Simulating fermions with a digital quantum computer - Nature

AbstractQuantum computers are expected to become a powerful tool for studying physical quantum systems. Consequently, a number of quantum algorithms to determine the physical properties of such systems have been developed. Although qubit-based quantum computers are naturally suited to the study of spin-1/2 systems, systems containing other degrees of freedom must first be encoded into qubits. Transformations to and from fermionic degrees of freedom have long been an important tool in physics and chemistry, which is now finding another application in the simulation of fermionic systems on quantum computers based on qubits. In this Review, we discuss methods for encoding fermionic degrees of freedom into qubits.Key points Physical systems have complex interactions that can involve fermions, and computing physically relevant quantities is classically challenging; quantum simulation algorithms to digitally prepare and estimate observables are actively researched. The study of physical systems beyond the reach of classical methods has been identified as one of the most important applications of the emerging technology. Simulating fermions on a quantum computer requires encoding the antisymmetric exchange into qubits, the basic memory elements of most quantum computers. First quantization, in which the number of fermions and their antisymmetrization correlation are explicitly encoded in the many-particle states, provides a compact representation of many-electron systems restricted to a fixed particle-number subspace. Second quantization, a generically applicable formalism natural in many applications areas, is amenable to a variety of encoding methods depending on the structure of the problem and available computing resources. Access through your institution Buy or subscribe This is a preview of subscription content, access via your institution Access options Access through your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription 27,99 € / 30 days cancel any time Learn more Subscribe to this journal Receive 12 digital issues and online access to articles 111,21 € per year only 9,27 € per issue Learn more Buy this articlePurchase on SpringerLinkInstant access to the full article PDF.39,95 €Prices may be subject to local taxes which are calculated during checkout Fig. 1: Overview of the four typical simulation steps.Fig. 2: Two Hamiltonian simulation algorithms to approximate the time evolution of fermionic systems.Fig. 3: The typical form for state antisymmetrization as part of fermionic simulation in first quantization.Fig. 4: Second-quantized encoding methods.Fig. 5: Ancilla-free encoding on a linear tree.Fig. 6: Local encoding of operators.Fig. 7: Factors that influence mapping choices. ReferencesManin, Y. Computable and Uncomputable (Sovetskoye Radio, 1980).Feynman, R. P. in Feynman and Computation, 133–153 (CRC Press, 2018).Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).Article ADS Google Scholar Arute, F. et al. Observation of separated dynamics of charge and spin in the Fermi–Hubbard model. Preprint at https://doi.org/10.48550/arXiv.2010.07965 (2020).Farrell, R. C., Illa, M., Ciavarella, A. N. & Savage, M. J. Scalable circuits for preparing ground states on digital quantum computers: the Schwinger model vacuum on 100 qubits. PRX Quantum 5, 020315 (2024).Article ADS Google Scholar Acharya, R. et al. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).Article ADS Google Scholar Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).Article ADS Google Scholar Paetznick, A. et al. Demonstration of logical qubits and repeated error correction with better-than-physical error rates. Preprint at https://doi.org/10.48550/arXiv.2404.02280 (2024).Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).Article ADS Google Scholar Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997).Article ADS Google Scholar Ortiz, G., Gubernatis, J., Knill, E. & Laflamme, R. Quantum algorithms for fermionic simulations. Phys. Rev. A 64, 022319 (2001).Article ADS Google Scholar Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010).Article Google Scholar Wecker, D., Bauer, B., Clark, B. K., Hastings, M. B. & Troyer, M. Gate-count estimates for performing quantum chemistry on small quantum computers. Phys. Rev. A 90, 022305 (2014).Article ADS Google Scholar Reiher, M., Wiebe, N., Svore, K. M., Wecker, D. & Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl Acad. Sci. USA 114, 7555–7560 (2017).Article ADS Google Scholar Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).Article ADS Google Scholar Whitfield, J. D., Biamonte, J. & Aspuru-Guzik, A. Simulation of electronic structure Hamiltonians using quantum computers. Mol. Phys. 109, 735–750 (2011).Article ADS Google Scholar Cao, Y. et al. Quantum chemistry in the age of quantum computing. Chem. Rev. 119, 10856–10915 (2019).Article Google Scholar Bauer, B., Bravyi, S., Motta, M. & Chan, G. K.