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  • Open Access

Measurement-induced phase transitions in quantum inference problems and quantum hidden Markov models

Sun Woo P. Kim* and Curt von Keyserlingk

Austen Lamacraft

  • *Contact author: swk34@cantab.ac.uk
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Phys. Rev. Research 8, 023155 – Published 11 May, 2026

DOI: https://doi.org/10.1103/b3s8-y1wb

Abstract

Recently, there is interest in coincident “sharpening” and “learnability” transitions in monitored quantum systems. In the latter, an outside observer's ability to infer properties of a quantum system from measurements undergoes a phase transition. Such transitions appear to be related to the decodability transition in quantum error correction, but the precise connection is not clear. Here, we study these problems under one framework we call the general quantum inference problem. In cases as above where the system has a Markov structure, we say that the inference is on a quantum hidden Markov model. We show a formal connection to classical hidden Markov models and that they coincide for certain setups. For example, we prove this for those involving Haar-random unitaries and measurements. We introduce the notion of Bayes nonoptimality, where parameters used for inference differs from true ones. This allows us to expand the phase diagrams of above models. At Bayes optimality, we obtain an explicit relation between “sharpening” and “learnability” order parameters, explicitly showing that the two transitions coincide. Next, we study concrete examples. We review quantum error correction on the toric and repetition code and their mapping to two-dimensional (2D) random-bond Ising model (RBIM) through our framework. We study the Haar-random U(1)-symmetric monitored quantum circuit and tree, mapping each to inference models that we call the planted simple symmetric exclusion process and planted xor, respectively, and expanding the phase diagram to Bayes nonoptimality. For the circuit, we deduce the phase boundary numerically and analytically argue that it is of a single universality class. For the tree, we present an exact solution of the entire phase boundary, which displays re-entrance as does the 2D RBIM. We discuss these phase diagrams, with their interpretations for quantum inference problems and rigorous arguments on their shapes.

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References (39)

  1. U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Phys. Rev. X 12, 041002 (2022).
  2. U. Agrawal, J. Lopez-Piqueres, R. Vasseur, S. Gopalakrishnan, and A. C. Potter, Observing quantum measurement collapse as a learnability phase transition, Phys. Rev. X 14, 041012 (2024).
  3. F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Phys. Rev. Lett. 129, 120604 (2022).
  4. F. Barratt, U. Agrawal, A. C. Potter, S. Gopalakrishnan, and R. Vasseur, Transitions in the learnability of global charges from local measurements, Phys. Rev. Lett. 129, 200602 (2022).
  5. Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  6. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  7. L. Rabiner and B. Juang, An introduction to hidden Markov models, Ieee Assp Magazine 3, 4 (1986).
  8. S. W. P. Kim and A. Lamacraft, Planted directed polymer: Inferring a random walk from noisy images, Phys. Rev. E 111, 024135 (2025).
  9. R. L. Stratonovich and V. Tikhonov, Non-Linear Transformations of Stochastic Processes (Pergamon Press, Oxford, London, 1965).
  10. A. Nahum and J. L. Jacobsen, Bayesian critical points in classical lattice models, Phys. Rev. B 112, 235113 (2025).
  11. L. Zdeborová and F. Krzakala, Statistical physics of inference: Thresholds and algorithms, Adv. Phys. 65, 453 (2016).
  12. C. M. Bishop and N. M. Nasrabadi, Pattern Recognition and Machine Learning (Springer Science+Business Media, LLC, New York, NY, 2006), Vol. 4.
  13. H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes (Kappa Research, LLC, Blue Bell, PA, 2014), Chap. 9.1.5.
  14. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  15. A. Monras, A. Beige, and K. Wiesner, Hidden quantum Markov models and non-adaptive read-out of many-body states, arXiv:1002.2337.
  16. B. Collins, S. Matsumoto, and J. Novak, The weingarten calculus, Not. Ame. Math. Soc. 69, 1 (2022).
  17. A. Barchielli, L. Lanz, and G. M. Prosperi, A model for the macroscopic description and continual observations in quantum mechanics, Nuovo Cim. B 72, 79 (1982).
  18. K. Binder and D. W. Heermann, Theoretical foundations of the Monte Carlo method and its applications in statistical physics, in Monte Carlo Simulation in Statistical Physics, edited by W. T. Rhodes and H. E. Stanley, Graduate Texts in Physics (Springer, Berlin, Heidelberg, 2010), pp. 5–67.
  19. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  20. F. Merz and J. T. Chalker, Two-dimensional random-bond Ising model, free fermions, and the network model, Phys. Rev. B 65, 054425 (2002).
  21. M. Ohzeki and J. L. Jacobsen, High-precision phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials, J. Phys. A: Math. Theor. 48, 095001 (2015).
  22. K. Fujii and Y. Tokunaga, Error and loss tolerances of surface codes with general lattice structures, Phys. Rev. A 86, 020303(R) (2012).
  23. J. Edmonds, Paths, trees, and flowers, Can. J. Math. 17, 449 (1965).
  24. H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Clarendon Press, Oxford, UK, 2001), Vol. 111, p. 17.
  25. D. S. Fisher and D. A. Huse, Equilibrium behavior of the spin-glass ordered phase, Phys. Rev. B 38, 386 (1988).
  26. T. M. Liggett, Interacting Particle Systems (Springer Science & Business Media, Berlin, Heidelberg, 1985), Vol. 2, pp. 361–417.
  27. A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, UK, 2010), pp. 134–142.
  28. D. Tong, Lectures on Statistical Field Theory (Cambridge University, Cambridge, UK, 2017), p. 125.
  29. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  30. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  31. A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in landau-ginsburg theory, PRX Quantum 2, 010352 (2021).
  32. X. Feng, B. Skinner, and A. Nahum, Measurement-induced phase transitions on dynamical quantum trees, PRX Quantum 4, 030333 (2023).
  33. X. Feng, N. Fishchenko, S. Gopalakrishnan, and M. Ippoliti, Charge and spin sharpening transitions on dynamical quantum trees, Quantum 9, 1692 (2025).
  34. B. Derrida and H. Spohn, Polymers on disordered trees, spin glasses, and traveling waves, J. Stat. Phys. 51, 817 (1988).
  35. This is the solution to (1−2λ)λ+(2+λ−λ2)ln(3/[2+λ−λ2]), which comes from minimizing Eq. (102) while ignoring the first term in the RHS.
  36. G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori's cat: Stable long-range entanglement from finite-depth unitaries and weak measurements, Phys. Rev. Lett. 131, 200201 (2023).
  37. M. Christandl and R. Renner, Reliable quantum state tomography, Phys. Rev. Lett. 109, 120403 (2012).
  38. A. deMarti iOlius, P. Fuentes, R. Orús, P. M. Crespo, and J. Etxezarreta Martinez, Decoding algorithms for surface codes, Quantum 8, 1498 (2024).
  39. J. F. Kam, S. Gicev, K. Modi, A. Southwell, and M. Usman, Detrimental non-Markovian errors for surface code memory, Quantum Sci. Technol. 10, 035060 (2025).

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Phys. Rev. Research 8, 023155– Published 11 May, 2026
  • Received 8 May 2025
  • Accepted 15 January 2026
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