansätze in this paper. We investigate two approaches in particular. Similar to Wahlen-Strothman et al. [Phys. Rev. B 91, 041114(R) (2015)], we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting the Schrödinger equation. The second approach is based on approximating the unitary pair-hopper Ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.
Abstract For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose and implement a unitary correlator on AGP and benchmark it on the ground state of the pairing Hamiltonian. The results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.
Achieving materials-specific predictions for large, realistic molecules and solids with strong electron correlations remains a long-standing challenge in quantum chemistry. This dissertation develops scalable ab initio quantum many-body methods for crystalline materials, demonstrating that rigorous optimization of theoretical approximations and numerical algorithms enables predictive computation even in strongly correlated systems. The work tackles three interconnected challenges, namely describing material-specific properties of strongly correlated superconductors, achieving linear scaling with k-points for large-scale periodic calculations, and establishing reliable methods for electron-phonon interactions. The following chapters address each challenge in turn. Chapter 2 demonstrates that ab initio density matrix embedding theory, combined with symmetry-breaking strategies for superconducting order parameters, reproduces experimental trends in cuprate superconductivity, including pressure and layer effects on transition temperatures. By directly solving the electronic Schrödinger equation from material structures, these calculations capture multi-orbital covalency effects that simplified model Hamiltonians neglect, revealing their essential role in the spin fluctuations driving pairing. This predictive capability establishes a foundation for efficient computational screening of superconducting materials. Extending these calculations to larger k-point meshes and broader phase diagram explorations, however, demands more efficient computational algorithms. To overcome this computational bottleneck, Chapter 3 introduces interpolative separable density fitting framework. This development enables thermodynamic limit calculations with up to 1000 k-points, as validated on diamond, carbon dioxide, nickel monoxide, and cuprate systems. Integration with density matrix embedding theory and local natural orbital methods further yields converged ground-state energies from accurate correlated wavefunction methods such as CCSD(T) for crystalline materials in the thermodynamic limit. With computational efficiency established, the remaining challenge lies in extending the theoretical framework beyond purely electronic degrees of freedom. Chapter 4 addresses this need by establishing exponential ansatz wavefunctions as competitive approaches for electron-phonon systems. Unlike previous treatments based on perturbation theory, systematic benchmarking of coupled cluster theory and variational Lang-Firsov methods on the Holstein model demonstrates, for the first time, that these approaches accurately describe polaron formation across coupling regimes when combined with appropriate reference state optimization. These results provide a foundation for incorporating phonon effects into ab initio studies of superconducting materials. In addition to these methodological advances, GPU acceleration and MPI parallelization enhance computational throughput, enabling efficient large-scale calculations. Together, these developments constitute a computational framework for predictive ab initio quantum many-body calculations in crystalline materials, opening new avenues for understanding and designing strongly correlated quantum materials.
The antisymmetrized geminal power (AGP) wave function has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP-based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J -CI provides excellent accuracy in energies when applied to the pairing Hamiltonian.