Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang; Amir Kalev; Wojciech Mruczkiewicz; Hartmut Neven

doi:10.22331/q-2020-06-04-276

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang¹, Amir Kalev², Wojciech Mruczkiewicz¹, and Hartmut Neven¹

¹Google Research, Venice, CA 90291
²Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA

Published:	2020-06-04, volume 4, page 276
Eprint:	arXiv:1910.10746v2
Doi:	https://doi.org/10.22331/q-2020-06-04-276
Citation:	Quantum 4, 276 (2020).

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Abstract

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an $n$-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on $\lceil \log_3(2n+1)\rceil$ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than $\log_3(2n)$ qubits on average. We apply it to the problem of learning $k$-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all $k$-fermion RDMs in parallel, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim (2n+1)^k \epsilon^{-2}$ times. This result is based on a method we develop here that allows one to determine individual elements of all $k$-qubit RDMs in parallel, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim 3^k \epsilon^{-2}$ times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

► BibTeX data

@article{Jiang2020optimalfermionto,
  doi = {10.22331/q-2020-06-04-276},
  url = {https://doi.org/10.22331/q-2020-06-04-276},
  title = {Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning},
  author = {Jiang, Zhang and Kalev, Amir and Mruczkiewicz, Wojciech and Neven, Hartmut},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {276},
  month = jun,
  year = {2020}
}

► References

[1] R. P. Feynman, ``Simulating physics with computers,'' International Journal of Theoretical Physics 21, 467 (1982).
https://doi.org/10.1007/BF02650179

[2] S. Lloyd, ``Universal Quantum Simulators,'' Science 273, 1073 (1996).
https://doi.org/10.1126/science.273.5278.1073

[3] I. Georgescu, S. Ashhab, and F. Nori, ``Quantum simulation,'' Reviews of Modern Physics 86, 153 (2014).
https://doi.org/10.1103/RevModPhys.86.153

[4] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, ``Solving strongly correlated electron models on a quantum computer,'' Physical Review A 92, 062318 (2015).
https://doi.org/10.1103/PhysRevA.92.062318

[5] R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, ``Low-Depth Quantum Simulation of Materials,'' Physical Review X 8, 011044 (2018).
https://doi.org/10.1103/PhysRevX.8.011044

[6] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, ``Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions,'' Physical Review Applied 9, 044036 (2018).
https://doi.org/10.1103/PhysRevApplied.9.044036

[7] J. I. Cirac and P. Zoller, ``Quantum computations with cold trapped ions,'' Physical Review Letters 74, 4091 (1995).
https://doi.org/10.1103/PhysRevLett.74.4091

[8] D. Kielpinski, C. Monroe, and D. J. Wineland, ``Architecture for a large-scale ion-trap quantum computer,'' Nature 417, 709 (2002).
https://doi.org/10.1038/nature00784

[9] H. Häffner, C. F. Roos, and R. Blatt, ``Quantum computing with trapped ions,'' Physics Reports 469, 155 (2008).
https://doi.org/10.1016/j.physrep.2008.09.003

[10] M. H. Devoret, A. Wallraff, and J. M. Martinis, ``Superconducting Qubits: A Short Review,'' arXiv:0411174 (2004).
arXiv:cond-mat/0411174

[11] G. Wendin, ``Quantum information processing with superconducting circuits: a review,'' Reports on Progress in Physics. Physical Society (Great Britain) 80, 106001 (2017).
https://doi.org/10.1088/1361-6633/aa7e1a

[12] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'ÄôBrien, ``A variational eigenvalue solver on a photonic quantum processor,'' Nature Communications 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213

[13] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, ``The theory of variational hybrid quantum-classical algorithms,'' New Journal of Physics 18, 023023 (2016).
https://doi.org/10.1088/1367-2630/18/2/023023

[14] M. A. Nielsen, ``The Fermionic canonical commutation relations and the Jordan-Wigner transform,'' Tech. Rep. University of Queensland (2005).
http://michaelnielsen.org/blog/archive/nones/fermions_and_jordan_wigner.pdf

[15] J. T. Seeley, M. J. Richard, and P. J. Love, ``The Bravyi-Kitaev transformation for quantum computation of electronic structure,'' The Journal of Chemical Physics 137, 224109 (2012).
https://doi.org/10.1063/1.4768229