-L. Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev. 120, 12685–12717 (2020).Article Google Scholar McArdle, S., Endo, S., Aspuru-Guzik, A., Benjamin, S. C. & Yuan, X. Quantum computational chemistry. Rev. Mod. Phys. 92, 015003 (2020).Article ADS MathSciNet Google Scholar Motta, M. & Rice, J. E. Emerging quantum computing algorithms for quantum chemistry. Wiley Interdiscip. Rev. Comput. Mol. Sci. 12, e1580 (2022).Article Google Scholar Kassal, I., Jordan, S. P., Love, P. J., Mohseni, M. & Aspuru-Guzik, A. Polynomial-time quantum algorithm for the simulation of chemical dynamics. Proc. Natl Acad. Sci. USA 105, 18681–18686 (2008).Article ADS Google Scholar Sawaya, N. P. D. & Huh, J. Quantum algorithm for calculating molecular vibronic spectra. J. Phys. Chem. Lett. 10, 3586–3591 (2019).Article Google Scholar Ollitrault, P. J., Mazzola, G. & Tavernelli, I. Nonadiabatic molecular quantum dynamics with quantum computers. Phys. Rev. Lett. 125, 260511 (2020).Article ADS Google Scholar Miessen, A., Ollitrault, P. J., Tacchino, F. & Tavernelli, I. Quantum algorithms for quantum dynamics. Nat. Comp. Sci. 3, 25–37 (2023).Article Google Scholar O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).

Google Scholar Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242 (2017).Article ADS Google Scholar Colless, J. I. et al. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X 8, 011021 (2018).

Google Scholar Hempel, C. et al. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X 8, 031022 (2018).

Google Scholar Google AI Quantum and Collaborators et al. Hartree-Fock on a superconducting qubit quantum computer. Science 369, 1084–1089 (2020).Article MathSciNet Google Scholar Navickas, T. et al. Experimental quantum simulation of chemical dynamics. J. Am. Chem. Soc. 147, 23566–23573 (2024).Article ADS Google Scholar Hubbard, J. Electron correlations in narrow energy bands. Proc. Roy. Soc. Lond. A 276, 238–257 (1963).Article ADS Google Scholar Oftelie, L. B. et al. Simulating quantum materials with digital quantum computers. Quantum Sci. Technol. 6, 043002 (2021).Article Google Scholar Wecker, D. et al. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A 92, 062318 (2015).Article ADS Google Scholar Cade, C., Mineh, L., Montanaro, A. & Stanisic, S. Strategies for solving the Fermi-Hubbard model on near-term quantum computers. Phys. Rev. B 102, 235122 (2020).Article ADS Google Scholar García-Ripoll, J. J., Solano, E. & Martin-Delgado, M. A. Quantum simulation of Anderson and Kondo lattices with superconducting qubits. Phys. Rev. B 77, 024522 (2008).Article ADS Google Scholar Bravyi, S. & Gosset, D. Complexity of quantum impurity problems. Commun. Math. Phys. 356, 451–500 (2017).Article ADS MathSciNet Google Scholar Jamet, F. et al. Anderson impurity solver integrating tensor network methods with quantum computing. APL Quantum 2, 016121 (2025).Article Google Scholar Rubin, N. C. et al. Fault-tolerant quantum simulation of materials using Bloch orbitals. PRX Quantum 4, 040303 (2023).Article ADS Google Scholar Clinton, L. et al. Towards near-term quantum simulation of materials. Nat. Commun. 15, 211 (2024).Article ADS Google Scholar Berry, D. W. et al. Quantum simulation of realistic materials in first quantization using non-local pseudopotentials. npj Quantum Inf. 10, 130 (2024).Article ADS Google Scholar Bauer, B., Wecker, D., Millis, A. J., Hastings, M. B. & Troyer, M. Hybrid quantum-classical approach to correlated materials. Phys. Rev. X 6, 031045 (2016).

Google Scholar Haah, J., Hastings, M. B., Kothari, R. & Low, G. H. Quantum algorithm for simulating real time evolution of lattice Hamiltonians. SIAM J. Comput. 52, FOCS18–250 (2021).MathSciNet Google Scholar Babbush, R. et al. Low-depth quantum simulation of materials. Phys. Rev. X 8, 011044 (2018).

