- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Scaling Laws of Quantum Information Lifetime in Monitored Quantum Dynamics

Bingzhi Zhang1,*, Fangjun Hu2, Runzhe Mo1, Tianyang Chen2, Hakan E. Türeci2, and Quntao Zhuang1,3,†

  • *Contact author: bingzhiz@usc.edu
  • †Contact author: qzhuang@usc.edu
PDF

Phys. Rev. X 16, 021027 – Published 6 May, 2026

DOI: https://doi.org/10.1103/7717-1mw2

Abstract

Quantum information is typically fragile under measurements and environmental coupling. Remarkably, we find that its lifetime can scale exponentially with system size when the environment is continuously monitored via midcircuit measurements—regardless of bath size. Starting from a maximally entangled state with a reference, we analytically prove this exponential scaling for typical Haar-random unitaries and confirm it through numerical simulations in both random unitary circuits and chaotic Hamiltonian systems. In the absence of bath monitoring, the lifetime exhibits a markedly different scaling: It grows at most linearly—or remains constant—with system size and decays inversely with the bath size. We further extend our findings numerically to a broad class of initial states. In the intermediate regime of partial monitoring, we identify and prove a two-scale transition, where the quantum mutual information decays logarithmically at microscopic timescales but linearly at macroscopic timescales. We discuss implications for monitored quantum circuits in the weak-measurement limit, quantum algorithms such as quantum diffusion models and quantum reservoir computing, and quantum communication. Finally, we experimentally verify the gap of persisted information on IBM Quantum hardware.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum information is typically fragile and prone to rapid decay when exposed to measurements and environmental coupling. By analyzing the scaling laws of quantum information lifetime in monitored quantum dynamics, we find that continuous monitoring via midcircuit measurements allows information to persist for a duration that scales exponentially with system size. We identify a fundamental distinction in decay behaviors: quantum mutual information vanishes logarithmically when measurement trajectories are recorded but decays linearly when they are discarded. These findings suggest that properly utilizing measurement data can significantly enhance the efficiency of generative learning models and optimize memory time in quantum reservoir computing. Our work provides a theoretical framework for preserving coherence in logical quantum spaces, which is essential for realizing scalable, fault-tolerant quantum computation in the presence of noise.

Article Text

References (54)

  1. W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
  2. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernandez, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
  3. Y. Li, X. Chen, and Matthew P. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  4. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  5. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  6. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  7. B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Holographic deep thermalization for secure and efficient quantum random state generation, Nat. Commun. 16, 6341 (2025).
  8. B. Zhang and Q. Zhuang, Anticoncentrated n-bit distribution from log(n) qubits, arXiv:2511.05433.
  9. A. D. Córcoles, M. Takita, K. Inoue, S. Lekuch, Z. K. Minev, J. M. Chow, and J. M. Gambetta, Exploiting dynamic quantum circuits in a quantum algorithm with superconducting qubits, Phys. Rev. Lett. 127, 100501 (2021).
  10. M. DeCross, E. Chertkov, M. Kohagen, and M. Foss-Feig, Qubit-reuse compilation with mid-circuit measurement and reset, Phys. Rev. X 13, 041057 (2023).
  11. E. Bäumer, V. Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, Efficient long-range entanglement using dynamic circuits, PRX Quantum 5, 030339 (2024).
  12. L. Piroli, G. Styliaris, and J. I. Cirac, Approximating many-body quantum states with quantum circuits and measurements, Phys. Rev. Lett. 133, 230401 (2024).
  13. H. Buhrman, M. Folkertsma, B. Loff, and N. M. Neumann, State preparation by shallow circuits using feed forward, Quantum 8, 1552 (2024).
  14. K. C. Smith, A. Khan, B. K. Clark, S. M. Girvin, and T.-C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024).
  15. M. Iqbal, N. Tantivasadakarn, R. Verresen, S. L. Campbell, J. M. Dreiling, C. Figgatt, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses et al., Non-Abelian topological order and anyons on a trapped-ion processor, Nature (London) 626, 505 (2024).
  16. C. Cao and J. Eisert, Measurement-driven quantum advantages in shallow circuits, Phys. Rev. Lett. 136, 080601 (2026).
  17. B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion probabilistic models, Phys. Rev. Lett. 132, 100602 (2024).
  18. J. Chen, H. I. Nurdin, and N. Yamamoto, Temporal information processing on noisy quantum computers, Phys. Rev. Appl. 14, 024065 (2020).
  19. F. Hu, S. A. Khan, N. T. Bronn, G. Angelatos, G. E. Rowlands, G. J. Ribeill, and H. E. Türeci, Overcoming the coherence time barrier in quantum machine learning on temporal data, Nat. Commun. 15, 7491 (2024).
  20. C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem, IEEE Trans. Inf. Theory 48, 2637 (2002).
  21. K. Nakajima, K. Fujii, M. Negoro, K. Mitarai, and M. Kitagawa, Boosting computational power through spatial multiplexing in quantum reservoir computing, Phys. Rev. Appl. 11, 034021 (2019).
  22. K. Fujii and K. Nakajima, Harnessing disordered-ensemble quantum dynamics for machine learning, Phys. Rev. Appl. 8, 024030 (2017).
  23. P. Mujal, R. Martínez-Peña, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Time-series quantum reservoir computing with weak and projective measurements, npj Quantum Inf. 9, 16 (2023).
  24. T. Yasuda, Y. Suzuki, T. Kubota, K. Nakajima, Q. Gao, W. Zhang, S. Shimono, H. I. Nurdin, and N. Yamamoto, Quantum reservoir computing with repeated measurements on superconducting devices, arXiv:2310.06706.
  25. R. Martínez-Peña and J.-P. Ortega, Quantum reservoir computing in finite dimensions, Phys. Rev. E 107, 035306 (2023).
  26. H. Jaeger, Short term memory in echo state networks. GMD-Report 152, GMD-German National Research Institute for Computer Science (2002).
  27. S. Ganguli, D. Huh, and H. Sompolinsky, Memory traces in dynamical systems, Proc. Natl. Acad. Sci. U.S.A. 105, 18970 (2008).
  28. J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, Information processing capacity of dynamical systems, Sci. Rep. 2, 514 (2012).
  29. D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, Memory versus non-linearity in reservoirs, in Proceedings of the 2010 International Joint Conference on Neural Networks (IJCNN) (IEEE, New York, 2010), pp. 1–8.
  30. M. Inubushi and K. Yoshimura, Reservoir computing beyond memory-nonlinearity trade-off, Sci. Rep. 7, 10199 (2017).
  31. E. P. Wigner, Remarks on the mind-body question, in Philosophical Reflections and Syntheses (Springer, New York, 1995), pp. 247–260.
  32. K. Kobayashi, K. Fujii, and N. Yamamoto, Feedback-driven quantum reservoir computing for time-series analysis, PRX Quantum 5, 040325 (2024).