[16] S. B. Bravyi and A. Y. Kitaev, ``Fermionic Quantum Computation,'' Annals of Physics 298, 210 (2002).
https://doi.org/10.1006/aphy.2002.6254

[17] A. Tranter, S. Sofia, J. Seeley, M. Kaicher, J. McClean, R. Babbush, P. V. Coveney, F. Mintert, F. Wilhelm, and P. J. Love, ``The Bravyi-Kitaev transformation: Properties and applications,'' International Journal of Quantum Chemistry 115, 1431 (2015).
https://doi.org/10.1002/qua.24969

[18] V. Havlíček, M. Troyer, and J. D. Whitfield, ``Operator locality in the quantum simulation of fermionic models,'' Physical Review A 95, 032332 (2017).
https://doi.org/10.1103/PhysRevA.95.032332

[19] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandr??, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, ``Quantum supremacy using a programmable superconducting processor,'' Nature 574, 505 (2019).
https://doi.org/10.1038/s41586-019-1666-5

[20] A. Y. Vlasov, ``Clifford algebras, Spin groups and qubit trees,'' arXiv:1904.09912 (2019).
arXiv:1904.09912

[21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 1st ed. (Cambridge University Press, 2000).

[22] S. Sharma, J. K. Dewhurst, N. N. Lathiotakis, and E. K. U. Gross, ``Reduced density matrix functional for many-electron systems,'' Physical Review B 78, 201103 (2008).
https://doi.org/10.1103/PhysRevB.78.201103

[23] M. Fagotti and F. H. L. Essler, ``Reduced density matrix after a quantum quench,'' Physical Review B 87, 245107 (2013).
https://doi.org/10.1103/PhysRevB.87.245107

[24] N. C. Rubin, R. Babbush, and J. McClean, ``Application of fermionic marginal constraints to hybrid quantum algorithms,'' New Journal of Physics 20, 053020 (2018).
https://doi.org/10.1088/1367-2630/aab919

[25] G. Gidofalvi and D. A. Mazziotti, ``Molecular properties from variational reduced-density-matrix theory with three-particle N-representability conditions,'' The Journal of Chemical Physics 126, 024105 (2007).
https://doi.org/10.1063/1.2423008

[26] C. Overy, G. H. Booth, N. S. Blunt, J. J. Shepherd, D. Cleland, and A. Alavi, ``Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo,'' The Journal of Chemical Physics 141, 244117 (2014).
https://doi.org/10.1063/1.4904313

[27] T. E. O'Brien, B. Senjean, R. Sagastizabal, X. Bonet-Monroig, A. Dutkiewicz, F. Buda, L. DiCarlo, and L. Visscher, ``Calculating energy derivatives for quantum chemistry on a quantum computer,'' npj Quantum Information 5, 1 (2019).
https://doi.org/10.1038/s41534-019-0213-4

[28] J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong, ``Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states,'' Physical Review A 95, 042308 (2017).
https://doi.org/10.1103/PhysRevA.95.042308

[29] T. Takeshita, N. C. Rubin, Z. Jiang, E. Lee, R. Babbush, and J. R. McClean, ``Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources,'' Physical Review X 10, 011004 (2020).
https://doi.org/10.1103/PhysRevX.10.011004

[30] J. Cotler and F. Wilczek, ``Quantum Overlapping Tomography,'' Physical Review Letters 124, 100401 (2020).
https://doi.org/10.1103/PhysRevLett.124.100401

[31] X. Bonet-Monroig, R. Babbush, and T. E. O'Brien, ``Nearly optimal measurement scheduling for partial tomography of quantum states,'' arXiv:1908.05628 (2019).
arXiv:1908.05628

[32] J. Řeháček, B.-G. Englert, and D. Kaszlikowski, ``Minimal qubit tomography,'' Phys. Rev. A 70, 052321 (2004).
https://doi.org/10.1103/PhysRevA.70.052321

[33] I. Hamamura and T. Imamichi, ``Efficient evaluation of quantum observables using entangled measurements,'' arXiv:1909.09119 (2019).
arXiv:1909.09119

[34] A. F. Izmaylov, T.-C. Yen, R. A. Lang, and V. Verteletskyi, ``Unitary Partitioning Approach to the Measurement Problem in the Variational Quantum Eigensolver Method,'' Journal of Chemical Theory and Computation 16, 190 (2020).
https://doi.org/10.1021/acs.jctc.9b00791

