Quantum Simulation of the Fermi–Hubbard Model Using Variational Quantum Algorithms with a Hybrid HEA–HVA Ansatz
Main Article Content
Abstract
Simulating the Fermi–Hubbard model is a key problem in condensed matter physics, yet it often exceeds the capabilities of classical computational methods as the system size increases. This study proposes a variational quantum simulation workflow that employs VQE and VQD to estimate the ground-state energy and the first excited-state energy. The hybrid HEA–HVA ansatz is designed to flexibly combine their advantages to optimize the balance between state representation capabilities and quantum circuit resource costs. Through numerical simulations on three lattice configurations a 4×1 chain, a 2×2 square lattice, and a 6-site ring over an interaction ratio ranging from 0 to 4, the results show that Hybrid HEA–HVA converges stably and accurately reproduces both the ground-state energy and the lowest excited-state energy. Compared with the pure ansatzes, the hybrid structure yields smaller or comparable energy deviations, being particularly effective in the intermediate-to-strong interaction regime, while maintaining a circuit depth compatible with the NISQ setting. These findings highlight the potential of hybrid ansatzes for simulating strongly correlated fermionic systems and provide a solid foundation for extending the approach to larger systems with more complex energy spectra
Keywords
Variational Quantum Eigensolver (VQE), Hybrid Ansatz, Fermi–Hubbard Model, Quantum Simulation, Ground-State Energy, First Excited-State Energy, Variational Quantum Deflation (VQD
Article Details
References
Beach, M. J. S., Melko, R. G., Grover, T., & Hsieh, T. H. (2019). Making trotters sprint: A variational imaginary time ansatz for quantum many-body systems. Physical Review B, 100(9), 094434.
Bethe, H. (1931). Zur theorie der metalle: I. Eigenwerte und eigenfunktionen der linearen atomkette. Zeitschrift für Physik, 71(3), 205-226.
Cai, Z. (2020). Resource Estimation for Quantum Variational Simulations of the Hubbard Model. Physical Review Applied, 14(1), 014059.
Ciliberto, C., Herbster, M., Ialongo, A. D., Pontil, M., Rocchetto, A., Severini, S., & Wossnig, L. (2018). Quantum machine learning: a classical perspective. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2209), 20170551.
Claudino, D. (2022). The Basics of Quantum Computing for Chemists. International Journal of Quantum Chemistry, 122(23), e26990.
Dagotto, E. (2005). Complexity in strongly correlated electronic systems. Science, 309(5732), 257-262.
Dev, M., Behra, B. K., Vyas, V., & Panigrahi, P. K. (2025). Excitation Gaps in the Fermi-Hubbard Model via Variational Quantum Eigensolver.
Devadas, R. M., & Sowmya, T. (2025). Quantum machine learning: A comprehensive review of integrating AI with quantum computing for computational advancements. MethodsX, 103318.
Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm.
Farhi, E., Goldstone, J., Gutmann, S., & Sipser, M. (2000). Quantum Computation by Adiabatic Evolution.
Fradkin, Eduardo, Kivelson, S. A., & Tranquada, J. M. (2015). Colloquium: Theory of intertwined orders in high temperature superconductors. Reviews of Modern Physics, 87(2), 457-482.
Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to quantum mechanics. Cambridge university press.
Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing.
Gulacsi, Z. (2024). Jordan-Wigner transformation constructed for spinful fermions at spin-1/2 in two dimensions. Philosophical Magazine, 1-19.
Higgott, O., Wang, D., & Brierley, S. (2019). Variational quantum computation of excited states. Quantum, 3, 156.
Hubbard, J. (1963). Electron Correlations in Narrow Energy Bands. Proceedings of the Royal Society of London. Series A, Mathematical and Physical SciencesProceedings of the Royal Society of London. Series A, 276(1365), 238–257.
Iskakov, S., Katsnelson, M. I., & Lichtenstein, A. I. (2024). Perturbative solution of fermionic sign problem in quantum Monte Carlo computations. npj Computational Materials, 10(1), 36.
