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Distributions of noisy expectation values over sets of measurement operators

Matthew Duschenes1,2,3,*, Roger G. Melko1,3, Juan Carrasquilla1,4, and Raymond Laflamme1,2,3,†

  • *Contact author: mduschen@uwaterloo.ca
  • †Deceased.
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Phys. Rev. A 114, 022434 – Published 17 August, 2026

DOI: https://doi.org/10.1103/qp99-qqgy

Abstract

Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum information sciences. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding nonunitary dynamics, we generalize previous derivations for distributions of expectation values [Campos et al., Phys. Lett. A 377, 1854 (2013)] to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behavior across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of nonsymmetric measurement operators are shown to exhibit distinct multimodal distributions relative to unimodal distributions for symmetric measurement operators, opening up questions about their simulability.

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References (100)

  1. D. M. Appleby, Å. Ericsson, and C. A. Fuchs, Properties of QBist state spaces, Found. Phys. 41, 564 (2011).
  2. M. Appleby, C. A. Fuchs, B. C. Stacey, and H. Zhu, Introducing the Qplex: A novel arena for quantum theory, Eur. Phys. J. D 71, 197 (2017).
  3. A. J. Scott, Tight informationally complete quantum measurements, J. Phys. A: Math. Gen. 39, 13507 (2006).
  4. V. I. Yashin, E. O. Kiktenko, A. S. Mastiukova, and A. K. Fedorov, Minimal informationally complete measurements for probability representation of quantum dynamics, New J. Phys. 22, 103026 (2020).
  5. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004).
  6. H. J. D. Miller, Statistical mechanics of random mixed state ensembles with fixed energy, arXiv:2508.00809.
  7. K. Życzkowski and H.-J. Sommers, Average fidelity between random quantum states, Phys. Rev. A 71, 032313 (2005).
  8. H. Sommers and K. Zyczkowski, Bures volume of the set of mixed quantum states, J. Phys. A: Math. Gen. 36, 10083 (2003).
  9. J. J. Meyer, Fisher information in noisy intermediate-scale quantum applications, Quantum 5, 539 (2021).
  10. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What limits the simulation of quantum computers? Phys. Rev. X 10, 041038 (2020).
  11. H.-J. Sommers and K. Zyczkowski, Statistical properties of random density matrices, J. Phys. A: Math. Gen. 37, 8457 (2004).
  12. B. Fefferman, S. Ghosh, M. Gullans, K. Kuroiwa, and K. Sharma, Effect of nonunital noise on random-circuit sampling, PRX Quantum 5030317 (2024).
  13. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  14. M. Duschenes, D. García-Martín, Z. Holmes, and M. Cerezo, Moments of quantum channel ensembles, arXiv:2511.12700.
  15. J. Emerson, R. Alicki, and K. Zyczkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).
  16. Y. Quek, D. Stilck França, S. Khatri, J. J. Meyer, and J. Eisert, Exponentially tighter bounds on limitations of quantum error mitigation, Nat. Phys. 20, 1648 (2024).
  17. S. Wang, P. Czarnik, A. Arrasmith, M. Cerezo, L. Cincio, and P. J. Coles, Can error mitigation improve trainability of noisy variational quantum algorithms? Quantum 8, 1287 (2024).
  18. J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. Laflamme, Symmetrized characterization of noisy quantum processes, Science 317, 1893 (2007).
  19. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  20. M. Zhang, C. Wang, and Y. Han, Noisy random quantum circuit sampling and its classical simulation, Adv. Quantum Technol. 6, 2300030 (2023).
  21. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  22. Z. Cheng and M. Ippoliti, Efficient sampling of noisy shallow circuits via monitored unraveling, PRX Quantum 4, 040326 (2023).