[35] A. Zhao, A. Tranter, W. M. Kirby, S. F. Ung, A. Miyake, and P. Love, ``Measurement reduction in variational quantum algorithms,'' arXiv:1908.08067 (2019).
arXiv:1908.08067

[36] W. J. Huggins, J. McClean, N. Rubin, Z. Jiang, N. Wiebe, K. B. Whaley, and R. Babbush, ``Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers,'' arXiv:1907.13117 (2019).
arXiv:1907.13117

[37] D. A. Mazziotti, ed., Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (Wiley-Interscience, 2009).

[38] M. Steudtner and S. Wehner, ``Quantum codes for quantum simulation of fermions on a square lattice of qubits,'' Physical Review A 99, 022308 (2019).
https://doi.org/10.1103/PhysRevA.99.022308

[39] K. Setia and J. D. Whitfield, ``Bravyi-Kitaev Superfast simulation of electronic structure on a quantum computer,'' The Journal of Chemical Physics 148, 164104 (2018).
https://doi.org/10.1063/1.5019371

[40] Z. Jiang, J. McClean, R. Babbush, and H. Neven, ``Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations,'' Physical Review Applied 12, 064041 (2019).
https://doi.org/10.1103/PhysRevApplied.12.064041

[41] C. A. Fuchs, M. C. Hoang, and B. C. Stacey, ``The SIC question: History and state of play,'' Axioms 6, 21 (2017).
https://doi.org/10.3390/axioms6030021

[42] H. B. Dang, K. Blanchfield, I. Bengtsson, and D. M. Appleby, ``Linear dependencies in Weyl–Heisenberg orbits,'' Quantum Information Processing 12, 3449 (2013).
https://doi.org/10.1007/s11128-013-0609-6

Cited by

[1] Irma Avdic, Yuchen Wang, Michael Rose, Lillian I. Payne Torres, Anna O. Schouten, Kevin J. Sung, and David A. Mazziotti, "Constrained shadow tomography for molecular simulation on quantum devices", Chemical Science 17 19, 9633 (2026).

[2] Guillermo García-Pérez, Oskari Kerppo, Matteo A. C. Rossi, and Sabrina Maniscalco, "Experimentally accessible nonseparability criteria for multipartite-entanglement-structure detection", Physical Review Research 5 1, 013226 (2023).

[3] Valentin Heyraud, Héloise Chomet, and Jules Tilly, "Unified framework for matchgate classical shadows", npj Quantum Information 11 1, 65 (2025).

[4] Jacob Bringewatt and Zohreh Davoudi, "Parallelization techniques for quantum simulation of fermionic systems", Quantum 7, 975 (2023).

[5] Matteo Ippoliti, "Classical shadows based on locally-entangled measurements", Quantum 8, 1293 (2024).

[6] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, "Fermionic Partial Tomography via Classical Shadows", Physical Review Letters 127 11, 110504 (2021).

[7] Mitchell Chiew, Cameron Ibrahim, Ilya Safro, and Sergii Strelchuk, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 560 (2025) ISBN:979-8-3315-5736-2.

[8] Zohreh Davoudi, Indrakshi Raychowdhury, and Andrew Shaw, "Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories", Physical Review D 104 7, 074505 (2021).

[9] Michael P. Kaicher, Davide Vodola, and Simon B. Jäger, "Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states", Physical Review B 107 16, 165144 (2023).

[10] Ilya G. Ryabinkin, Seyyed Mehdi Hosseini Jenab, and Scott N. Genin, "Optimization of the Qubit Coupled Cluster Ansatz on Classical Computers", Journal of Chemical Theory and Computation 21 13, 6421 (2025).

[11] Anton Nykänen, Matteo A. C. Rossi, Elsi-Mari Borrelli, Sabrina Maniscalco, Guillermo García-Pérez, and Adam Glos, "Mitigating the measurement overhead of ADAPT-VQE with optimized informationally complete generalized measurements", Physical Review Research 7 4, 043114 (2025).

