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  • Open Access

Large-Scale Implementation of Quantum Subspace Expansion with Classical Shadows

Laurin E. Fischer1,2,*, Daniel Bultrini3, Timothée Dao1,4, Ivano Tavernelli1, and Francesco Tacchino1,†,‡

  • *Contact author: aur@zurich.ibm.com
  • †Contact author: francesco.tacchino@unipv.it
  • ‡Present address: Università di Pavia, Pavia, Italy.
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Phys. Rev. Lett. 137, 160602 – Published 15 October, 2026

DOI: https://doi.org/10.1103/2w21-qs38

Abstract

Quantum subspace expansion (QSE) offers promising avenues to perform spectral calculations on quantum processors but comes with a large measurement overhead. Informationally complete measurements, such as classical shadows, were recently proposed to overcome this bottleneck. Here, we report the first large-scale implementation of QSE with informationally complete measurements. In particular, we probe the quantum phase transition of a spin model with three-body interactions, for which we observe accurate ground state energy recovery and mitigation of local order parameters across system sizes of up to 80 qubits. We achieve this by reformulating QSE as a constrained optimization problem, obtaining rigorous statistical error estimates and avoiding numerical ill conditioning. With over 3×104 measurement basis randomizations per circuit and the evaluation of O(1014) Pauli traces, this represents one of the most significant realizations of classical shadows to date.

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References (53)