  23. S. Czischek, G. Torlai, S. Ray, R. Islam, and R. G. Melko, Simulating a measurement-induced phase transition for trapped ion circuits, Phys. Rev. A 104, 062405 (2021).
  24. 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, et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  25. S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near-term devices, Nat. Phys. 14, 595 (2018).
  26. R. LaRose, A brief history of quantum vs classical computational advantage, arXiv:2412.14703.
  27. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  28. Q. Zhu, S. Cao, F. Chen, M. C. Chen, X. Chen, T. H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, C. Guo, C. Guo, S. Guo, L. Han, L. Hong, H. L. Huang, Y. H. Huo, L. Li, N. Li, S. Li, et al., Quantum computational advantage via 60-qubit 24-cycle random circuit sampling, Sci. Bull. 67, 240 (2022).
  29. G. Torlai, C. J. Wood, A. Acharya, G. Carleo, J. Carrasquilla, and L. Aolita, Quantum process tomography with unsupervised learning and tensor networks, Nat. Commun. 14, 2858 (2023).
  30. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  31. J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nat. Mach. Intell. 1, 155 (2019).
  32. H. Y. Hu, S. Choi, and Y. Z. You, Classical shadow tomography with locally scrambled quantum dynamics, Phys. Rev. Res. 5, 023027 (2023).
  33. F. Hu, G. Angelatos, S. A. Khan, M. Vives, E. Türeci, L. Bello, G. E. Rowlands, G. J. Ribeill, and H. E. Türeci, Tackling sampling noise in physical systems for machine learning applications: Fundamental limits and eigentasks, Phys. Rev. X 13, 041020 (2023).
  34. 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).
  35. V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Real-time quantum error correction beyond break-even, Nature (London) 616, 50 (2023).
  36. M. Duschenes, J. Carrasquilla, and R. Laflamme, Characterization of overparametrization in the simulation of realistic quantum systems, Phys. Rev. A 109, 062607 (2024).
  37. K. Georgopoulos, C. Emary, and P. Zuliani, Modelling and simulating the noisy behavior of near-term quantum computers, Phys. Rev. A 104, 062432 (2021).
  38. X. Ge, R. B. Wu, and H. Rabitz, The optimization landscape of hybrid quantum–classical algorithms: From quantum control to NISQ applications, Annu. Rev. Control 54, 314 (2022).
  39. K. Tsubouchi, Y. Mitsuhashi, K. Sharma, and N. Yoshioka, Symmetric Clifford twirling for cost-optimal quantum error mitigation in early FTQC regime, npj Quantum Inf. 11, 104 (2025).
  40. H. Jnane, J. Steinberg, Z. Cai, H. C. Nguyen, and B. Koczor, Quantum error mitigated classical shadows, PRX Quantum 5, 010324 (2024).
  41. V. B. Bulchandani, S. L. Sondhi, and J. T. Chalker, Random-matrix models of monitored quantum circuits, J. Stat. Phys. 191, 55 (2024).
  42. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  43. Y. Li, X. Chen, and M. P. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  44. T. Heinosaari, M. A. Jivulescu, and I. Nechita, Random positive operator valued measures, J. Math. Phys. 61, 042202 (2020).
  45. A. Acharya, S. Saha, and A. M. Sengupta, Shadow tomography based on informationally complete positive operator-valued measure, Phys. Rev. A 104, 052418 (2021).
  46. A. Anshu and S. Arunachalam, A survey on the complexity of learning quantum states, Nat. Rev. Phys. 6, 59 (2023).
  47. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit quantum-state tomography, Phys. Rev. A 66, 012303 (2002).
  48. D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum state tomography via compressed sensing, Phys. Rev. Lett. 105, 150401 (2010).
  49. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  50. D. E. Koh and S. Grewal, Classical shadows with noise, Quantum 6, 776 (2020).