[12] Riley W. Chien, Mitchell Chiew, Brent Harrison, Jason Necaise, Weishi Wang, Maryam Mudassar, Campbell McLauchlan, Thomas M. Henderson, Gustavo E. Scuseria, Sergii Strelchuk, and James D. Whitfield, "Simulating fermions with a digital quantum computer", Nature Reviews Physics 8 3, 131 (2026).

[13] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[14] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[15] Lane G. Gunderman, "Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers", Physical Review A 107 6, 062416 (2023).

[16] Jeffery Yu, Yuan Liu, Sho Sugiura, Troy Van Voorhis, and Sina Zeytinoğlu, "Clifford Circuit-Based Heuristic Optimization of Fermion-To-Qubit Mappings", Journal of Chemical Theory and Computation 21 19, 9430 (2025).

[17] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[18] Ludwig Nützel, Michael J Hartmann, Henrik Dreyer, and Etienne Granet, "Ground state energy via adiabatic evolution and phase measurement for a molecular Hamiltonian on an ion-trap quantum computer", Quantum Science and Technology 11 3, 035038 (2026).

[19] Robbie King, Guang Hao Low, Ryan Babbush, Rolando D. Somma, and Nicholas C. Rubin, "Quantum Simulation with Sum-of-Squares Spectral Amplification", Physical Review Letters 136 11, 110601 (2026).

[20] Qingfeng Wang, Ze-Pei Cian, Ming Li, Igor L. Markov, and Yunseong Nam, 2023 60th ACM/IEEE Design Automation Conference (DAC) 1 (2023) ISBN:979-8-3503-2348-1.

[21] Guillermo García-Pérez, Matteo A.C. Rossi, Boris Sokolov, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Guglielmo Mazzola, Ivano Tavernelli, and Sabrina Maniscalco, "Learning to Measure: Adaptive Informationally Complete Generalized Measurements for Quantum Algorithms", PRX Quantum 2 4, 040342 (2021).

[22] Robbie King, David Gosset, Robin Kothari, and Ryan Babbush, "Triply Efficient Shadow Tomography", PRX Quantum 6 1, 010336 (2025).

[23] Yiyuan Chen, Jonas Helsen, and Maris Ozols, "Trotter error and gate complexity of the SYK and sparse SYK models", Quantum 10, 1999 (2026).

[24] Aaron Miller, Zoltán Zimborás, Stefan Knecht, Sabrina Maniscalco, and Guillermo García-Pérez, "Bonsai Algorithm: Grow Your Own Fermion-to-Qubit Mappings", PRX Quantum 4 3, 030314 (2023).

[25] Cristian A. Galvis-Florez, Daniel Reitzner, and Simo Särkkä, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 111 (2023) ISBN:979-8-3503-4323-6.

[26] Emiliia Dyrenkova, Raymond Laflamme, and Michael Vasmer, "Scalable simulation of fermionic encoding performance on noisy quantum computers", Physical Review A 113 2, 022607 (2026).

[27] Aaron Miller, Adam Glos, and Zoltán Zimborás, "Treespilation: architecture- and state-optimised fermion-to-qubit mappings", npj Quantum Information 12 1, 26 (2026).

[28] Samuel J. Elman, Adrian Chapman, and Steven T. Flammia, "Free Fermions Behind the Disguise", Communications in Mathematical Physics 388 2, 969 (2021).

[29] Nils Herrmann, Daanish Arya, Marcus W. Doherty, Angus Mingare, Jason C. Pillay, Florian Preis, and Stefan Prestel, 2023 IEEE International Conference on Quantum Software (QSW) 162 (2023) ISBN:979-8-3503-0479-4.

[30] Joonas Malmi, Keijo Korhonen, Daniel Cavalcanti, and Guillermo García-Pérez, "Enhanced observable estimation through classical optimization of informationally overcomplete measurement data: Beyond classical shadows", Physical Review A 109 6, 062412 (2024).

[31] Thomas M. Henderson, Brent Harrison, Ilias Magoulas, Jason Necaise, Andrew M. Projansky, Francesco A. Evangelista, James D. Whitfield, and Gustavo E. Scuseria, "Fermionic mean-field theory as a tool for studying spin Hamiltonians", The Journal of Chemical Physics 161 23, 234112 (2024).

[32] Xianzhi Huang, Lili Niu, Jie Chen, Lichao Li, Kashif Hayat, and Weiping Liu, "Quantum and classical computational synergy for emerging contaminants management: Advanced insights into cytochrome P450 metabolic mechanisms", Critical Reviews in Environmental Science and Technology 54 24, 1827 (2024).

