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.
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.
We present a symmetry-adapted extension of sample-based quantum diagonalization (SQD) that rigorously embeds space-group symmetry into the many-body subspace sampled by quantum hardware. The method is benchmarked on the two-leg ladder Hubbard model using both molecular orbital and momentum bases. Energy convergence is shown to be improved in the momentum basis compared to the molecular orbital basis for both the spin-quintet ground state and the spin-singlet excited state. We clarify the relationship between the compactness of the many-body wave function and the sparsity of the representation matrices of symmetry operations. Furthermore, the enhancement of the superconducting correlation function due to the Coulomb interaction is demonstrated. Our method highlights the importance of symmetry structure in random-sampling quantum simulation of correlated systems
Sample-based quantum diagonalization (SQD) constructs subspaces from computational-basis configurations obtained via measurements of a quantum state, with the goal of approximating low-energy eigenspaces of many-body Hamiltonians. The effectiveness of this approach relies on the assumption that physically relevant states admit a compact representation in the computational basis. We investigate this assumption by analyzing SQD subspaces constructed directly from configurations of exact ground states of Heisenberg and Hubbard model lattices. By eliminating state-preparation and measurement inefficiencies, we isolate the intrinsic configuration-space structure of the wavefunction. We determine the minimal number of configurations required to reproduce the ground-state energy within fixed accuracy thresholds and find that this number grows exponentially with the system size. Notably, this scaling persists even under optimal inclusion of configurations in order of decreasing probability, demonstrating that it originates from intrinsic delocalization of the wavefunction rather than sampling inefficiencies. Our results indicate that SQD effectively probes the configuration-space entropy but faces fundamental scalability limitations for these models.
Fermionic atoms in optical lattices provide a native implementation of Fermi-Hubbard (FH) models that can be used as analog quantum simulators of many-body fermionic systems. Recent experimental advances include the time-dependent local control of chemical potentials and tunnelings, and thus enable us to operate this platform digitally as a programmable quantum simulator. Here, we explore these opportunities and develop ground-state preparation algorithms for different fermionic models, based on the ability to implement both single-particle and many-body high-fidelity fermionic gates, as provided by the native FH Hamiltonian. In particular, we first design variational precompiled quantum circuits to prepare the ground state of the natively implemented FH model, with significant speedups relative to competing adiabatic protocols. Besides, the versatility of this variational approach enables us to target extended FH models, i.e., including terms that are not natively realized on the platform. As an illustration, we include next-nearest-neighbor tunnelings at finite dopings, relevant in the context of d -wave superconductivity. Furthermore, we discuss how to approximate the imaginary-time evolution using variational fermionic circuits, both as an alternative state-preparation strategy and as a subroutine for the quantum Lanczos algorithm to further improve the energy estimation. We benchmark our protocols for ladder geometries, though they can be readily applied to two-dimensional experimental setups to address regimes beyond the capabilities of current classical methods. These results pave the way for more efficient and comprehensive explorations of relevant many-body phases with existing programmable fermionic quantum simulators.
Sample-based quantum diagonalization can reduce the classical cost of many-body calculations by restricting diagonalization to configurations identified through quantum sampling, but its efficiency depends on how compactly the target state is represented in the sampled basis. We investigate this dependence for sample-based Krylov quantum diagonalization of the half-filled Fermi-Hubbard ladder using physically motivated one-body orbital rotations. Across position, momentum, bonding/antibonding, and leg-mode representations, the most compressible basis varies systematically with interaction strength and hopping anisotropy, with momentum sampling favored in the weakly interacting regime and geometry- or interaction-adapted representations becoming advantageous as the dominant energy scale changes. Near intermediate interaction and isotropic hopping, however, even the most favorable representation remains poorly compressible. Statevector SKQD calculations further show that basis choice strongly affects sampling efficiency. Hardware calculations on systems containing up to 120 qubits retain qualitative signatures of this basis dependence while revealing increasing particle-number leakage and reconstruction sensitivity with system size. These results show that scalable sample-based diagonalization requires co-design of the sampled representation, quantum circuit, and classical reconstruction procedure.
Quantum computation offers a promising path towards efficient calculations of the electronic structure of correlated materials. The properties of iron-based materials, which are key for advancing geological and astrophysical understanding, are notoriously difficult to calculate with classical computers. However, despite the many recent advances in quantum computing on near intermediate scale quantum hardware, a full treatment of the electronic structure of high-Z atoms or molecules remains challenging. Here, we investigate the ground-state properties of molecular FeO and Fe2O2 model Hamiltonians using classical exact diagonalization, Variational Quantum Eigensolver (VQE), and Sample-based Quantum Diagonalization (SQD). The experimental quintet ground state of FeO is recovered classically and with quantum algorithms, while a nonet ground state is recovered for Fe2O2. Although raw VQE does not achieve chemical accuracy for the ground state of FeO or Fe2O2, SQD with two connectivity shells recovers the exact ground energy for both molecules utilizing remarkably few quantum computational resources. Overall, our results indicate that abinitio-informed low-energy models of correlated materials, combined with SQD, offer a new alternative for scaling up calculations of ground-state properties of strongly correlated solids on existing quantum hardware.
The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-21 Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-21 wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin-21 state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.