Quantum simulation and near-term quantum algorithms have gained significant momentum over the past few years, with researchers exploring quantum-based approaches to solve classically intractable problems. Out of many research problems, certain problems are well suited for quantum algorithms. Fermi Hubbard model based on lattice structures is one of the quantum-mechanical models used in many-body and condensed matter physics to capture the essence of electron-correlated systems. In this work, we show one of the first kinds of experiments on simulating the ground state properties of 1D and 2D Fermi Hubbard-based lattice models compared with the reference ground state energy values using Qiskit Nature. We utilize the Variational Quantum Eigensolver (VQE) and different gradient-based optimizers to achieve results that closely converge towards the exact values, showcasing the potential of variational quantum algorithms.
Quantum simulation and near-term quantum algorithms have gained significant momentum over the past few years, with researchers exploring quantum-based approaches to solve classically intractable problems. Out of many research problems, certain problems are well suited for quantum algorithms. Fermi Hubbard model based on lattice structures is one of the quantum-mechanical models used in many-body and condensed matter physics to capture the essence of electron-correlated systems. In this work, we show one of the first kinds of experiments on simulating the ground state properties of 1D and 2D Fermi Hubbard-based lattice models compared with the reference ground state energy values using Qiskit Nature. We utilize the Variational Quantum Eigensolver (VQE) and different gradient-based optimizers to achieve results that closely converge towards the exact values, showcasing the potential of variational quantum algorithms.
The famous, yet unsolved, Fermi-Hubbard model for strongly-correlated electronic systems is a prominent target for quantum computers. However, accurately representing the Fermi-Hubbard ground state for large instances may be beyond the reach of near-term quantum hardware. Here we show experimentally that an efficient, low-depth variational quantum algorithm with few parameters can reproduce important qualitative features of medium-size instances of the Fermi-Hubbard model. We address 1 × 8 and 2 × 4 instances on 16 qubits on a superconducting quantum processor, substantially larger than previous work based on less scalable compression techniques, and going beyond the family of 1D Fermi-Hubbard instances, which are solvable classically. Consistent with predictions for the ground state, we observe the onset of the metal-insulator transition and Friedel oscillations in 1D, and antiferromagnetic order in both 1D and 2D. We use a variety of error-mitigation techniques, including symmetries of the Fermi-Hubbard model and a recently developed technique tailored to simulating fermionic systems. We also introduce a new variational optimisation algorithm based on iterative Bayesian updates of a local surrogate model.
Simulating the Hubbard model is of great interest to a wide range of applications within condensed matter physics, however its solution on classical computers remains challenging in dimensions larger than one. The relative simplicity of this model, embodied by the sparseness of the Hamiltonian matrix, allows for its efficient implementation on quantum computers, and for its approximate solution using variational algorithms such as the variational quantum eigensolver. While these algorithms have been shown to reproduce the qualitative features of the Hubbard model, their quantitative accuracy in terms of producing true ground state energies and other properties, and the dependence of this accuracy on the system size and interaction strength, the choice of variational ansatz, and the degree of spatial inhomogeneity in the model, remains unknown. Here we present a rigorous classical benchmarking study, demonstrating the potential impact of these factors on the accuracy of the variational solution of the Hubbard model on quantum hardware, for systems with up to 32 qubits. We find that even when using the most accurate wavefunction ansätze for the Hubbard model, the error in its ground state energy and wavefunction plateaus for larger lattices, while stronger electronic correlations magnify this issue. Concurrently, spatially inhomogeneous parameters and the presence of off-site Coulomb interactions only have a small effect on the accuracy of the computed ground state energies. Our study highlights the capabilities and limitations of current approaches for solving the Hubbard model on quantum hardware, and we discuss potential future avenues of research.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids up to 6 sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver(VQE) called ADAPT-VQE. In contrast with non-adaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the non-adaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach, and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well-suited for NISQ devices.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids of up to six sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver (VQE) called ADAPT-VQE [H. R. Grimsley et al., Nat. Commun. 10, 3007 (2019)]. In contrast with nonadaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the nonadaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green's functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach [A. Peruzzo et al., Nat. Commun. 5, 4213 (2014); J. R. McLean et al., New J. Phys. 18, 023023 (2016); K. Sharma et al., New J. Phys. 22, 043006 (2020)], and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well suited for noisy intermediate-scale quantum devices.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.
It is expected that the simulation of correlated fermions in chemistry and material science will be one of the first practical applications of quantum processors. Given the rapid evolution of quantum hardware, it is increasingly important to develop robust benchmarking techniques to gauge the capacity of quantum hardware specifically for the purpose of fermionic simulation. Here we propose using the one-dimensional Fermi-Hubbard model as an application benchmark for variational quantum simulations on near-term quantum devices. Since the one-dimensional Hubbard model is both strongly correlated and exactly solvable with the Bethe ansatz, it provides a reference ground state energy that a given device with limited coherence will be able to approximate up to a maximal size. The length of the largest chain that can be simulated provides an effective fermionic length. We use variational quantum eigensolver to approximate the ground state energy values of Fermi-Hubbard instances and show how the fermionic length benchmark can be used in practice to assess the performance of bounded-depth devices in a scalable fashion.