Google Scholar Kogut, J. & Susskind, L. Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395 (1975).Article ADS Google Scholar Nielsen, H. B. & Ninomiya, M. Absence of neutrinos on a lattice: (I). Proof by homotopy theory. Nucl. Phys. B 185, 20–40 (1981).Article ADS MathSciNet Google Scholar Jordan, S. P., Lee, K. S. & Preskill, J. Quantum computation of scattering in scalar quantum field theories. Quant. Inf. Comput. 14, 1014–1080 (2011).MathSciNet Google Scholar Jordan, S. P., Lee, K. S. & Preskill, J. Quantum algorithms for quantum field theories. Science 336, 1130–1133 (2012).Article ADS Google Scholar Jordan, S. P., Krovi, H., Lee, K. S. & Preskill, J. BQP-completeness of scattering in scalar quantum field theory. Quantum 2, 44 (2018).Article Google Scholar Jordan, S. P., Lee, K. S. & Preskill, J. Quantum algorithms for fermionic quantum field theories. Preprint at https://doi.org/10.48550/arXiv.1404.7115 (2014).García-Álvarez, L. et al. Fermion-fermion scattering in quantum field theory with superconducting circuits. Phys. Rev. Lett. 114, 070502 (2015).Article ADS Google Scholar Zohar, E. & Cirac, J. I. Removing staggered fermionic matter in U(N) and SU(N) lattice gauge theories. Phys. Rev. D 99, 114511 (2019).Article ADS MathSciNet Google Scholar Lamm, H., Lawrence, S., Yamauchi, Y. & Collaboration, N. General methods for digital quantum simulation of gauge theories. Phys. Rev. D 100, 034518 (2019).Article ADS MathSciNet Google Scholar Banuls, M. C. et al. Simulating lattice gauge theories within quantum technologies. Eur. Phys. J. D 74, 1–42 (2020).Article Google Scholar Kan, A. & Nam, Y. Lattice quantum chromodynamics and electrodynamics on a universal quantum computer. Quant. Sci. Technol. https://doi.org/10.1088/2058-9565/aca0b8 (2021).Tong, Y., Albert, V. V., McClean, J. R., Preskill, J. & Su, Y. Provably accurate simulation of gauge theories and bosonic systems. Quantum 6, 816 (2022).Article Google Scholar Bauer, C. W. et al. Quantum simulation for high-energy physics. PRX Quantum 4, 027001 (2023).Article ADS Google Scholar Bauer, C. W., Davoudi, Z., Klco, N. & Savage, M. J. Quantum simulation of fundamental particles and forces. Nat. Rev. Phys. 5, 420–432 (2023).Article Google Scholar Irmejs, R., Bañuls, M.-C. & Cirac, J. I. Quantum simulation of \({{\rm{{\mathbb{Z}}}}}_{2}\) lattice gauge theory with minimal resources. Phys. Rev. D 108, 074503 (2023).Article ADS MathSciNet Google Scholar Lamm, H., Li, Y.-Y., Shu, J., Wang, Y.-L. & Xu, B. Block encodings of discrete subgroups on a quantum computer. Phys. Rev. D 110, 054505 (2024).Article ADS MathSciNet Google Scholar Rhodes, M. L., Kreshchuk, M. & Pathak, S. Exponential improvements in the simulation of lattice gauge theories using near-optimal techniques. PRX Quantum 5, 040347 (2024).Article ADS Google Scholar Watson, J. D. et al. Quantum algorithms for simulating nuclear effective field theories. Preprint at https://doi.org/10.48550/arXiv.2312.05344 (2023).Schwinger, J. Gauge invariance and mass. II. Phys. Rev. 128, 2425 (1962).Article ADS MathSciNet Google Scholar Kühn, S., Cirac, J. I. & Bañuls, M.-C. Quantum simulation of the Schwinger model: a study of feasibility. Phys. Rev. A 90, 042305 (2014).Article ADS Google Scholar Alexandru, A. et al. Gluon field digitization for quantum computers. Phys. Rev. D 100, 114501 (2019).Article ADS Google Scholar Davoudi, Z., Raychowdhury, I. & Shaw, A. Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories. Phys. Rev. D 104, 074505 (2021).Article ADS Google Scholar Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution. Preprint at https://doi.org/10.48550/arXiv.quant-ph/0001106 (2000).Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).Article ADS MathSciNet Google Scholar Kato, T. On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn 5, 435–439 (1950).Article ADS Google Scholar Lee, S. et al. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nat. Commun. 