Kübra Yeter-Aydeniz and Nora Bauer of The MITRE Corporation have demonstrated a new method for calculating ground state energies of the Fermi-Hubbard model, a system central to understanding high-temperature superconductivity. Their work uses Rydberg atom processors and a technique called sample-based quantum diagonalization, achieving convergence near the ground state for up to 56 qubits. The researchers observed a clear advantage with Rydberg atom sampling, outperforming random sampling even when using ten times more data for diagonalization. This approach also includes a gap analysis for studying emergent superconductivity within the 1-dimensional Fermi-Hubbard model.

Rydberg Atom Processors for Fermi-Hubbard Model Diagonalization

The Aquila Rydberg atom processor successfully converged on approximate ground states for the Fermi-Hubbard model with up to 56 qubits using a novel sample-based quantum diagonalization (SQD) technique, a scale previously challenging for this type of calculation. The team’s method prepares the Heisenberg model on the quantum processor and uses samples from it to estimate the ground state of the Fermi-Hubbard model, an important step in understanding strongly correlated electron systems. This efficiency stems from the targeted sampling of the Heisenberg model, which provides a more informed basis for approximating the Fermi-Hubbard ground state, particularly in regimes where the large-U limit applies.

Beyond the performance gains, the work also presents a gate-based implementation of the SQD algorithm on IBM Quantum hardware, serving as a valuable benchmark against the Rydberg atom processor results. This comparison allows for a broader assessment of the SQD method’s applicability across different quantum computing platforms and highlights the potential of Rydberg atom systems for tackling specific computational challenges. This analysis provides a pathway for using the SQD method to investigate the conditions under which superconductivity might arise in this system.

Fermi-Hubbard Model & Perturbative Link to Heisenberg Interactions

Exploiting the connection between the Fermi-Hubbard and Heisenberg models, researchers have demonstrated a method for calculating the ground state energy of the Fermi-Hubbard model using sample-based quantum diagonalization (SQD) on Rydberg atom processors, a technique previously untested on cloud-accessed analog quantum hardware. Calculations included anisotropy and next-nearest-neighbor interactions, focusing on regimes relevant to understanding quasi-superconductivity in one-dimensional systems.

This VQITE implementation represents, to the researchers’ knowledge, the first demonstration of its kind on a cloud-accessible analog quantum system. The methodology involves encoding a Hubbard site with a vertical pair of spins, where the upper spin represents a spin-up orbital and the lower spin represents a spin-down orbital.

Horizontal couplings, denoted as J, implement the effective Heisenberg interactions arising from second-order perturbation theory in t/U, while strong vertical coupling enforces the no-double-occupancy constraint. Extending this to include next-nearest-neighbor interactions, the researchers analyzed the spin-asymmetric NNN Fermi-Hubbard system, deriving the $J_{1} - J_{2}$ Heisenberg model. The resulting Hamiltonian incorporates exchange couplings that depend on the direction of the spins, allowing for the investigation of anisotropic systems.

The researchers found that Heisenberg model sampling in the studied regime is sufficient to converge near the ground state. The team’s analysis of the NNN anisotropic Heisenberg model revealed the effective exchange couplings, $J_{x y, 1}$ , $J_{x y, 2}$ , $J_{z, 1}$ , and $J_{z, 2}$ , which govern the interactions between spins. These couplings are derived through second-order perturbation theory and depend on the hopping amplitudes and on-site repulsion strength.

By carefully controlling these parameters, the researchers were able to explore the parameter regimes relevant to superconducting correlations in one dimension. The researchers also note that the isotropic Heisenberg model, where $J_{x y}$ equals $J_{z}$ , is not natively supported by the Aquila quantum processor.

They include anisotropy and next-nearest-neighbor interactions and discuss the relevant regime for quasi-superconductivity in the 1-dimensional Fermi-Hubbard model. The successful implementation of VQITE on the Aquila processor and the subsequent use of SQD for ground state calculations represent a step towards realizing the full potential of Rydberg atom processors for studying complex materials.

Sample-Based Quantum Diagonalization up to 56 Qubits

This scale surpasses previous demonstrations of similar techniques, offering a pathway to investigate strongly correlated electron systems that defy classical simulation. This implementation allows for direct comparison of the performance of Rydberg atom-based and gate-based approaches for solving the same problem. Quantum imaginary time evolution (QITE) algorithms, while promising, face challenges in implementation due to the need for non-unitary operations. Motta et al. proposed a QITE algorithm that approximates these operations with sequences of unitary operations, enabling implementation on quantum hardware.

However, these sequences can introduce degradation in the quantum state due to noise levels in current hardware. This allowed for a comprehensive comparison of the performance and limitations of different quantum computing architectures. The researchers note that quantum Monte Carlo (QMC) methods, while effective for large, high-temperature systems, can encounter the sign problem and struggle to resolve ground state properties. While fault-tolerant quantum computers are still under development, even noisy intermediate-scale quantum (NISQ) computers are already capable of simulating real-time dynamics for systems exceeding the size of exact classical solutions.

Efficient algorithms for Green’s functions calculations have been studied. The presented work builds upon these advancements, pushing the boundaries of quantum simulation for condensed matter physics.

Heisenberg Model Sampling vs. Random Sampling Performance

Rydberg atom processors achieved a demonstrable advantage over conventional random sampling methods when diagonalizing the Fermi-Hubbard model, converging near the ground state for systems as large as 56 qubits. This performance was observed even when random sampling was given a ten-fold advantage in the number of samples used, highlighting the efficiency of the sample-based quantum diagonalization (SQD) technique employed. The work, based on a perturbative relationship between the Fermi-Hubbard and Heisenberg models, presents a new approach to simulating complex materials and probing emergent phenomena.

The researchers directly compared SQD using states prepared via variational quantum imaginary time evolution (VQITE) with SQD initiated from a uniform superposition, revealing a clear benefit from utilizing the VQITE-prepared state for smaller fractions of the sampling subspace. This improvement in accuracy is quantified by a metric, the difference in ground state energy error, which demonstrated a performance gain when sampling from the VQITE state.

This gate-based implementation served as a benchmark for the analog approach on the Aquila processor, confirming the framework’s hardware agnosticism and potential for broader application. The ability to translate the method between analog and digital quantum platforms expands its accessibility and allows for validation across diverse computational resources. The study focused on the anisotropic Heisenberg model, a necessary adaptation for implementation on the Aquila processor, which does not natively support the isotropic version related to the spin-symmetric Hubbard model.

By tuning the interactions to achieve a reasonable fidelity up to $J_{x y} / J_{z}$ = 0. 5, the researchers were able to perform sampling and extract meaningful data. The researchers exploited the fact that, in the limit of strong coupling, the Hubbard model can be approximated by the Heisenberg model, allowing them to use the strengths of Rydberg atom processors in simulating magnetic interactions.

By focusing on the 1-dimensional Fermi-Hubbard model, they aimed to understand the regime relevant to quasi-superconductivity and develop new tools for investigating this complex behavior. The team’s success in adapting their framework to different quantum hardware further solidifies its potential as a versatile tool for quantum simulation.

Source: https://iopscience.iop.org/article/10.1088/2058-9565/aeb3a7/pdf

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