MÔ PHỎNG MÔ HÌNH FERMI–HUBBARD BẰNG THUẬT TOÁN LƯỢNG TỬ BIẾN PHÂN SỬ DỤNG ANSATZ HYBRID HEA–HVA
Nội dung chính của bài viết
Tóm tắt
Mô phỏng mô hình Fermi–Hubbard là bài toán then chốt trong vật lí vật chất ngưng tụ nhưng thường vượt quá khả năng của các phương pháp tính toán cổ điển khi quy mô hệ tăng. Nghiên cứu này đề xuất một quy trình mô phỏng lượng tử biến phân sử dụng VQE và VQD để ước lượng năng lượng trạng thái cơ bản và mức kích thích thứ nhất. Ansatz lai HEA–HVA được thiết kế và kết hợp linh hoạt ưu điểm của chúng nhằm tối ưu cân bằng giữa khả năng biểu diễn trạng thái và chi phí tài nguyên mạch lượng tử. Thông qua mô phỏng số trên các cấu hình chuỗi 4×1, mạng vuông 2×2 và vòng 6 vị trí với tỉ số tương tác U/t từ 0 đến 10, kết quả cho thấy Hybrid HEA–HVA hội tụ ổn định và tái tạo tốt năng lượng cơ bản cũng như mức kích thích thấp nhất. So với các ansatz thuần túy, cấu trúc lai cho sai lệch năng lượng nhỏ hơn hoặc tương đương, đặc biệt hiệu quả trong vùng tương tác trung bình mạnh, đồng thời vẫn duy trì độ sâu mạch phù hợp trong bối cảnh NISQ. Nghiên cứu khẳng định tiềm năng của ansatz lai trong mô phỏng các hệ fermion tương quan mạnh và tạo tiền đề để mở rộng sang các hệ có phổ năng lượng phức tạp hơn trên quy mô lớn.
Từ khóa
mô hình Fermi–Hubbard, mô phỏng lượng tử, thuật toán lượng tử biến phân (VQA), khử suy biến lượng tử biến phân (VQD), thuật toán giải trị riêng lượng tử biến phân (VQE)
Chi tiết bài viết
Tài liệu tham khảo
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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.
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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
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