14, 1952 (2023).Article ADS Google Scholar Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Preprint at https://doi.org/10.48550/arXiv.quant-ph/9511026 (1995).Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum metropolis sampling. Nature 471, 87–90 (2011).Article ADS Google Scholar Chen, C.-F., Kastoryano, M. J., Brandão, F. G. & Gilyén, A. Quantum thermal state preparation. Preprint at https://doi.org/10.48550/arXiv.2303.18224 (2023).Chen, C.-F., Kastoryano, M. J. & Gilyén, A. An efficient and exact noncommutative quantum Gibbs sampler. Preprint at https://doi.org/10.48550/arXiv.2311.09207 (2023).Cubitt, T. S. Dissipative ground state preparation and the dissipative quantum eigensolver. Preprint at https://doi.org/10.48550/arXiv.2303.11962 (2023).Zhang, D., Bosse, J. L. & Cubitt, T. Dissipative quantum Gibbs sampling. Preprint at https://doi.org/10.48550/arXiv.2304.04526 (2023).Endo, S., Kurata, I. & Nakagawa, Y. O. Calculation of the Green’s function on near-term quantum computers. Phys. Rev. Res. 2, 033281 (2020).Article Google Scholar Keen, T., Dumitrescu, E. & Wang, Y. Quantum algorithms for ground-state preparation and Green’s function calculation. Preprint at https://doi.org/10.48550/arXiv.2112.05731 (2021).Irmejs, R. & Santos, R. A. Approximating dynamical correlation functions with constant depth quantum circuits. Quantum 9, 1639 (2025).Article Google Scholar Piccinelli, S., Tacchino, F., Tavernelli, I. & Carleo, G. Efficient calculation of Green’s functions on quantum computers via simultaneous circuit perturbation. Preprint at https://doi.org/10.48550/arXiv.2505.05563 (2025).Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).Article Google Scholar Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 9–24 (2023).Article Google Scholar Busch, P. Informationally complete sets of physical quantities. Int. J. Theor. Phys. 30, 1217–1227 (1991).Article MathSciNet Google Scholar Zhao, A., Rubin, N. C. & Miyake, A. Fermionic partial tomography via classical shadows. Phys. Rev. Lett. 127, 110504 (2021).Article ADS MathSciNet Google Scholar Wan, K., Huggins, W. J., Lee, J. & Babbush, R. Matchgate shadows for fermionic quantum simulation. Commun. Math. Phys. 404, 629–700 (2023).Article ADS MathSciNet Google Scholar Low, G. H. Classical shadows of fermions with particle number symmetry. Preprint at https://doi.org/10.48550/arXiv.2208.08964 (2022).King, R., Gosset, D., Kothari, R. & Babbush, R. Triply efficient shadow tomography. PRX Quantum 6, 010336 (2025).Article ADS Google Scholar Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).Article ADS MathSciNet Google Scholar Suzuki, M. General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991).Article ADS MathSciNet Google Scholar Campbell, E. Random compiler for fast Hamiltonian simulation. Phys. Rev. Lett. 123, 070503 (2019).Article ADS Google Scholar Childs, A. M., Ostrander, A. & Su, Y. Faster quantum simulation by randomization. Quantum 3, 182 (2019).Article Google Scholar Chen, C.-F., Huang, H.-Y., Kueng, R. & Tropp, J. A. Concentration for random product formulas. PRX Quantum 2, 040305 (2021).Article ADS Google Scholar Faehrmann, P. K., Steudtner, M., Kueng, R., Kieferova, M. & Eisert, J. Randomizing multi-product formulas for Hamiltonian simulation. Quantum 6, 806 (2022).Article Google Scholar Cho, C.-H., Berry, D. W. & Hsieh, M.-H. Doubling the order of approximation via the randomized product formula. Phys. Rev. A 109, 062431 (2024).Article ADS MathSciNet Google Scholar Kivlichan, I. D. et al. Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via Trotterization. Quantum 4, 296 (2020).Article Google Scholar Bosse, J. L. et al. Efficient and practical Hamiltonian simulation from time-dependent product formulas. Nat. Commun. 16, 2673 (2025).Article ADS Google Scholar Childs, A. M. & Su, Y. Nearly optimal lattice simulation by product formulas. Phys. Rev. Lett. 123, 050503 (2019).Article ADS MathSciNet Google Scholar Childs, A. M., Su, Y., Tran, M. C., Wiebe, N. & Zhu, S. Theory of Trotter error with commutator scaling. Phys. Rev. X 11, 011020 (2021).

Google Scholar Low, G. H. & Chuang, I. L. Optimal Hamiltonian simulation by quantum signal processing. Phys. Rev. Lett. 118, 010501 (2017).Article ADS MathSciNet Google Scholar Low, G. H. & Chuang, I. L. Hamiltonian simulation by qubitization. Quantum 3, 163 (2019).Article Google Scholar Gilyén, A., Su, Y., Low, G. H. & Wiebe, N. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing 193–204 (ACM, 2019).Motlagh, D. & Wiebe, N. Generalized quantum signal processing. PRX Quantum 5, 020368 (2024).Article ADS Google Scholar Childs, A. M. & Wiebe, N. Hamiltonian simulation using linear combinations of unitary operations. Quant. Inf. Comput. 12, 901– 924 (2012).MathSciNet Google Scholar Babbush, R. et al. Encoding electronic spectra in quantum circuits with linear T complexity. Phys. Rev. X 8, 041015 (2018).

Google Scholar Berry, D. W. & Childs, A. M. Black-box Hamiltonian simulation and unitary implementation. Quant. Inf. Comput. 12, 29–62 (2009).MathSciNet Google Scholar Berry, D. W. et al. Improved techniques for preparing eigenstates of fermionic Hamiltonians. npj Quantum Inf. 4, 1–7 (2018).Article Google Scholar Babbush, R., Berry, D. W., McClean, J. R. & Neven, H. Quantum simulation of chemistry with sublinear scaling in basis size. npj Quantum Inf. 5, 92 (2019).Article ADS Google Scholar Su, Y., Berry, D. W., Wiebe, N., Rubin, N. & Babbush, R. Fault-tolerant quantum simulations of chemistry in first quantization. PRX Quantum 2, 040332 (2021).Article ADS Google Scholar Babbush, R. et al. Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods. Nat. Commun. 14, 4058 (2023).Article ADS Google Scholar Georges, T. N. et al. Quantum simulations of chemistry in first quantization with any basis set. npj Quantum Inf. 11, 55 (2025).Article ADS Google Scholar Wick, G. C., Wightman, A. S. & Wigner, E. P.

The Intrinsic Parity of Elementary Particles 102–106 (Springer, 1997).Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002).Article ADS MathSciNet Google Scholar Chien, R. W. & Klassen, J. Optimizing fermionic encodings for both Hamiltonian and hardware. Preprint at https://doi.org/10.48550/arXiv.2210.05652 (2022).Cobanera, E., Ortiz, G. & Nussinov, Z. The bond-algebraic approach to dualities. Adv. Phys. 60, 679–798 (2011).Article ADS Google Scholar Jordan, P. & Wigner, E. About the Pauli exclusion principle. Z. Phys. 47, 631–651 (1928).Article ADS Google Scholar Lieb, E., Schultz, T. & Mattis, D. Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961).Article ADS MathSciNet Google Scholar Chiew, M. & Strelchuk, S. Discovering optimal fermion–qubit mappings through algorithmic enumeration. Quantum 7, 1145 (2023).Article Google Scholar Parella-Dilmé, T. et al. Reducing entanglement with physically inspired fermion-to-qubit mappings. PRX Quantum 5, 030333 (2024).Article ADS Google Scholar Jones, N. C. et al. Faster quantum chemistry simulation on fault-tolerant quantum computers. New J. Phys. 14, 115023 (2012).Article Google Scholar Wan, K. Exponentially faster implementations of Select(H) for fermionic Hamiltonians. Quantum 5, 380 (2021).Article Google Scholar Seeley, J. T., Richard, M. J., Love, P. J. & Love, P. J. The Bravyi-Kitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 137, 224109 (2012).Article ADS Google Scholar Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New J. Phys. 20, 063010 (2018).Article ADS Google Scholar Bravyi, S., Gambetta, J. M., Mezzacapo, A. & Temme, K. Tapering off qubits to simulate fermionic Hamiltonians. Preprint at https://doi.org/10.48550/arXiv.1701.08213 (2017).Wang, Q., Li, M., Monroe, C. & Nam, Y. Resource-optimized fermionic local-Hamiltonian simulation on a quantum computer for quantum chemistry. Quantum 5, 509 (2021).Article Google Scholar Wang, Q., Cian, Z.