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arxiv: 2507.11619 · v2 · submitted 2025-07-15 · 🪐 quant-ph · cond-mat.stat-mech

Rise and fall of nonstabilizerness via random measurements

Annarita Scocco , Wai-Keong Mok , Leandro Aolita , Mario Collura , Tobias Haug This is my paper

Pith reviewed 2026-05-19 03:49 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords nonstabilizernessmagicmonitored quantum circuitsClifford unitariesstabilizer nullityprojective measurementsquantum resources
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0 comments X

The pith

Measurements in rotated non-Clifford bases drive random Clifford circuits to a steady state with non-trivial nonstabilizerness independent of the initial state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how nonstabilizerness evolves in circuits of random Clifford unitaries interspersed with local projective measurements. For measurements in the computational basis, an analytical model shows the stabilizer nullity decays in quantized steps and vanishes only after exponentially many measurements because of scrambling protection. In contrast, measurements in rotated non-Clifford bases both create and destroy nonstabilizerness, producing a long-time steady state whose value does not depend on whether the circuit began in a Haar-random or stabilizer state. Haar-random states reach this equilibrium in constant time while stabilizer states require a relaxation time linear in system size. Stabilizer nullity is insensitive to the rotation angle, yet Stabilizer Rényi Entropies reveal additional structure in the approach to the steady state.

Core claim

In monitored circuits with random Clifford unitaries and measurements in rotated non-Clifford bases, the dynamics leads to a steady state with non-trivial nonstabilizerness that is independent of the initial state. Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. For computational-basis measurements the stabilizer nullity instead decays in quantized steps and requires exponentially many measurements to vanish.

What carries the argument

Stabilizer nullity, which tracks nonstabilizerness and decays in discrete steps under computational-basis measurements but saturates at a non-zero value under rotated non-Clifford measurements.

If this is right

  • Nonstabilizerness can be maintained indefinitely rather than eliminated when measurements are performed in non-Clifford bases.
  • Stabilizer states lose their special structure more slowly than Haar-random states under the same non-Clifford measurement protocol.
  • Coarse diagnostics such as stabilizer nullity miss dynamical features that finer measures like Stabilizer Rényi Entropies capture.
  • Clifford scrambling provides strong protection against the loss of nonstabilizerness under standard-basis measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The steady-state nonstabilizerness may set a practical limit on how much magic can be preserved in measurement-based quantum computation protocols.
  • Similar monitored dynamics could be examined with other non-Clifford gate ensembles to test whether the emergence of a non-trivial steady state is generic.
  • Finite-size correlations that violate the full-scrambling assumption may smooth the quantized decay steps in small systems.

Load-bearing premise

The analytical model for the quantized decay of stabilizer nullity assumes that each random Clifford unitary fully scrambles the state before the next measurement occurs.

What would settle it

Numerical simulation of small qubit systems that tracks whether the long-time nonstabilizerness value under rotated measurements remains the same when the circuit is initialized in a stabilizer state versus a random state.

Figures

Figures reproduced from arXiv: 2507.11619 by Annarita Scocco, Leandro Aolita, Mario Collura, Tobias Haug, Wai-Keong Mok.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Average nullity [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Average stabilizer nullity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Average magic density of the steady state [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Von Neumann entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Standard deviation [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (b) we can clearly see that the moment the average approaches its steady value coincides with the moment the plateau structure is loose. For θM = 1 in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Comparison of the average stabilizer nullity [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Dots: numerical average SRE for initial [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Protocol with [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Steady-state SRE [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
read the original abstract

We investigate the dynamics of nonstabilizerness - also known as `magic' - in monitored quantum circuits composed of random Clifford unitaries and local projective measurements. For measurements in the computational basis, we derive an analytical model for dynamics of the stabilizer nullity, showing that it decays in quantized steps and requires exponentially many measurements to vanish, which reveals the strong protection through Clifford scrambling. On the other hand, for measurements performed in rotated non-Clifford bases, measurements can both create and destroy nonstabilizerness. Here, the dynamics leads to a steady-state with non-trivial nonstabilizerness, independent of the initial state. We find that Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. While the stabilizer nullity is insensitive to the rotation angle, Stabilizer R\'enyi Entropies expose a richer structure in their dynamics. Our results uncover sharp distinctions between coarse and fine-grained nonstabilizerness diagnostics and demonstrate how measurements can both suppress and sustain quantum computational resources.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the dynamics of nonstabilizerness (magic) in monitored quantum circuits of random Clifford unitaries followed by local projective measurements. For computational-basis measurements, an analytical model is derived for the decay of stabilizer nullity, which proceeds in quantized steps and requires exponentially many measurements to reach zero owing to Clifford scrambling. For measurements in rotated non-Clifford bases, both creation and destruction of nonstabilizerness occur, driving the system to a steady state with non-trivial nonstabilizerness that is independent of the initial state; Haar-random states equilibrate in constant time while stabilizer states exhibit linear-in-size relaxation. Stabilizer Rényi entropies reveal additional dynamical structure not captured by the nullity alone.

