Towards Studying Superconductivity in the Fermi-Hubbard Model on Rydberg Atoms

Abstract

We present a method for calculating the ground state energy of the Fermi-Hubbard model leveraging Rydberg atom processors and sample-based quantum diagonalization (SQD). By exploiting the perturbative relationship between the Fermi-Hubbard and Heisenberg models, the procedure samples from the Heisenberg model as prepared on the Rydberg atom processor, and uses the samples to diagonalize the Fermi-Hubbard model for large U. We include anisotropy and next-nearest-neighbor interactions and discuss the relevant regime for quasi-superconductivity in the 1-dimensional Fermi- Hubbard model. Numerical and experimental results on the Aquila quantum processor are presented for ground state energy calculations as well as the chemical potential. We find that the Heisenberg model sampling in the studied regime is sufficient to converge near to the ground state for up to 56 qubits, and we see a clear advantage of Rydberg atom sampling as opposed to random sampling even with 10x more samples for diagonalization. We also present a gate-based implementation of the gate-based SQD algorithm on IBM Quantum hardware for 56-qubit Hubbard model as a benchmark. Finally, we provide a gap analysis for studying emergent superconductivity using this method.

Figures (9)

• Figure 1: Schematic representation of the mapping between the two Heisenberg chains and the Hubbard model to the second order in large-$U$ ($U_\infty$) perturbation theory. Each vertical pair of spins encodes a single Hubbard site (upper: spin-up orbital, lower: spin-down orbital). The horizontal couplings $J$ implement the effective Heisenberg interactions that arise at second order in $t/U$, while a strong vertical coupling enforces the no–double-occupancy constraint.
• Figure 2: Schematic diagram of dominant correlations of the undoped (upper panel) and doped (lower panel) NNN neighbor 1-dimensional Fermi Hubbard model at zero temperature.
• Figure 3: Energy error between ground state energy and VQITE calculated energy and fidelity between the ground state and the quantum state at given VQITE step for the $N=4$-qubit XXZ model as a function of off-diagonal coupling $J_{xy}/J_z=0.0, 0.1, 0.2, 0.3, 0.4, 0.5$, respectively.
• Figure 4: Convergence of the model hyperparameters for VQITE on the anisotropic NNN Heisenberg model as a function of the problem size.
• Figure 5: SQD results for the ground state energy of a 20-orbital spin-asymmetric Hubbard model at half-filling with $U=10$, $t_\uparrow=1$, $t_\downarrow=0.25$, $t_\uparrow'=0.25$, and $t_\downarrow'=0.0625$. The SQD subspaces are sampled from VQITE-prepared anisotropic NNN XXZ model ground state approximation on a simulated Aquila Rydberg processor. The upper (lower) panel demonstrates convergence to the exact ground state energy (GSE) as a function of SQD iteration (subspace dimension).