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Preparing the Gutzwiller wave function for attractive SU(3) fermions on a quantum computer

Han Xu1,*, Kazuhiro Seki2, and Seiji Yunoki1,2,3,4

  • *Contact author: han.xu@riken.jp
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Phys. Rev. Research 7, 033290 – Published 29 September, 2025

DOI: https://doi.org/10.1103/rc6z-837q

Abstract

We implement the Gutzwiller wave function for attractive SU(3) fermion systems on a quantum computer using a quantum-classical hybrid scheme based on the discrete Hubbard-Stratonovich transformation. In this approach, the nonunitary Gutzwiller operator is decomposed into a linear combination of unitaries constructed from two-qubit fermionic Givens-rotation gates, whose rotation angles are dictated by the auxiliary fields. We develop and reformulate two complementary methods to perform the sum over these auxiliary fields. In the first method, the Gutzwiller wave function is probabilistically prepared on the register qubits by projectively postselecting the desired state via measurements of ancilla qubits. We analyze the success rate both analytically and numerically as a function of the Gutzwiller variational parameter g for the Fermi-sea and BCS-like trial states at half filling. The success rate is found to decay exponentially for small |g|, but remains finite in the |g|→∞ limit, with increasing |g|. In the second method, we employ importance sampling to address the Gutzwiller variational problem, where the central objective is to estimate the expectation values of observables. We demonstrate the proposed scheme by calculating the energy and triple occupancy of the attractive SU(3) Hubbard model in the framework of digital quantum simulation. Moreover, we present experimental results obtained on a trapped-ion quantum computer for the two-site attractive SU(3) Hubbard model, showing good agreement with exact values within statistical errors.

