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 central challenge in condensed-matter physics and often exceeds the capabilities of classical computational methods as system size increases. This study proposes a variational quantum simulation workflow employing VQE and VQD to estimate the ground-state and first excited-state energies. The hybrid HEA–HVA ansatz combines the advantages of both approaches to balance state-representation capacity against quantum-circuit resource requirements. Numerical simulations were conducted on three lattice configurations—a 4 × 1 chain, a 2 × 2 square lattice, and a six-site ring—with the interaction ratio ranging from 0 to 10. The results show that the hybrid HEA–HVA ansatz converges reliably and accurately reproduces both the ground-state and lowest excited-state energies. Compared with the individual ansatzes, the hybrid structure produces smaller or comparable energy deviations, particularly in the intermediate-to-strong interaction regime, while maintaining a circuit depth compatible with noisy intermediate-scale quantum (NISQ) devices. These findings highlight the potential of hybrid ansatzes for simulating strongly correlated fermionic systems and provide a foundation for extending the approach to larger systems with more complex energy spectra.
Keywords
Fermi–Hubbard model, quantum simulation, variational quantum algorithm (VQA), variational quantum deflation (VQD), variational quantum eigensolver (VQE)
Article Details
References
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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. https://doi.org/10.1038/s41467-018-07090-4
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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. https://doi.org/10.1103/PhysRevResearch.1.033062
Orús, R., Mugel, S., & Lizaso, E. (2019). Quantum computing for finance: Overview and prospects. Reviews in Physics, 4, 100028. https://doi.org/10.1016/j.revip.2019.100028
Park, C.-Y. (2024). Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers. APL Quantum, 1(1), 016101. https://doi.org/10.1063/5.0186205
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. https://doi.org/10.1103/PhysRevLett.122.230401
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. https://doi.org/10.1038/ncomms5213
Poilblanc, D. (2014). Entanglement Hamiltonian of the quantum Néel state. Journal of Statistical Mechanics: Theory and Experiment, 2014(10), P10026. https://doi.org/10.1088/1742-5468/2014/10/P10026
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Sachdev, S., Sengupta, K., & Girvin, S. M. (2002). Mott insulators in strong electric fields. Physical Review B, 66(7), 075128. https://doi.org/10.1103/PhysRevB.66.075128
Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 124–134. https://doi.org/10.1109/SFCS.1994.365700
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), 044117. https://doi.org/10.1063/5.0161057
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. https://doi.org/10.1021/acs.jctc.9b01125
Wecker, D., Hastings, M. B., & Troyer, M. (2015). Progress towards practical quantum variational algorithms. Physical Review A, 92(4), 042303. https://doi.org/10.1103/PhysRevA.92.042303
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. https://doi.org/10.1109/SC.2005.1
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. https://doi.org/10.1103/PhysRevB.100.094434
Bethe, H. (1931). Zur theorie der metalle: I. Eigenwerte und eigenfunktionen der linearen atomkette. Zeitschrift für Physik, 71(3), 205-226. https://doi.org/10.1007/BF01341708
Cai, Z. (2020). Resource Estimation for Quantum Variational Simulations of the Hubbard Model. Physical Review Applied, 14(1), 014059. https://doi.org/10.1103/PhysRevApplied.14.014059
Claudino, D. (2022). The Basics of Quantum Computing for Chemists. International Journal of Quantum Chemistry, 122(23), e26990. https://doi.org/10.1002/qua.26990
Dagotto, E. (2005). Complexity in strongly correlated electronic systems. Science, 309(5732), 257-262. https://doi.org/10.1126/science.1107559
Dev, M., Behera, B. K., Vyas, V., & Panigrahi, P. K. (2025). Excitation Gaps in the Fermi-Hubbard Model via Variational Quantum Eigensolver. arXiv preprint arXiv:2508.12307.
Devadas, R. M., & Sowmya, T. (2025). Quantum machine learning: A comprehensive review of integrating AI with quantum computing for computational advancements. MethodsX, 14, 103318. https://doi.org/10.1016/j.mex.2025.103318
Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm. arXiv preprint arXiv:1411.4028.