Jha, N., Parakh, A., & Subramaniam, M. (2025). Quantum Key Distribution: Bridging Theoretical Security Proofs, Practical Attacks, and Error Correction for Quantum-Augmented Networks. arXiv preprint arXiv:2511.20602.
Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., & Gambetta, J. M. (2017). Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549(7671), 242-246.
LeBlanc, J. P. F., Antipov, A. E., Becca, F., Bulik, I. W., Chan, G. K.-L., Chung, C.-M., Deng, Y., Ferrero, M., Henderson, T. M., Jiménez‑Hoyos, C. A., Kozik, E., Liu, X.-W., Millis, A. J., Prokof’e, N. V., Qin, M., Scuseria, G. E., Shi, H., Svistunov, B. V., Tocchio, L. F., . . . Gull., E. (2015). Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms. Physical Review X, 5(4), 041041.
McArdle, S., Endo, S., Aspuru-Guzik, A., Benjamin, S., & Yuan, X. (2020). Quantum computational chemistry. Reviews of Modern Physics, 92(1), 015003.
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R., & Neven, H. (2018). Barren plateaus in quantum neural network training landscapes. Nature communications, 9(1), 4812.
McClean, J. R., Kimchi-Schwartz, M. E., Carter, J., & Jong, W. A. d. (2017). Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Physical Review A, 95(4), 042308.
McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A. (2016). The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2), 023023.
Montanaro, A., & Stanisic, S. (2020). Compressed variational quantum eigensolver for the Fermi-Hubbard model. arXiv preprint arXiv:2006.01179
Nakanishi, K. M., Mitarai, K., & Fujii, K. (2019). Subspace-search variational quantum eigensolver for excited states. Physical Review Research, 1(3), 033062.
Park, C.-Y. (2024). Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers. APL Quantum, 1(1), 016101.
Parrish, R. M., Hohenstein, E. G., McMahon, P. L., & Martínez, T. J. (2019). Quantum computation of electronic transitions using a variational quantum eigensolver. Physical Review Letters, 122(23), 230401.
Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P. J., Aspuru-Guzik, A., & O’Brien, J. L. (2014). A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5(1), 4213.
Pirandola, S., Andersen, U. L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C., Pereira, J., Razavi, M., Shaari, J. S., Tomamichel, M., Usenko, V. C., Vallone, G., Villoresi, P., & Wallden, P. (2020). Advances in quantum cryptography. Advances in optics and photonics, 12(4), 1012-1236.
Poilblanc, D. (2014). Entanglement Hamiltonian of the quantum Néel state. Journal of Statistical Mechanics: Theory and Experiment, 10, P10026.
Rieffel, E. G., & Polak, W. (2000). An Introduction to Quantum Computing for Non-Physicists. ACM Comput.Surveys, 32(3), 300-335.
Sachdev, S., Sengupta, K., & Girvin, S. M. (2002). Mott insulators in strong electric fields. Physical Review B, 66(7), 075128.
Shor, P. W. (1999). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41(2), 303-332.
Singh, H., Majumder, S., & Mishra, S. (2023). Benchmarking of different optimizers in the variational quantum algorithms for applications in quantum chemistry. The Journal of Chemical Physics, 159(4).
Stair, N. H., Huang, R., & Evangelista, F. A. (2020). A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons Journal of Chemical Theory and Computation, 16(4), 2236–2245.
Wecker, D., Hastings, M. B., & Troyer, M. (2015). Progress towards practical quantum variational algorithms. Physical Review A, 92(4), 042303.
Wolf, R. (2021). Quantum key distribution. Berlin/Heidelberg, Germany: Springer International Publishing, 988.
Yamada, S., Imamura, T., & Machida, M. (2005). 16.447 tflops and 159-billion-dimensional exact-diagonalization for trapped fermion-hubbard model on the earth simulator. SC'05: Proceedings of the 2005 ACM/IEEE Conference on Supercomputing, Washington, DC, USA.