[33] Rahul Sarkar and Theodore J. Yoder, "The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems", Quantum 8, 1307 (2024).

[34] Andrew M Projansky, Jason Necaise, and James D Whitfield, "Gaussianity and simulability of Cliffords and matchgates", Journal of Physics A: Mathematical and Theoretical 58 19, 195302 (2025).

[35] Chenfeng Cao, Hiroshi Yano, and Yuya O. Nakagawa, "Accelerated variational quantum eigensolver with joint Bell measurement", Physical Review Research 6 1, 013205 (2024).

[36] Daniel McNulty, Susane Calegari, and Michał Oszmaniec, "Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables", Quantum 9, 1914 (2025).

[37] Yu-An Chen, Alexey V. Gorshkov, and Yijia Xu, "Error-correcting codes for fermionic quantum simulation", SciPost Physics 16 1, 033 (2024).

[38] Yuhao Liu, Shize Che, Junyu Zhou, Yunong Shi, and Gushu Li, Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 3 382 (2024) ISBN:9798400703867.

[39] Rodolfo Carobene, Andrea Giachero, Stefano Barison, and Jannes Nys, "Local fermion-to-qudit mappings: A practical recipe for four-level systems", Physical Review A 112 3, 032619 (2025).

[40] Yutaka Shikano, Hiroshi C. Watanabe, Ken M. Nakanishi, and Yu-ya Ohnishi, "Post-Hartree–Fock method in quantum chemistry for quantum computer", The European Physical Journal Special Topics 230 4, 1037 (2021).

[41] Kade Head-Marsden, Johannes Flick, Christopher J. Ciccarino, and Prineha Narang, "Quantum Information and Algorithms for Correlated Quantum Matter", Chemical Reviews 121 5, 3061 (2021).

[42] Nicolas Poirier, Jakob S. Kottmann, Alán Aspuru‐Guzik, Luc Mongeau, and Alireza Najafi‐Yazdi, "Range‐separated density functional theory using multiresolution analysis and quantum computing", Journal of Computational Chemistry 45 23, 1987 (2024).

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[44] Leonardo Zambrano, Sergi Ramos-Calderer, and Richard Kueng, "Fast quantum measurement tomography with optimal error bounds", Quantum 10, 2162 (2026).

[45] Tommaso Guaita, "On the locality of qubit encodings of local fermionic modes", Quantum 9, 1644 (2025).

[46] Bruna G. M. Araújo, Márcio M. Taddei, Daniel Cavalcanti, and Antonio Acín, "Local quantum overlapping tomography", Physical Review A 106 6, 062441 (2022).

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[50] James Brown, Marc P Coons, Erika Lloyd, Alexandre Fleury, Krzysztof Bieniasz, Valentin Senicourt, and Arman Zaribafiyan, "Iterative qubit coupled cluster using only clifford circuits", Physica Scripta 100 10, 105401 (2025).

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[57] Daniele Morrone, N. Walter Talarico, Marco Cattaneo, and Matteo A. C. Rossi, "Estimating Molecular Thermal Averages with the Quantum Equation of Motion and Informationally Complete Measurements", Entropy 26 9, 722 (2024).

[58] Wei Zhao, Tangjie Weng, Yue Ruan, Zhi Liu, Xuangou Wu, Xiao Zheng, and Nei Kato, "Quantum Computing in Wireless Communications and Networking: A Tutorial-cum-Survey", IEEE Communications Surveys & Tutorials 27 4, 2378 (2025).

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[60] ACS In Focus (2025) ISBN:9780841295964.

[61] Joris Kattemölle and Jasper van Wezel, "Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice", Physical Review B 106 21, 214429 (2022).

[62] Etienne Granet, Yuta Kikuchi, Henrik Dreyer, and Enrico Rinaldi, "Simulating sparse SYK model with a randomized algorithm on a trapped-ion quantum computer", npj Quantum Information 12 1, 43 (2026).

[63] Ikko Hamamura and Takashi Imamichi, "Efficient evaluation of quantum observables using entangled measurements", npj Quantum Information 6 1, 56 (2020).