Fradkin, E., Kivelson, S. A., & Tranquada, J. M. (2015). Colloquium: Theory of intertwined orders in high temperature superconductors. Reviews of Modern Physics, 87(2), 457-482. https://doi.org/10.1103/RevModPhys.87.457
Georges, T. N., Bothe, M., Sünderhauf, C., Berntson, B. K., Izsák, R., & Ivanov, A. V. (2025). Quantum simulations of chemistry in first quantization with any basis set. npj Quantum Information, 11, 55. https://doi.org/10.1038/s41534-025-00987-1
Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to quantum mechanics. Cambridge University Press. https://doi.org/10.1017/9781316995433
Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, 212-219. https://doi.org/10.1145/237814.237866
Gulacsi, Z. (2025). Jordan-Wigner transformation constructed for spinful fermions at spin-1/2 in two dimensions. Philosophical Magazine, 105(14), 794-812. https://doi.org/10.1080/14786435.2025.2465731
Higgott, O., Wang, D., & Brierley, S. (2019). Variational quantum computation of excited states. Quantum, 3, 156. https://doi.org/10.22331/q-2019-07-01-156
Hubbard, J. (1963). Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 276(1365), 238–257. https://doi.org/10.1098/rspa.1963.0204
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. https://doi.org/10.1038/s41524-024-01221-w
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. https://doi.org/10.1038/nature23879
Kivlichan, I. D., McClean, J., Wiebe, N., Gidney, C., Aspuru-Guzik, A., Chan, G. K.-L., & Babbush, R. (2018). Quantum simulation of electronic structure with linear depth and connectivity. Physical Review Letters, 120(11), 110501. https://doi.org/10.1103/PhysRevLett.120.110501
Kumar, M., & Mondal, B. (2025). A brief review on quantum key distribution protocols. Multimedia Tools and Applications, 84(27), 33267–33306. https://doi.org/10.1007/s11042-024-20535-x
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’ev, 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. https://doi.org/10.1103/PhysRevX.5.041041
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. https://doi.org/10.1038/s41467-018-07090-4
McClean, J. R., Kimchi-Schwartz, M. E., Carter, J., & de Jong, W. A. (2017). Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Physical Review A, 95(4), 042308. https://doi.org/10.1103/PhysRevA.95.042308
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. https://doi.org/10.1088/1367-2630/18/2/023023
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. https://doi.org/10.1103/PhysRevResearch.1.033062
Orús, R., Mugel, S., & Lizaso, E. (2019). Quantum computing for finance: Overview and prospects. Reviews in Physics, 4, 100028. https://doi.org/10.1016/j.revip.2019.100028
Park, C.-Y. (2024). Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers. APL Quantum, 1(1), 016101. https://doi.org/10.1063/5.0186205
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. https://doi.org/10.1103/PhysRevLett.122.230401
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. https://doi.org/10.1038/ncomms5213
Poilblanc, D. (2014). Entanglement Hamiltonian of the quantum Néel state. Journal of Statistical Mechanics: Theory and Experiment, 2014(10), P10026. https://doi.org/10.1088/1742-5468/2014/10/P10026
Rietsche, R., Dremel, C., Bosch, S., Steinacker, L., Meckel, M., & Leimeister, J.-M. (2022). Quantum computing. Electronic Markets, 32(4), 2525–2536. https://doi.org/10.1007/s12525-022-00570-y
Sachdev, S., Sengupta, K., & Girvin, S. M. (2002). Mott insulators in strong electric fields. Physical Review B, 66(7), 075128. https://doi.org/10.1103/PhysRevB.66.075128
Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 124–134. https://doi.org/10.1109/SFCS.1994.365700
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), 044117. https://doi.org/10.1063/5.0161057
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. https://doi.org/10.1021/acs.jctc.9b01125
Wecker, D., Hastings, M. B., & Troyer, M. (2015). Progress towards practical quantum variational algorithms. Physical Review A, 92(4), 042303. https://doi.org/10.1103/PhysRevA.92.042303
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. https://doi.org/10.1109/SC.2005.1