© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2408.00836 · Quantum

Classical Benchmarks for Variational Quantum Eigensolver Simulations of the Hubbard Model

Antonios M. Alvertis, Abid Khan, Thomas Iadecola, Peter P. Orth, Norm M. Tubman

Published May 20, 2025• 4 citations

Abstract

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.

DOI: 10.22331/q-2025-05-20-1748

Related papers

Solving the Hubbard model using density matrix embedding theory and the variational quantum eigensolver

Lana Mineh, Ashley Montanaro

Calculating the ground state properties of a Hamiltonian can be mapped to the problem of finding the ground state of a smaller Hamiltonian through the use of embedding methods. These embedding techniques have the ability to drastically reduce the problem size, and hence the number of qubits required when running on a quantum computer. However, the embedding process can produce a relatively complicated Hamiltonian, leading to a more complex quantum algorithm. In this paper we carry out a detailed study into how density matrix embedding theory (DMET) could be implemented on a quantum computer to solve the Hubbard model. We consider the variational quantum eigensolver (VQE) as the solver for the embedded Hamiltonian within the DMET algorithm. We derive the exact form of the embedded Hamiltonian and use it to construct efficient ansatz circuits and measurement schemes. We conduct detailed numerical simulations up to 16 qubits, the largest to date, for a range of Hubbard model parameters and find that the combination of DMET and VQE is effective for reproducing ground state properties of the model.

arXiv:2108.08611•Physical review. B./Physical review. B•Mar 14, 2022•19 citations

Simulating strongly interacting Hubbard chains with the variational Hamiltonian ansatz on a quantum computer

Baptiste Anselme Martin, Pascal Simon, Marko J. Rančić

Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the variational quantum eigensolver has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this paper we try to answer the question: How much of the underlying physics of a 1D Hubbard chain is described by a problem-inspired variational Hamiltonian ansatz in a broad range of parameter values? We start by probing how much the solution increases fidelity with increasing ansatz complexity. Our findings suggest that even low fidelity solutions capture energy and number of doubly occupied sites well, while spin-spin correlations are not well captured even when the solution is of high fidelity. Our powerful simulation platform allows us to incorporate a realistic noise model and shows a successful implementation of noise-mitigation strategies---postselection and the Richardson extrapolation. Finally, we compare our results with an experimental realization of the algorithm on IBM Quantum's ibmq_quito device.

arXiv:2111.11996•Physical Review Research•Jun 6, 2022•20 citations

Variational Hamiltonian Ansatz for 1D Hubbard chains in a broad range of parameter values

Baptiste Anselme Martin, Pascal Simon, Marko J. Rančić

Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the Variational Quantum Eigensolver (VQE) has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this manuscript, we try to answer the question: how much of the underlying physics of a 1D Hubbard chain is described by a problem-inspired Variational Hamiltonian Ansatz (VHA) in a broad range of parameter values ? We start by probing how much does the solution increases fidelity with increasing ansatz complexity. Our findings suggest that even low fidelity solutions capture energy and number of doubly occupied sites well, while spin-spin correlations are not well captured even when the solution is of high fidelity. Our powerful simulation platform allows us to incorporate a realistic noise model and show a successful implementation of a noise-mitigation strategy - the Richardson extrapolation.

•Nov 23, 2021•0 citations

Quantum Simulation of 1D & 2D Lattice-based Fermi Hubbard Model using Variational Quantum Algorithms

D Shalini, Srinjoy Ganguly, Prateek Jain

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.

International Conference on Trends in Quantum Computing and Emerging Business Technologies (TQCEBT)•Mar 22, 2024•1 citation

Unraveling correlated material properties with noisy quantum computers: Natural orbitalized variational quantum eigensolving of extended impurity models within a slave-boson approach

Pauline Besserve, Thomas Ayral

We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a noisy, intermediate scale quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the variational quantum eigensolver algorithm. This iterative scheme, dubbed natural orbitalization (NOization), gradually transforms the single-particle basis to the approximate natural-orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) Ansatz circuits faster and more accurate, and we propose an Ansatz, the multireference excitation-preserving (MREP) Ansatz, that achieves great expressivity without requiring a prohibitive gate count, thus bridging the gap between hardware-efficient and physically motivated strategies in variational Ansatz design. The one-impurity version of the Ansatz has only one parameter, making the ground-state preparation a trivial step, which supports the optimal character of our approach. Within a rotationally invariant slave boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP Ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.