  1. J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
  2. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  3. J. F. Gonthier, M. D. Radin, C. Buda, E. J. Doskocil, C. M. Abuan, and J. Romero, Measurements as a roadblock to near-term practical quantum advantage in chemistry: Resource analysis, Phys. Rev. Res. 4, 033154 (2022).
  4. M. Motta, W. Kirby, I. Liepuoniute, K. J. Sung, J. Cohn, A. Mezzacapo, K. Klymko, N. Nguyen, N. Yoshioka, and J. E. Rice, Subspace methods for electronic structure simulations on quantum computers, Electron. Struct. 6, 013001 (2024).
  5. K. Kanno, M. Kohda, R. Imai, S. Koh, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, Quantum-selected configuration interaction: Classical diagonalization of hamiltonians in subspaces selected by quantum computers, Phys. Rev. Res. 8, 023268 (2026).
  6. J. Robledo-Moreno, M. Motta, H. Haas, A. Javadi-Abhari, P. Jurcevic, W. Kirby, S. Martiel, K. Sharma, S. Sharma, T. Shirakawa, I. Sitdikov, R.-Y. Sun, K. J. Sung, M. Takita, M. C. Tran, S. Yunoki, and A. Mezzacapo, Chemistry beyond the scale of exact diagonalization on a quantum-centric supercomputer, Sci. Adv. 11, eadu9991 (2025).
  7. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  8. G. García-Pérez, E.-M. Borrelli, M. Leahy, J. Malmi, S. Maniscalco, M. A. C. Rossi, B. Sokolov, and D. Cavalcanti, Virtual linear map algorithm for classical boost in near-term quantum computing, arXiv:2207.01360.
  9. S. Filippov, M. Leahy, M. A. Rossi, and G. García-Pérez, Scalable tensor-network error mitigation for near-term quantum computing, arXiv:2307.11740.
  10. S.-X. Zhang, Z.-Q. Wan, C.-K. Lee, C.-Y. Hsieh, S. Zhang, and H. Yao, Variational quantum-neural hybrid eigensolver, Phys. Rev. Lett. 128, 120502 (2022).
  11. N. Yoshioka, H. Hakoshima, Y. Matsuzaki, Y. Tokunaga, Y. Suzuki, and S. Endo, Generalized quantum subspace expansion, Phys. Rev. Lett. 129, 020502 (2022).
  12. 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, Phys. Rev. X 10, 011004 (2020).
  13. M. Urbanek, D. Camps, R. Van Beeumen, and W. A. De Jong, Chemistry on quantum computers with virtual quantum subspace expansion, J. Chem. Theory Comput. 16, 5425 (2020).
  14. X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O’Brien, Low-cost error mitigation by symmetry verification, Phys. Rev. A 98, 062339 (2018).
  15. Z. Cai, Quantum error mitigation using symmetry expansion, Quantum 5, 548 (2021).
  16. P. Suchsland, F. Tacchino, M. H. Fischer, T. Neupert, P. K. Barkoutsos, and I. Tavernelli, Algorithmic error mitigation scheme for current quantum processors, Quantum 5, 492 (2021).
  17. P. J. Ollitrault, A. Kandala, C.-F. Chen, P. K. Barkoutsos, A. Mezzacapo, M. Pistoia, S. Sheldon, S. Wörner, J. M. Gambetta, and I. Tavernelli, Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor, Phys. Rev. Res. 2, 043140 (2020).
  18. J. Selisko, M. Amsler, C. Wever, Y. Kawashima, G. Samsonidze, R. U. Haq, F. Tacchino, I. Tavernelli, and T. Eckl, Dynamical mean field theory for real materials on a quantum computer, arXiv:2404.09527.
  19. G. García-Pérez, M. A. Rossi, B. Sokolov, F. Tacchino, P. K. Barkoutsos, G. Mazzola, I. Tavernelli, and S. Maniscalco, Learning to measure: Adaptive informationally complete generalized measurements for quantum algorithms, PRX Quantum 2, 040342 (2021).
  20. S. Choi and A. F. Izmaylov, Measurement optimization techniques for excited electronic states in near-term quantum computing algorithms, J. Chem. Theory Comput. 19, 3184 (2023).
  21. L. E. Fischer, M. Leahy, A. Eddins, N. Keenan, D. Ferracin, M. A. Rossi, Y. Kim, A. He, F. Pietracaprina, B. Sokolov et al., Dynamical simulations of many-body quantum chaos on a quantum computer, Nat. Phys. 1 (2026).
  22. M. Votto, M. Ljubotina, C. Lancien, J. I. Cirac, P. Zoller, M. Serbyn, L. Piroli, and B. Vermersch, Learning mixed quantum states in large-scale experiments, Phys. Rev. Lett. 136, 090801 (2026).
  23. G. Boyd, B. Koczor, and Z. Cai, High-dimensional subspace expansion using classical shadows, Phys. Rev. A 111, 022423 (2025).
  24. 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).
  25. M. Scheer, A. Baiardi, E. Bäumer Marty, Z.-Y. Wei, and D. Malz, Renormalization-group-based preparation of matrix product states on up to 80 qubits, arXiv:2510.24681.
  26. L. E. Fischer, D. Miller, F. Tacchino, P. K. Barkoutsos, D. J. Egger, and I. Tavernelli, Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space, Phys. Rev. Res. 4, 033027 (2022).
  27. L. Innocenti, S. Lorenzo, I. Palmisano, F. Albarelli, A. Ferraro, M. Paternostro, and G. M. Palma, Shadow tomography on general measurement frames, PRX Quantum 4, 040328 (2023).
  28. E. N. Epperly, L. Lin, and Y. Nakatsukasa, A theory of quantum subspace diagonalization, SIAM J. Matrix Anal. Appl. 43, 1263 (2022).
  29. See Supplemental Material at http://link.aps.org/supplemental/10.1103/2w21-qs38 for details on the hardware demonstration, classical postprocessing, bootstrap validation of statistical uncertainties, and an application to molecular systems with fermionic shadows, which includes Refs. [30–35].
  30. 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.
  31. S. K. Lam, A. Pitrou, and S. Seibert, Numba: A LLVM-based Python JIT compiler, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC (Association for Computing Machinery, New York, NY, USA, 2015), pp. 1–6.
  32. J. Lee, W. J. Huggins, M. Head-Gordon, and K. B. Whaley, Generalized unitary coupled cluster wave functions for quantum computation, J. Chem. Theory Comput. 15, 311 (2019).
  33. P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Z. Phys. 51, 521 (1928).
  34. 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).
  35. A. Zhao and A. Miyake, Group-theoretic error mitigation enabled by classical shadows and symmetries, npj Quantum Inf. 10, 57 (2024).
  36. G. van Kempen and L. van Vliet, Mean and variance of ratio estimators used in fluorescence ratio imaging, Cytometry 39, 300 (2000).
  37. W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Phys. Rev. X 11, 041036 (2021).
  38. M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum phase transitions in matrix product systems, Phys. Rev. Lett. 97, 110403 (2006).
  39. A. Smith, B. Jobst, A. G. Green, and F. Pollmann, Crossing a topological phase transition with a quantum computer, Phys. Rev. Res. 4, L022020 (2022).
  40. M. Motta, C. Sun, A. T. K. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandão, and G. K.-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205 (2020).
  41. K. Klymko, C. Mejuto-Zaera, S. J. Cotton, F. Wudarski, M. Urbanek, D. Hait, M. Head-Gordon, K. B. Whaley, J. Moussa, N. Wiebe, W. A. de Jong, and N. M. Tubman, Real-time evolution for ultracompact hamiltonian eigenstates on quantum hardware, PRX Quantum 3, 020323 (2022).
  42. W. Kirby, M. Motta, and A. Mezzacapo, Exact and efficient Lanczos method on a quantum computer, Quantum 7, 1018 (2023).
  43. M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, in Advances in Optimization and Numerical Analysis, edited by S. Gomez and J.-P. Hennart (Springer Netherlands, Dordrecht, 1994), pp. 51–67.
  44. E. Van Den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nat. Phys. 19, 1116 (2023).
  45. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme et al., Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  46. L. E. Fischer, T. Dao, I. Tavernelli, and F. Tacchino, Dual-frame optimization for informationally complete quantum measurements, Phys. Rev. A 109, 062415 (2024).
  47. J. Malmi, K. Korhonen, D. Cavalcanti, and G. García-Pérez, Enhanced observable estimation through classical optimization of informationally overcomplete measurement data: Beyond classical shadows, Phys. Rev. A 109, 062412 (2024).
  48. A. Caprotti, J. Morris, and B. Dakić, Optimizing quantum tomography via shadow inversion, Phys. Rev. Res. 6, 033301 (2024).
  49. C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eisert, and H. Pashayan, Shallow shadows: Expectation estimation using low-depth random Clifford circuits, Phys. Rev. Lett. 133, 020602 (2024).
  50. H.-Y. Hu, A. Gu, S. Majumder, H. Ren, Y. Zhang, D. S. Wang, Y.-Z. You, Z. Minev, S. F. Yelin, and A. Seif, Demonstration of robust and efficient quantum property learning with shallow shadows, Nat. Commun. 16, 2943 (2025).
  51. A. Zhao, N. C. Rubin, and A. Miyake, Fermionic partial tomography via classical shadows, Phys. Rev. Lett. 127, 110504 (2021).
  52. K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Matchgate shadows for fermionic quantum simulation, Commun. Math. Phys. 404, 629 (2023).
  53. L. E. Fischer, B. Daniel, T. Dao, I. Tavernelli, and F. Tacchino Raw data for “Large-scale implementation of quantum subspace expansion with classical shadows,” (2026), 10.5281/zenodo.20323161.

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Phys. Rev. Lett. 137, 160602– Published 15 October, 2026
  • Received 4 December 2025
  • Revised 22 May 2026
  • Accepted 24 August 2026
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