- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Decoherence-free quantum error mitigation by density matrix vectorization

Zhong-Xia Shang1,2,3,*, Zi-Han Chen1,2,3,†, and Cai-Sheng Cheng4,‡

  • *Contact author: ustcszx@mail.ustc.edu.cn
  • †Contact author: czh007@mail.ustc.edu.cn
  • ‡Contact author: ccs112202@mail.ustc.edu.cn
PDF

Phys. Rev. Research 8, 023206 – Published 26 May, 2026

DOI: https://doi.org/10.1103/kzyj-j3g8

Abstract

Fighting against noise is crucial for Noisy Intermediate Scale Quantum (NISQ) devices to demonstrate practical quantum applications. In this work, we give a paradigm of quantum error mitigation based on the vectorization of density matrices for variational quantum algorithms. Different from the ideas of existing quantum error mitigation methods that try to distill noiseless information from noisy quantum states, our proposal directly changes the way of encoding information and maps the density matrices of noisy quantum states to noiseless virtual pure states, which is realized by an NISQ-friendly measurement protocol and a classical postprocessing procedure. Our protocol requires no knowledge of the noise model, no ability to tune the noise strength, and no ancilla qubits for complicated controlled unitaries. Under our encoding, NISQ devices are always preparing virtual pure quantum states that are highly desired resources for variational quantum algorithms to have good performance in many tasks. We show how this protocol can be well fitted into variational quantum algorithms. We give several concrete ansatz constructions that are suitable for our proposal and do theoretical analysis on the sampling complexity, the expressibility, and the trainability. We also give a discussion on how this protocol is influenced by large noise and how it can be well combined with other quantum error mitigation protocols. The effectiveness of our proposal is demonstrated by various numerical experiments.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (86)