-P., Li, M., Markov, I. L. & Nam, Y. Ever more optimized simulations of fermionic systems on a quantum computer. In 2023 60th ACM/IEEE Design Automation Conference (DAC) 1–6 (IEEE, 2023).Miller, A., Zimborás, Z., Knecht, S., Maniscalco, S. & García-Pérez, G. Bonsai algorithm: grow your own fermion-to-qubit mappings. PRX Quantum 4, 030314 (2023).Article ADS Google Scholar Miller, A., Glos, A. & Zimborás, Z. Treespilation: architecture- and state-optimised fermion-to-qubit mappings. npj Quant. Inf. https://doi.org/10.1038/s41534-025-01170-2 (2024).Jiang, Z., Kalev, A., Mruczkiewicz, W. & Neven, H. Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning. Quantum 4, 276 (2020).Article Google Scholar Vlasov, A. Y. Clifford algebras, spin groups and qubit trees. Quanta 11, 97–114 (2022).Article Google Scholar Chiew, M., Harrison, B. & Strelchuk, S. Ternary tree transformations are equivalent to linear encodings of the Fock basis. Preprint at https://doi.org/10.48550/arXiv.2412.07578 (2024).McDowall-Rose, H., Shaikh, R. A. & Yeh, L. From fermions to qubits: a ZX-calculus perspective. Preprint at https://doi.org/10.48550/arXiv.2505.06212 (2025).Havlíček, V., Troyer, M. & Whitfield, J. D. Operator locality in the quantum simulation of fermionic models. Phys. Rev. A 95, 032332 (2017).Article ADS Google Scholar Harrison, B. et al. A Sierpinski triangle fermion-to-qubit transform. Preprint at https://doi.org/10.48550/arXiv.2409.04348 (2024).Yu, J., Liu, Y., Sugiura, S., Voorhis, T. V. & Zeytinoğlu, S. Clifford circuit based heuristic optimization of fermion-to-qubit mappings. J. Chem. Theory Comput. 21, 9430–9443 (2025).Article Google Scholar Arcos, M., Apel, H. & Cubitt, T. Encodings of observable subalgebras. Preprint at https://doi.org/10.48550/arXiv.2502.20591 (2025).Setia, K. et al. Reducing qubit requirements for quantum simulations using molecular point group symmetries. J. Chem. Theory Comput. 16, 6091–6097 (2020).Article Google Scholar Shee, Y., Tsai, P.-K., Hong, C.-L., Cheng, H.-C. & Goan, H.-S. Qubit-efficient encoding scheme for quantum simulations of electronic structure. Phys. Rev. Res. 4, 023154 (2022).Article Google Scholar Harrison, B., Nelson, D., Adamiak, D. & Whitfield, J. Reducing the qubit requirement of Jordan-Wigner encodings of N-mode, K-fermion systems from N to \(\lceil {\log }_{2}(\begin{array}{c}N\\ K\end{array})\rceil \). Preprint at https://doi.org/10.48550/arXiv.2211.04501 (2023).Kirby, W., Fuller, B., Hadfield, C. & Mezzacapo, A. Second-quantized fermionic operators with polylogarithmic qubit and gate complexity. PRX Quantum 3, 020351 (2022).Article ADS Google Scholar Carolan, J. & Schaeffer, L. Succinct fermion data structures. In 16th Innovations in Theoretical Computer Science Conference (ed. Meka, R.) Vol. 325 (ITCS, 2025).Lieb, E. H. & Robinson, D. W. The finite group velocity of quantum spin system