Correlating AGP on a quantum computer

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.

Geminal replacement models based on AGP

The antisymmetrized geminal power (AGP) wavefunction 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.

Correlating the Antisymmetrized Geminal Power Wave Function

Strong pairing correlations are responsible for superconductivity and off-diagonal long range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.

On a dual representation of the Goldstone manifold

An intrinsic wavefunction with a broken continuous symmetry can be rotated with no energy penalty leading to an infinite set of degenerate states known as a Goldstone manifold. In this work, we show that a dual representation of such manifold exists that is sampled by an infinite set of non-degenerate states. A proof that both representations are equivalent is provided. From the work of Peierls and Yoccoz (Proc. Phys. Soc. A {bf 70}, 381 (1957)), it is known that collective states with good symmetries can be obtained from the Goldstone manifold using a generator coordinate trial wavefunction. We show that an analogous generator coordinate can be used in the dual representation; we provide numerical evidence using an intrinsic wavefunction with particle number symmetry-breaking for the electronic structure of the Be atom and one with $hat{S}^z$ symmetry-breaking for a H$_5$ ring. We discuss how the dual representation can be used to evaluate expectation values of symmetry-projected states when the norm $|langle Phi | hat{P}^q | Phi rangle|$ becomes very small.

Exact Parameterization of Fermionic Wave Functions via Unitary Coupled Cluster Theory

A formal analysis is conducted on the exactness of various forms of unitary coupled cluster (UCC) theory based on particle-hole excitation and de-excitation operators. Both the conventional single exponential UCC parameterization and a disentangled (factorized) version are considered. We formulate a differential cluster analysis to determine the UCC amplitudes corresponding to a general quantum state. The exactness of conventional UCC (ability to represent any state) is explored numerically and it is formally shown to be determined by the structure of the critical points of the UCC exponential mapping. A family of disentangled UCC wave functions are shown to exactly parameterize any state, thus showing how to construct Trotter-error-free parameterizations of UCC for applications in quantum computing. From these results, we derive an exact disentangled UCC parameterization that employs an infinite sequence of particle-hole or general one- and two-body substitution operators.

Thermofield theory for finite-temperature coupled cluster

We present a coupled cluster and linear response theory to compute properties of many-electron systems at non-zero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wavefunction representation of the thermal density matrix, and extend our recently developed framework [J. Chem. Phys. 150, 154109 (2019)] to parameterize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) as well as ab-initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.

On the difference between variational and unitary coupled cluster theories

There have been assertions in the literature that the variational and unitary forms of coupled cluster theory lead to the same energy functional. Numerical evidence from previous authors was inconsistent with this claim, yet the small energy differences found between the two methods and the relatively large number of variational parameters precluded an unequivocal conclusion. Using the Lipkin Hamiltonian, we here present conclusive numerical evidence that the two theories yield different energies. The ambiguities arising from the size of the cluster parameter space are absent in the Lipkin model, particularly when truncating to double excitations. We show that in the symmetry adapted basis under strong correlation the differences between the variational and unitary models are large, whereas they yield quite similar energies in the weakly correlated regime previously explored. We also provide a qualitative argument rationalizing why these two models cannot be the same. Additionally, we study a generalized non-unitary and non-hermitian variant that contains excitation, de-excitation and mixed operators with different amplitudes and show that it works best when compared to the traditional, variational, unitary, and extended forms of coupled cluster doubles theories.

On pair functions for strong correlations

The UHF wave function may be written as a spin-contaminated textit{pair} wave function of the APSG form, and the overlap of the alpha and beta corresponding orbitals of the UHF solution can be taken as a proxy for the strength of the correlation captured by breaking symmetry. We demonstrate this with calculations on one- and two-dimensional hydrogen clusters and make contact with the well studied Hubbard model. The UHF corresponding orbitals pair in a manner that allows a smooth evolution from doubly occupied orbitals at small distance to one in which wave function breaks symmetry, segregating the $alpha$ and $beta$ electrons onto distinct sublattices at large distances. By performing spin projection on these UHF solutions, we address strong correlations that are difficult to capture at intermediate distances using a single determinant. Approved for public release: LA-UR-13-22691.