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  2. R. P. Feynman, Simulating physics with computers, in Feynman and Computation, edited by A. J. G. Hey (CRC Press, Boca Raton, FL, 2018), pp. 133–153.
  3. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev. 41, 303 (1999).
  4. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 1996), pp. 212–219.
  5. W. G. Unruh, Maintaining coherence in quantum computers, Phys. Rev. A 51, 992 (1995).
  6. P. W. Shor, Fault-tolerant quantum computation, in Proceedings of the 37th Conference on Foundations of Computer Science (IEEE, 1996), pp. 56–65.
  7. D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 1997), pp. 176–188.
  8. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  9. S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  10. H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).
  11. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  12. D. Gottesman, Fault-tolerant quantum computation with constant overhead, Quantum Inf. Comput. 14, 1338 (2014).
  13. D. Litinski, A game of surface codes: Large-scale quantum computing with lattice surgery, Quantum 3, 128 (2019).
  14. Z. Chen, K. J. Satzinger, J. Atalaya, A. N. Korotkov, A. Dunsworth, D. Sank, C. Quintana, M. McEwen, R. Barends, P. V Klimov, et al., Exponential suppression of bit or phase flip errors with repetitive error correction, Nature (London) 595, 383 (2021).
  15. A. Erhard, H. P. Nautrup, M. Meth, L. Postler, R. Stricker, M. Stadler, V. Negnevitsky, M. Ringbauer, P. Schindler, H. J. Briegel, et al., Entangling logical qubits with lattice surgery, Nature (London) 589, 220 (2021).
  16. C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Realization of real-time fault-tolerant quantum error correction, Phys. Rev. X 11, 041058 (2021).
  17. S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, et al., Realizing repeated quantum error correction in a distance-three surface code, Nature (London) 605, 669 (2022).
  18. Y. Zhao, Y. Ye, H.-L. Huang, Y. Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao, et al., Realization of an error-correcting surface code with superconducting qubits, Phys. Rev. Lett. 129, 030501 (2022).
  19. Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  20. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2024).
  21. M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, A. Chernoguzov, J. M. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, D. Hayes, N. Hewitt, et al., Demonstration of logical qubits and repeated error correction with better-than-physical error rates, arXiv:2404.02280.
  22. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  23. A. Montanaro, Quantum algorithms: An overview, npj Quantum Inf. 2, 15023 (2016).
  24. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. GSL Brandao, D. A. Buell, et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  25. H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  26. K. Bharti, A. Cervera-Lierta, T. Ha Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S Kottmann, T. Menke, et al., Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  27. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  28. 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).
  29. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  30. M. Schuld and N. Killoran, Quantum machine learning in feature Hilbert spaces, Phys. Rev. Lett. 122, 040504 (2019).
  31. X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019).
  32. D. S. França and R. Garcia-Patron, Limitations of optimization algorithms on noisy quantum devices, Nat. Phys. 17, 1221 (2021).
  33. G. De Palma, M. Marvian, C. Rouzé, and D. S. França, Limitations of variational quantum algorithms: A quantum optimal transport approach, PRX Quantum 4, 010309 (2023).
  34. S. Sim, P. D. Johnson, and A. Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technol. 2, 1900070 (2019).
  35. S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97, 050401 (2006).
  36. S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nat. Commun. 12, 6961 (2021).
  37. 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).
  38. Z.-X. Shang, M.-C. Chen, X. Yuan, C.-Y. Lu, and J.-W. Pan, Schrödinger-Heisenberg variational quantum algorithms, Phys. Rev. Lett. 131, 060406 (2023).
  39. J. Huang, W. He, Y. Zhang, Y. Wu, B. Wu, and X. Yuan, Tensor-network-assisted variational quantum algorithm, Phys. Rev. A 108, 052407 (2023).
  40. K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  41. S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018).
  42. Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
  43. S. McArdle, X. Yuan, and S. Benjamin, Error-mitigated digital quantum simulation, Phys. Rev. Lett. 122, 180501 (2019).
  44. 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, Phys. Rev. A 95, 042308 (2017).
  45. 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).
  46. Z. Cai, Quantum error mitigation using symmetry expansion, Quantum 5, 548 (2021).
  47. Z. Liu, X. Zhang, Y.-Y. Fei, and Z. Cai, Virtual channel purification, PRX Quantum 6, 020325 (2025).
  48. Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  49. R. Takagi, S. Endo, S. Minagawa, and M. Gu, Fundamental limits of quantum error mitigation, npj Quantum Inf. 8, 114 (2022).
  50. R. Takagi, H. Tajima, and M. Gu, Universal sampling lower bounds for quantum error mitigation, Phys. Rev. Lett. 131, 210602 (2023).
  51. K. Tsubouchi, T. Sagawa, and N. Yoshioka, Universal cost bound of quantum error mitigation based on quantum estimation theory, Phys. Rev. Lett. 131, 210601 (2023).
  52. Y. Quek, D. S. França, S. Khatri, J. J. Meyer, and J. Eisert, Exponentially tighter bounds on limitations of quantum error mitigation, Nat. Phys. 20, 1648 (2024).
  53. Z.-X. Shang, Z.-H. Chen, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, A polynomial-time quantum algorithm for solving the ground states of a class of classically hard Hamiltonians, arXiv:2401.13946.
  54. N. Yoshioka, Y. O. Nakagawa, K. Mitarai, and K. Fujii, Variational quantum algorithm for nonequilibrium steady states, Phys. Rev. Res. 2, 043289 (2020).
  55. Z.-X. Shang, Hermitian-preserving ansatz and variational open quantum eigensolver, Phys. Rev. A 109, 062608 (2024).
  56. A. Gilchrist, D. R. Terno, and C. J. Wood, Vectorization of quantum operations and its use, arXiv:0911.2539.
  57. D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 2006), pp. 427–436.
  58. H. Buhrman, R. Cleve, J. Watrous, and R. De Wolf, Quantum fingerprinting, Phys. Rev. Lett. 87, 167902 (2001).
  59. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
  60. J. C. Garcia-Escartin and P. Chamorro-Posada, Swap test and Hong-Ou-Mandel effect are equivalent, Phys. Rev. A 87, 052330 (2013).
  61. J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation (California Institute of Technology, Pasadena, CA, 1998), Vol. 16, pp. 1–8.
  62. M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Rényi entropies: A new generalization and some properties, J. Math. Phys. 54, 122203 (2013).
  63. A. A. Mele, Introduction to Haar measure tools in quantum information: A beginner's tutorial, Quantum 8, 1340 (2024).
  64. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems (Springer, New York, 2019).
  65. W. J. Huggins, J. Lee, U. Baek, B. O’Gorman, and K. B. Whaley, A non-orthogonal variational quantum eigensolver, New J. Phys. 22, 073009 (2020).
  66. K. Bharti and T. Haug, Iterative quantum-assisted eigensolver, Phys. Rev. A 104, L050401 (2021).
  67. K. Bharti and T. Haug, Quantum-assisted simulator, Phys. Rev. A 104, 042418 (2021).
  68. T. Peng, A. W. Harrow, M. Ozols, and X. Wu, Simulating large quantum circuits on a small quantum computer, Phys. Rev. Lett. 125, 150504 (2020).
  69. A. W. Harrow and A. Lowe, Optimal quantum circuit cuts with application to clustered Hamiltonian simulation, PRX Quantum 6, 010316 (2025).
  70. A. Bouland, T. Giurgică-Tiron, and J. Wright, The state hidden subgroup problem and an efficient algorithm for locating unentanglement, in Proceedings of the 57th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 2025), pp. 463–470.
  71. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  72. T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic-Structure Theory (John Wiley & Sons, England, 2013).
  73. S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
  74. J.-G. Liu, L. Mao, P. Zhang, and L. Wang, Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits, Mach. Learn.: Sci. Technol. 2, 025011 (2021).
  75. Z. Webb, The Clifford group forms a unitary 3-design, Quantum Inf. Comput. 16, 1379 (2016).
  76. V. Balasubramanian, M. DeCross, A. Kar, and O. Parrikar, Binding complexity and multiparty entanglement, J. High Energy Phys. 02 (2019) 069.
  77. A. N. Kolmogorov and V. M. Tikhomirov, ɛ-entropy and ɛ-capacity of sets in function spaces, Usp. Mat. Nauk 14, 3 (1959).
  78. Y. Du, Z. Tu, X. Yuan, and D. Tao, Efficient measure for the expressivity of variational quantum algorithms, Phys. Rev. Lett. 128, 080506 (2022).
  79. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  80. Y. Yan, Z. Du, J. Chen, and X. Ma, Limitations of noisy quantum devices in computational and entangling power, npj Quantum Inf. 11, 188 (2025).
  81. S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Phys. Rev. A 103, 042605 (2021).
  82. A. Cross, The IBM Q experience and Qiskit open-source quantum computing software, in APS March Meeting Abstracts, Vol. 2018 (American Physical Society, College Park, MD, 2018), pp. L58–003.
  83. Z. Chen, DMVQE-h, 4-molecules, GitHub, 2024, https://github.com/Zihan-Chen-PhMA/dmvqe_H4_molecule.
  84. T. Barthel and J. Lu, Fundamental limitations for measurements in quantum many-body systems, Phys. Rev. Lett. 121, 080406 (2018).
  85. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1791 (2021).
  86. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo, A unified theory of barren plateaus for deep parametrized quantum circuits, Nat. Commun. 15, 7172 (2024).

Outline

Information

Phys. Rev. Research 8, 023206– Published 26 May, 2026
  • Received 9 December 2024
  • Accepted 3 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation