Correlating AGP on a quantum computer

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.

Construction of linearly independent non-orthogonal AGP states

“…[28][29][30] In contrast, AGP and AGP-based methods are able to capture most of the correlation energies systematically. [2][3][4][5][6] We need to compute overlaps between AGPs to build the LC-AGP metric and need the following transition reduced density matrix (RDM) elements…”

“…In a series of papers, [2][3][4][5][6] our group has explored abandoning Slater determinants as the many-electron basis and proposed instead working with wave functions where the basic building blocks are two-electron functions called geminals, 7 instead of one-electron spin-orbitals. Specifically, the exact wave function can be written in the basis of identical geminal product states known as the antisymmetrized geminal power (AGP), 8 which can be obtained as the number-projected Bardeen-Cooper-Schrieffer (PBCS) wave function.…”

“…9 AGP is variationally more flexible than a Slater determinant and is the simplest wave function supporting off-diagonal long-range order in a number-conserving framework. 10 Correlated methods on an AGP reference accurately describe strong pairing interactions, [2][3][4][5][6] and many correlated AGP models based on different correlators are equivalent since the wave function is essentially expanded in AGP and states replacing one or more geminals. 4 While these constructions work well, they become increasingly challenging as we replace more geminals.…”

Exploring non-linear correlators on AGP

“…Recently, we proposed to use the AGP wavefunction as the initial reference for this and potentially more general problems [25,26,[29][30][31][32]. The main advantage of AGP for the seniority-zero sector is that on the one hand, it puts coefficients on each Slater determinant in the DOCI while still conserving seniority, and on the other hand, AGP is computationally straightforward.…”

“…The second ansatz is the unitary pair-hopper that we proposed recently for applications in quantum computers [32]. Unitary ansätze can be implemented efficiently on a quantum computer, [48] but some approximations must be made in order to implement them on a classical computer.…”