arXiv:2108.10780•Physical review. B./Physical review. B•Mar 7, 2022•14 citations

Compressed variational quantum eigensolver for the Fermi-Hubbard model

Ashley Montanaro, Stasja Stanisic

The Fermi-Hubbard model is a plausible target to be solved by a quantum computer using the variational quantum eigensolver algorithm. However, problem sizes beyond the reach of classical exact diagonalisation are also beyond the reach of current quantum computing hardware. Here we use a simple method which compresses the first nontrivial subcase of the Hubbard model -- with one spin-up and one spin-down fermion -- enabling larger instances to be addressed using current quantum computing hardware. We implement this method on a superconducting quantum hardware platform for the case of the $2 \times 1$ Hubbard model, including error-mitigation techniques, and show that the ground state is found with relatively high accuracy.

arXiv:2006.01179•Jun 1, 2020•8 citations

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.

Calculating the ground state properties of a Hamiltonian can be mapped to the problem of finding the ground state of a smaller Hamiltonian through the use of embedding methods. These embedding techniques have the ability to drastically reduce the problem size, and hence the number of qubits required when running on a quantum computer. However, the embedding process can produce a relatively complicated Hamiltonian, leading to a more complex quantum algorithm. In this paper we carry out a detailed study into how density matrix embedding theory (DMET) could be implemented on a quantum computer to solve the Hubbard model. We consider the variational quantum eigensolver (VQE) as the solver for the embedded Hamiltonian within the DMET algorithm. We derive the exact form of the embedded Hamiltonian and use it to construct efficient ansatz circuits and measurement schemes. We conduct detailed numerical simulations up to 16 qubits, the largest to date, for a range of Hubbard model parameters and find that the combination of DMET and VQE is effective for reproducing ground state properties of the model.

Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the variational quantum eigensolver has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this paper we try to answer the question: How much of the underlying physics of a 1D Hubbard chain is described by a problem-inspired variational Hamiltonian ansatz in a broad range of parameter values? We start by probing how much the solution increases fidelity with increasing ansatz complexity. Our findings suggest that even low fidelity solutions capture energy and number of doubly occupied sites well, while spin-spin correlations are not well captured even when the solution is of high fidelity. Our powerful simulation platform allows us to incorporate a realistic noise model and shows a successful implementation of noise-mitigation strategies---postselection and the Richardson extrapolation. Finally, we compare our results with an experimental realization of the algorithm on IBM Quantum's ibmq_quito device.

Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the Variational Quantum Eigensolver (VQE) has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this manuscript, we try to answer the question: how much of the underlying physics of a 1D Hubbard chain is described by a problem-inspired Variational Hamiltonian Ansatz (VHA) in a broad range of parameter values ? We start by probing how much does the solution increases fidelity with increasing ansatz complexity. Our findings suggest that even low fidelity solutions capture energy and number of doubly occupied sites well, while spin-spin correlations are not well captured even when the solution is of high fidelity. Our powerful simulation platform allows us to incorporate a realistic noise model and show a successful implementation of a noise-mitigation strategy - the Richardson extrapolation.

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.

We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a noisy, intermediate scale quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the variational quantum eigensolver algorithm. This iterative scheme, dubbed natural orbitalization (NOization), gradually transforms the single-particle basis to the approximate natural-orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) Ansatz circuits faster and more accurate, and we propose an Ansatz, the multireference excitation-preserving (MREP) Ansatz, that achieves great expressivity without requiring a prohibitive gate count, thus bridging the gap between hardware-efficient and physically motivated strategies in variational Ansatz design. The one-impurity version of the Ansatz has only one parameter, making the ground-state preparation a trivial step, which supports the optimal character of our approach. Within a rotationally invariant slave boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP Ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.

The Fermi-Hubbard model is a plausible target to be solved by a quantum computer using the variational quantum eigensolver algorithm. However, problem sizes beyond the reach of classical exact diagonalisation are also beyond the reach of current quantum computing hardware. Here we use a simple method which compresses the first nontrivial subcase of the Hubbard model -- with one spin-up and one spin-down fermion -- enabling larger instances to be addressed using current quantum computing hardware. We implement this method on a superconducting quantum hardware platform for the case of the

2 \times 1

Hubbard model, including error-mitigation techniques, and show that the ground state is found with relatively high accuracy.