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References (79)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. P. Benioff, The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by turing machines, J. Stat. Phys. 22, 563 (1980).
  3. G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Quantum algorithms for fermionic simulations, Phys. Rev. A 64, 022319 (2001).
  4. D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).
  5. D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Solving strongly correlated electron models on a quantum computer, Phys. Rev. A 92, 062318 (2015).
  6. Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Quantum algorithms to simulate many-body physics of correlated fermions, Phys. Rev. Appl. 9, 044036 (2018).
  7. P. J. J. O’Malley et al., Scalable quantum simulation of molecular energies, Phys. Rev. X 6, 031007 (2016).
  8. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  9. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  10. S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf. 5, 75 (2019).
  11. M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205 (2020).
  12. B. Murta and J. Fernández-Rossier, Gutzwiller wave function on a digital quantum computer, Phys. Rev. B 103, L241113 (2021).
  13. M. C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett. 10, 159 (1963).
  14. M. C. Gutzwiller, Correlation of electrons in a narrow s band, Phys. Rev. 137, A1726 (1965).
  15. J. Bünemann, W. Weber, and F. Gebhard, Multiband Gutzwiller wave functions for general on-site interactions, Phys. Rev. B 57, 6896 (1998).
  16. M. Fabrizio, Gutzwiller description of non-magnetic Mott insulators: Dimer lattice model, Phys. Rev. B 76, 165110 (2007).
  17. F. D. M. Haldane, Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-12 antiferromagnetic Heisenberg chain with 1/r2 exchange, Phys. Rev. Lett. 60, 635 (1988).
  18. B. S. Shastry, Exact solution of an s = 1/2 Heisenberg antiferromagnetic chain with long-ranged interactions, Phys. Rev. Lett. 60, 639 (1988).
  19. H. Yokoyama and H. Shiba, Hubbard model in strong correlation regime—variational Monte-Carlo studies on singlet liquid and Néel state, J. Phys. Soc. Jpn. 56, 3570 (1987).
  20. Y. Kuramoto and H. Yokoyama, Exactly soluble supersymmetric t−J-type model with long-range exchange and transfer, Phys. Rev. Lett. 67, 1338 (1991).
  21. H. Yokoyama and M. Ogata, Variational wave functions and ground-state properties in the one-dimensional t−J model, Phys. Rev. Lett. 67, 3610 (1991).
  22. A. Himeda and M. Ogata, Spontaneous deformation of the Fermi surface due to strong correlation in the two-dimensional t−J model, Phys. Rev. Lett. 85, 4345 (2000).
  23. A. Himeda, T. Kato, and M. Ogata, Stripe states with spatially oscillating d-wave superconductivity in the two-dimensional t−t′−J model, Phys. Rev. Lett. 88, 117001 (2002).
  24. S. Yunoki, Coherent inverse photoemission spectrum for Gutzwiller projected superconductors, Phys. Rev. B 72, 092505 (2005).
  25. S. Yunoki, Single-particle anomalous excitations of Gutzwiller-projected BCS superconductors and Bogoliubov quasiparticle characteristics, Phys. Rev. B 74, 180504 (2006).
  26. P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
  27. G. Mazzola, P. J. Ollitrault, P. K. Barkoutsos, and I. Tavernelli, Nonunitary operations for ground-state calculations in near-term quantum computers, Phys. Rev. Lett. 123, 130501 (2019).
  28. Y. Yao, F. Zhang, C.-Z. Wang, K.-M. Ho, and P. P. Orth, Gutzwiller hybrid quantum-classical computing approach for correlated materials, Phys. Rev. Res. 3, 013184 (2021).
  29. K. Seki, Y. Otsuka, and S. Yunoki, Gutzwiller wave function on a quantum computer using a discrete Hubbard-Stratonovich transformation, Phys. Rev. B 105, 155119 (2022).
  30. J. P. T. Stenger, C. S. Hellberg, and D. Gunlycke, Implementing Jastrow-Gutzwiller operators on a quantum computer using the cascaded variational quantum eigensolver algorithm, Phys. Rev. A 107, 062606 (2023).
  31. Z. Fodor and S. D. Katz, Lattice determination of the critical point of QCD at finite t and μ, J. High Energy Phys. 03 (2002) 014.
  32. Y. Aoki, G. Endrődi, Z. Fodor, S. Katz, and K. Szabo, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature (London) 443, 675 (2006).
  33. F. Wilczek, Quantum chromodynamics: Lifestyles of the small and simple, Nat. Phys. 3, 375 (2007).
  34. C. Honerkamp and W. Hofstetter, Ultracold fermions and the SU(N) Hubbard model, Phys. Rev. Lett. 92, 170403 (2004).
  35. C. Honerkamp and W. Hofstetter, BCS pairing in Fermi systems with N different hyperfine states, Phys. Rev. B 70, 094521 (2004).
  36. E. R. I. Abraham, W. I. McAlexander, J. M. Gerton, R. G. Hulet, R. Côté, and A. Dalgarno, Triplet s-wave resonance in Li6 collisions and scattering lengths of Li6 and Li7, Phys. Rev. A 55, R3299 (1997).
  37. M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne, Precise determination of Li6 cold collision parameters by radio-frequency spectroscopy on weakly bound molecules, Phys. Rev. Lett. 94, 103201 (2005).
  38. T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, and S. Jochim, Collisional stability of a three-component degenerate Fermi gas, Phys. Rev. Lett. 101, 203202 (2008).
  39. J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, and K. M. O’Hara, Three-body recombination in a three-state Fermi gas with widely tunable interactions, Phys. Rev. Lett. 102, 165302 (2009).
  40. A. Rapp, G. Zaránd, C. Honerkamp, and W. Hofstetter, Color superfluidity and “baryon” formation in ultracold fermions, Phys. Rev. Lett. 98, 160405 (2007).
  41. A. Rapp, W. Hofstetter, and G. Zaránd, Trionic phase of ultracold fermions in an optical lattice: A variational study, Phys. Rev. B 77, 144520 (2008).
  42. K. Inaba and S.-I. Suga, Finite-temperature properties of attractive three-component fermionic atoms in optical lattices, Phys. Rev. A 80, 041602 (2009).