[64] Qing-Song Li, Huan-Yu Liu, Qingchun Wang, Yu-Chun Wu, and Guo-Ping Guo, "Huffman-Code-Based Ternary Tree Transformation", Chinese Physics Letters 42 10, 100001 (2025).

[65] Mitchell Chiew and Sergii Strelchuk, "Discovering optimal fermion-qubit mappings through algorithmic enumeration", Quantum 7, 1145 (2023).

[66] Stefano Battaglia, Max Rossmannek, Vladimir V. Rybkin, Ivano Tavernelli, and Jürg Hutter, "A general framework for active space embedding methods with applications in quantum computing", npj Computational Materials 10 1, 297 (2024).

[67] Michael Lubasch, Yuta Kikuchi, Lewis Wright, and Conor Mc Keever, "Quantum circuits for partial differential equations in Fourier space", Physical Review Research 7 4, 043326 (2025).

[68] Zhou You, Qing Liu, and You Zhou, "Circuit optimization of informationally complete positive operator–valued qubit measurements for shadow estimation", Physical Review Applied 23 1, 014021 (2025).

[69] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", Reviews of Modern Physics 92 1, 015003 (2020).

[70] Adrian Chapman, Samuel J. Elman, and Ryan L. Mann, "A Unified Graph-Theoretic Framework for Free-Fermion Solvability", arXiv:2305.15625, (2023).

[71] Adrian Chapman and Steven T. Flammia, "Characterization of solvable spin models via graph invariants", Quantum 4, 278 (2020).

[72] Andrew J. Landahl and Benjamin C. A. Morrison, "Logical fermions for fault-tolerant quantum simulation", arXiv:2110.10280, (2021).

[73] Alexander Yu. Vlasov, "Clifford algebras, Spin groups and qubit trees", arXiv:1904.09912, (2019).

[74] Adrian Chapman, Steven T. Flammia, and Alicia J. Kollár, "Free-Fermion Subsystem Codes", PRX Quantum 3 3, 030321 (2022).

[75] Yuhao Liu, Kevin Yao, Jonathan Hong, Julien Froustey, Ermal Rrapaj, Costin Iancu, Gushu Li, and Yunong Shi, "HATT: Hamiltonian Adaptive Ternary Tree for Optimizing Fermion-to-Qubit Mapping", arXiv:2409.02010, (2024).

[76] Yiyuan Chen, Jonas Helsen, and Maris Ozols, "Trotter error and gate complexity of the SYK and sparse SYK models", arXiv:2502.18420, (2025).

[77] Mitchell Chiew, Cameron Ibrahim, Ilya Safro, and Sergii Strelchuk, "Optimal fermion-qubit mappings via quadratic assignment", arXiv:2504.21636, (2025).

[78] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

[79] Haytham McDowall-Rose, Razin A. Shaikh, and Lia Yeh, "From Fermions to Qubits: A ZX-Calculus Perspective", arXiv:2505.06212, (2025).

[80] James Brown, Tarini S Hardikar, Kenny Heitritter, and Kanav Setia, "Efficient and Noise-Resilient Molecular Quantum Simulation with the Generalized Superfast Encoding", arXiv:2511.09322, (2025).

[81] Laurin E. Fischer, "Enabling large-scale digital quantum simulations with superconducting qubits", arXiv:2602.04719, (2026).

[82] Alexander Yu. Vlasov, "Mutual transformations of arbitrary ternary qubit trees by Clifford gates", arXiv:2404.16693, (2024).

[83] D. E. Fisher, S. A. Fldzhyan, D. V. Minaev, S. S. Straupe, and M. Yu. Saygin, "Improving fermionic variational quantum eigensolvers with Majorana swap networks", arXiv:2509.07855, (2025).

[84] V. Anurag K. S., Ashish Kumar Patra, Vikas Dattatraya Ghevade, Sai Shankar P., Ruchika Bhat, V. Raghavendra, Rahul Maitra, and Jaiganesh G, "Resource Estimation for VQE on Small Molecules: Impact of Fermion Mappings and Hamiltonian Reductions", arXiv:2512.01605, (2025).

[85] James Brown, Erika Lloyd, Alexandre Fleury, Tarini S Hardikar, and Kenny Heitritter, "Reducing entanglement with a Hamiltonian derived Clifford transformation", arXiv:2607.18404, (2026).

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