  33. In the context of the NISQRC algorithm of Ref. [19], these are referred to as the Memory (M) and Readout (R) subsystems.

  34. Z. Webb, The Clifford group forms a unitary 3-design, Quantum Inf. Comput. 16, 1379 (2016).
  35. G. Adesso, D. Girolami, and A. Serafini, Measuring Gaussian quantum information and correlations using the Rényi entropy of order 2, Phys. Rev. Lett. 109, 190502 (2012).
  36. A. Hamma, S. M. Giampaolo, and F. Illuminati, Mutual information and spontaneous symmetry breaking, Phys. Rev. A 93, 012303 (2016).
  37. Google Quantum AI and Collaborators, Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature (London) 622, 481 (2023).
  38. R. Kukulski, I. Nechita, Ł. Pawela, Z. Puchała, and K. Życzkowski, Generating random quantum channels, J. Math. Phys. (N.Y.) 62 (2021).
  39. M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  40. S. Liu, M.-R. Li, S.-X. Zhang, S.-K. Jian, and H. Yao, Noise-induced phase transitions in hybrid quantum circuits, Phys. Rev. B 110, 064323 (2024).
  41. N. Hunter-Jones, Unitary designs from statistical mechanics in random quantum circuits, arXiv:1905.12053.
  42. M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, England, 2013).
  43. 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.
  44. A. Elben, B. Vermersch, C. F. Roos, and P. Zoller, Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states, Phys. Rev. A 99, 052323 (2019).
  45. Y. Li, Y. Zou, P. Glorioso, E. Altman, and Matthew P. Fisher, Cross entropy benchmark for measurement-induced phase transitions, Phys. Rev. Lett. 130, 220404 (2023).
  46. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  47. J. Feng and Q. Zhuang, Spectrum and quantum information in reset-driven Floquet Hamiltonian dynamics (to be published).
  48. B. Zhang, F. Hu, R. Mo, T. Chen, H. E. Türeci, and Q. Zhuang, QMI_monitored_dynamics, https://github.com/bzGit06/QMI_monitored_dynamics (2025).
  49. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  50. QuantumSavory, Quantumclifford.jl, https://github.com/QuantumSavory/QuantumClifford.jl (accessed: 2025-11-19).
  51. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09 (2007) 120.
  52. M. Ippoliti and W. W. Ho, Solvable model of deep thermalization with distinct design times, Quantum 6, 886 (2022).
  53. S.-X. Zhang, J. Allcock, Z.-Q. Wan, S. Liu, J. Sun, H. Yu, X.-H. Yang, J. Qiu, Z. Ye, Y.-Q. Chen et al., Tensorcircuit: A quantum software framework for the NISQ era, Quantum 7, 912 (2023).
  54. R. Belyansky, P. Bienias, Y. A. Kharkov, A. V. Gorshkov, and B. Swingle, Minimal model for fast scrambling, Phys. Rev. Lett. 125, 130601 (2020).

Outline

Information

Phys. Rev. X 16, 021027– Published 6 May, 2026
  • Received 28 June 2025
  • Accepted 1 April 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