  51. S. Mullane, Sampling random quantum circuits: A pedestrian's guide, arXiv:2007.07872.
  52. P. W. Claeys and G. De Tomasi, Fock-space delocalization and the emergence of the Porter-Thomas distribution from dual-unitary dynamics, Phys. Rev. Lett. 134, 050405 (2025).
  53. B. Collins, S. Matsumoto, and J. Novak, The Weingarten calculus, Not. Am. Math. Soc. 69, 734 (2022).
  54. R. Kukulski, I. Nechita, L. Pawela, Z. Puchala, and K. Zyczkowski, Generating random quantum channels, J. Math. Phys. 62, 062201 (2021).
  55. D. Belkin, J. Allen, S. Ghosh, C. Kang, S. Lin, J. Sud, F. T. Chong, B. Fefferman, and B. K. Clark, Approximate t-designs in generic circuit architectures, PRX Quantum 5, 040344 (2024).
  56. A. Deshpande, P. Niroula, O. Shtanko, A. V. Gorshkov, B. Fefferman, and M. J. Gullans, Tight bounds on the convergence of noisy random circuits to the uniform distribution, PRX Quantum 3, 040329 (2022).
  57. M. Kus, J. Mostowski, and F. Haake, Universality of eigenvector statistics of kicked tops of different symmetries, J. Phys. A: Math. Gen. 21, L1073 (1988).
  58. A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandão, Random quantum circuits anticoncentrate in log depth, PRX Quantum 3, 010333 (2022).
  59. L. Campos Venuti and P. Zanardi, Probability density of quantum expectation values, Phys. Lett. A 377, 1854 (2013).
  60. C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak, Z. Puchała, and K. Życzkowski, Numerical shadow and geometry of quantum states, J. Phys. A: Math. Theor. 44, 335301 (2011).
  61. E. Gutkin and K. Zyczkowski, Joint numerical ranges, quantum maps, and joint numerical shadows, Linear Algebra Appl. 438, 2394 (2013).
  62. L. Zhang, S. Luo, S.-M. Fei, and J. Wu, Uncertainty regions of observables and state-independent uncertainty relations, Quantum Info. Proc. 20, 357 (2021).
  63. A. Sauliere, B. Magni, G. Lami, X. Turkeshi, and J. De Nardis, Universality in the anticoncentration of chaotic quantum circuits, Phys. Rev. B 112, 134312 (2025).
  64. W. G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104, 250501 (2010).
  65. A. Sauliere, G. Lami, C. Boyer, J. De Nardis, and A. De Luca, Universality in the anticoncentration of noisy quantum circuits at finite depths, PRX Quantum 7, 020365 (2026).
  66. H. Lóio, G. Cecile, S. Gopalakrishnan, G. Lami, and J. De Nardis, Correlations, spectra, and entanglement transitions in ensembles of matrix product states, Phys. Rev. B 112, 035127 (2025).
  67. A. De Luca, C. Liu, A. Nahum, and T. Zhou, Universality classes for purification in nonunitary quantum processes, Phys. Rev. X 15, 041024 (2025).
  68. B. Collins and S. Matsumoto, Weingarten calculus via orthogonality relations: New applications, Lat. Am. J. Probab. Math. Stat. 14, 631 (2017).
  69. B. Collins and P. Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Commun. Math. Phys. 264, 773 (2006).
  70. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  71. S. U. Lee, C. Oh, Y. Wong, S. Chen, and L. Jiang, Universal spreading of conditional mutual information in noisy random circuits, Phys. Rev. Lett. 133, 200402 (2024).
  72. S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X 14, 031044 (2024).
  73. M. Zhang, C. Wang, S. Dong, H. Zhang, Y. Han, and L. He, Entanglement entropy scaling of noisy random quantum circuits in two dimensions, Phys. Rev. A 106, 052430 (2022).
  74. J. Denzler, J. Carrasco, J. Eisert, and T. Guaita, Simulation of noisy quantum circuits using frame representations, arXiv:2601.05131.
  75. J. Bai, J. Wang, and Z. Yin, Primitivity for random quantum channels, Quantum Info. Proc. 23, 47 (2024).
  76. B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57, 015215 (2016).
  77. K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, J. Phys. A: Math. Gen. 34, 7111 (2001).