  43. K. Inaba and S.-I. Suga, Color superfluid and trionic state of attractive three-component lattice fermionic atoms at finite temperatures, Mod. Phys. Lett. B 25, 987 (2011).
  44. I. Titvinidze, A. Privitera, S.-Y. Chang, S. Diehl, M. A. Baranov, A. Daley, and W. Hofstetter, Magnetism and domain formation in SU(3)-symmetric multi-species Fermi mixtures, New J. Phys. 13, 035013 (2011).
  45. A. Koga and H. Yanatori, Spontaneously symmetry-breaking states in the attractive SU(N) Hubbard model, J. Phys. Soc. Jpn. 86, 034702 (2017).
  46. H. Xu, X. Li, Z. Zhou, X. Wang, L. Wang, C. Wu, and Y. Wang, Trion states and quantum criticality of attractive SU(3) Dirac fermions, Phys. Rev. Res. 5, 023180 (2023).
  47. X. Li, H. Xu, and Y. Wang, Quantum Monte Carlo simulations of thermodynamic properties of attractive SU(3) Dirac fermions, Phys. Rev. B 108, 165102 (2023).
  48. W. J. Chetcuti, J. Polo, A. Osterloh, P. Castorina, and L. Amico, Probe for bound states of SU(3) fermions and colour deconfinement, Commun. Phys. 6, 128 (2023).
  49. F. Assaad and H. Evertz, World-line and determinantal quantum Monte Carlo methods for spins, phonons and electrons, in Computational Many-Particle Physics, edited by H. Fehske, R. Schneider, and A. Weiße (Springer, Berlin, 2008), pp. 277–356.
  50. K. Seki and S. Yunoki, Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer, Phys. Rev. A 105, 032419 (2022).
  51. M. Foss-Feig, D. Hayes, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, J. M. Pino, and A. C. Potter, Holographic quantum algorithms for simulating correlated spin systems, Phys. Rev. Res. 3, 033002 (2021).
  52. F. Verstraete, J. I. Cirac, and J. I. Latorre, Quantum circuits for strongly correlated quantum systems, Phys. Rev. A 79, 032316 (2009).
  53. N. H. Stair, R. Huang, and F. A. Evangelista, A multireference quantum Krylov algorithm for strongly correlated electrons, J. Chem. Theory Comput. 16, 2236 (2020).
  54. M. Bajdich, L. Mitas, G. Drobný, L. K. Wagner, and K. E. Schmidt, Pfaffian pairing wave functions in electronic-structure quantum Monte Carlo simulations, Phys. Rev. Lett. 96, 130201 (2006).
  55. M. Bajdich, L. Mitas, L. K. Wagner, and K. E. Schmidt, Pfaffian pairing and backflow wavefunctions for electronic structure quantum Monte Carlo methods, Phys. Rev. B 77, 115112 (2008).
  56. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, 2017).
  57. Qiskit contributors, Qiskit: An open-source framework for quantum computing, 2023, https://www.ibm.com/quantum/qiskit.
  58. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with qiskit, arXiv:2405.08810.
  59. S. Sivarajah, S. Dilkes, A. Cowtan, W. Simmons, A. Edgington, and R. Duncan, t|ket〉: A retargetable compiler for NISQ devices, Quantum Sci. Technol. 6, 014003 (2021).
  60. Quantinuum’s System Model H1, 2024, https://www.quantinuum.com/products-solutions/quantinuum-systems/system-model-h1.
  61. A. Summer, C. Chiaracane, M. T. Mitchison, and J. Goold, Calculating the many-body density of states on a digital quantum computer, Phys. Rev. Res. 6, 013106 (2024).
  62. M. DeCross et al., Computational power of random quantum circuits in arbitrary geometries, Phys. Rev. X 15, 021052 (2025).
  63. K. Seki, Y. Kikuchi, T. Hayata, and S. Yunoki, Simulating Floquet scrambling circuits on trapped-ion quantum computers, Phys. Rev. Res. 7, 023032 (2025).
  64. A. Chiesa, F. Tacchino, M. Grossi, P. Santini, I. Tavernelli, D. Gerace, and S. Carretta, Quantum hardware simulating four-dimensional inelastic neutron scattering, Nat. Phys. 15, 455 (2019).
  65. A. Francis, J. K. Freericks, and A. F. Kemper, Quantum computation of magnon spectra, Phys. Rev. B 101, 014411 (2020).
  66. Qiskit IBM Runtime’s Fake Torino backend, 2024, https://quantum.cloud.ibm.com/docs/en/api/qiskit-ibm-runtime/fake-provider-fake-torino.
  67. X. Mi et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  68. K. Shinjo, K. Seki, T. Shirakawa, R.-Y. Sun, and S. Yunoki, Unveiling clean two-dimensional discrete time quasicrystals on a digital quantum computer, arXiv:2403.16718.
  69. K. Kechedzhi, S. Isakov, S. Mandrà, B. Villalonga, X. Mi, S. Boixo, and V. Smelyanskiy, Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments, Future Gener. Comput. Syst. 153, 431 (2024).
  70. Z.-X. Li, Y.-F. Jiang, and H. Yao, Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation, Phys. Rev. B 91, 241117 (2015).
  71. Z.-X. Li, Y.-F. Jiang, and H. Yao, Majorana-time-reversal symmetries: A fundamental principle for sign-problem-free quantum Monte Carlo simulations, Phys. Rev. Lett. 117, 267002 (2016).
  72. Z. C. Wei, C. Wu, Y. Li, S. Zhang, and T. Xiang, Majorana positivity and the fermion sign problem of quantum Monte Carlo simulations, Phys. Rev. Lett. 116, 250601 (2016).
  73. Z.-Y. Han, Z.-Q. Wan, and H. Yao, Pfaffian quantum Monte Carlo: Solution to Majorana sign ambiguity and applications, arXiv:2408.10311.
  74. T. Shirakawa, H. Ueda, and S. Yunoki, Automatic quantum circuit encoding of a given arbitrary quantum state, Phys. Rev. Res. 6, 043008 (2024).
  75. D. Malz, G. Styliaris, Z.-Y. Wei, and J. I. Cirac, Preparation of matrix product states with log-depth quantum circuits, Phys. Rev. Lett. 132, 040404 (2024).
  76. K. C. Smith, A. Khan, B. K. Clark, S. Girvin, and T.-C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024).
  77. S.-J. Ran, Encoding of matrix product states into quantum circuits of one- and two-qubit gates, Phys. Rev. A 101, 032310 (2020).
  78. S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
  79. H. Xu, K. Seki, and S. Yunoki, Research data supporting “Preparing the Gutzwiller wave function for attractive SU(3) fermions on a quantum computer”, Zenodo (2025), https://doi.org/10.5281/zenodo.15285140.

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Phys. Rev. Research 7, 033290– Published 29 September, 2025
  • Received 25 April 2025
  • Accepted 28 July 2025
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