  78. B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Commun. Math. Phys. 297, 345 (2010).
  79. K. Życzkowski, K. A. Penson, I. Nechita, and B. Collins, Generating random density matrices, J. Math. Phys. 52, 062201 (2011).
  80. S. G. Bobkov and F. Götze, Concentration of empirical distribution functions with applications to non-i.i.d. models, Bernoulli 16, 1385 (2010).
  81. F. Perez-Cruz, Kullback-Leibler divergence estimation of continuous distributions, in 2008 IEEE International Symposium on Information Theory (IEEE, Toronto, Canada, 2008), pp. 1666–1670.
  82. D. S. Dimitrova, V. K. Kaishev, and S. Tan, Computing the Kolmogorov-Smirnov distribution when the underlying CDF is purely discrete, mixed, or continuous, J. Stat. Software 95, 1 (2020).
  83. S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, 1st ed. (Birkhäuser, Basel, 1992), pp. 1–184.
  84. M. Duschenes, Simulation: JAX based simulator for quantum systems (2022), https://github.com/mduschenes/simulation.
  85. M. Duschenes, R. Melko, J. Carrasquilla, and R. Laflamme, Data: Distributions of Noisy Expectation Values over Sets of Measurement Operators, Zenodo, 2026, https://doi.org/10.5281/zenodo.18671768.
  86. A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandão, Random quantum circuits transform local noise into global white noise, Commun. Math. Phys. 405 (2021).
  87. M. J. Schlosser, Multiple hypergeometric series: Appell series and beyond, in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, edited by C. Schneider and J. Blümlein (Springer, Vienna, 2013), pp. 305–324.
  88. A. A. Mele, Introduction to Haar measure tools in quantum information: A beginner's tutorial, Quantum 8, 1340 (2024).
  89. M. A. Stephens, Tests based on EDF statistics, in Goodness-of-Fit Techniques, edited by R. B. D'Agostino and M. A. Stephens (Marcel Dekker, New York, 1986), pp. 97–193.
  90. A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. Stilck França, and Y. Quek, Noise-induced shallow circuits and absence of barren plateaus, Nat. Phys. 22, 751 (2026).
  91. B. Magni, A. Christopoulos, A. De Luca, and X. Turkeshi, Anticoncentration in Clifford circuits and beyond: From random tensor networks to pseudo-magic states, Phys. Rev. X 15, 031071 (2025).
  92. M. Ragone, Q. T. Nguyen, L. Schatzki, P. Braccia, M. Larocca, F. Sauvage, P. J. Coles, and M. Cerezo, Representation theory for geometric quantum machine learning, arXiv:2210.07980.
  93. A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, On the complexity and verification of quantum random circuit sampling, Nat. Phys. 15, 159 (2019).
  94. D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Rev. Mod. Phys. 95, 035001 (2023).
  95. A. Zlokapa, B. Villalonga, S. Boixo, and D. A. Lidar, Boundaries of quantum supremacy via random circuit sampling, npj Quantum Inf. 9, 36 (2023).
  96. D. Aharonov, X. Gao, Z. Landau, Y. Liu, and U. Vazirani, A polynomial-time classical algorithm for noisy random circuit sampling, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023 (Association for Computing Machinery, Orlando, USA, 2023), pp. 945–957.
  97. F. Pan, K. Chen, and P. Zhang, Solving the sampling problem of the Sycamore quantum circuits, Phys. Rev. Lett. 129, 090502 (2022).
  98. https://www.vectorinstitute.ai/partnerships/current-partners/.
  99. Y. Virkar and A. Clauset, Power-law distributions in binned empirical data, Ann. Appl. Stat. 8, 89 (2014).
  100. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al., SciPy 1.0: fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).

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Phys. Rev. A 114, 022434– Published 17 August, 2026
  • Received 17 April 2026
  • Accepted 30 July 2026
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