Significance. If the central claims are confirmed, the work is significant for clarifying how measurements can both suppress and sustain quantum computational resources in scrambled circuits. The analytical treatment of nullity decay under Clifford protection and the numerical demonstration of an initial-state-independent attractor under rotated measurements provide concrete, falsifiable predictions that distinguish coarse (nullity) from fine-grained (SRE) diagnostics. These results bear on resource theories in monitored dynamics and on the design of circuits that preserve or control magic.

major comments (2)
  1. [§3] §3 (analytical model for nullity decay): the derivation of quantized-step behavior assumes that each random Clifford layer fully scrambles the state to an effectively Haar distribution before the subsequent measurement. For finite L and early times this assumption may leave residual correlations, which would modify the predicted step structure and the exponential scaling of the number of measurements required to reach zero nullity.
  2. [§4] §4 (rotated-basis numerics and steady-state claims): the reported linear-in-L relaxation time for stabilizer initial states and the independence of the steady-state nonstabilizerness from the initial state rest on numerical trajectories whose finite-size scaling, error bars, and convergence diagnostics are not shown. Without these, it remains unclear whether the linear scaling is robust or influenced by post-hoc fitting or incomplete scrambling.
minor comments (2)
  1. [Abstract/Introduction] The abstract and introduction would benefit from a single sentence explicitly contrasting the two measurement bases and the two diagnostics (nullity vs. SREs).
  2. [Figures] Figure captions should state the system sizes, number of trajectories, and whether analytical curves are overlaid on the numerical data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for the constructive comments, which help clarify the presentation of our results on nonstabilizerness dynamics in monitored Clifford circuits. We respond to each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (analytical model for nullity decay): the derivation of quantized-step behavior assumes that each random Clifford layer fully scrambles the state to an effectively Haar distribution before the subsequent measurement. For finite L and early times this assumption may leave residual correlations, which would modify the predicted step structure and the exponential scaling of the number of measurements required to reach zero nullity.

    Authors: We appreciate the referee's observation regarding the scrambling assumption. The analytical model relies on the rapid mixing properties of random Clifford circuits, which are known to scramble information on timescales logarithmic in system size. While residual correlations could in principle exist at very early times for finite L, our numerical simulations of the full circuit dynamics (not shown in the original manuscript) confirm that the quantized step structure and the exponential scaling of the number of measurements persist across the relevant parameter regime. In the revision we will add an explicit discussion of the approximation's validity, including a brief comparison between the analytical prediction and direct numerics for moderate L, to delineate the range where the model applies without modification. revision: partial

  2. Referee: [§4] §4 (rotated-basis numerics and steady-state claims): the reported linear-in-L relaxation time for stabilizer initial states and the independence of the steady-state nonstabilizerness from the initial state rest on numerical trajectories whose finite-size scaling, error bars, and convergence diagnostics are not shown. Without these, it remains unclear whether the linear scaling is robust or influenced by post-hoc fitting or incomplete scrambling.

    Authors: We agree that additional numerical diagnostics are necessary to substantiate the claims of linear-in-L relaxation for stabilizer initial states and initial-state independence of the steady state. In the revised manuscript we will include (i) finite-size scaling plots of the relaxation time versus L, (ii) error bars obtained by averaging over at least 100 independent trajectories, and (iii) explicit convergence checks showing that the steady-state value is reached and remains stable after the reported timescale. These additions will confirm that the observed linear scaling is robust and not an artifact of fitting or incomplete scrambling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations use standard Clifford scrambling and random averaging.

full rationale

The paper's analytical model for stabilizer nullity decay under computational-basis measurements follows from the established scrambling action of random Clifford layers, producing quantized decay steps via averaging over the Clifford group without any parameter fitted to the output data itself. The steady-state nonstabilizerness for rotated non-Clifford measurements is obtained by balancing creation and destruction channels, yielding initial-state independence as a direct consequence of the effective Haar-like distribution after scrambling; this does not reduce to a self-definition or a fitted input renamed as prediction. Relaxation timescales (constant-time for Haar states, linear-in-size for stabilizers) are likewise derived from the same mixing properties. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem imported from the authors' prior work appears in the central claims; the results rest on externally standard properties of monitored Clifford circuits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum mechanics and Clifford group properties; no new entities are postulated and no free parameters are introduced beyond the measurement basis choice.

axioms (2)
  • domain assumption Random Clifford unitaries fully scramble the state on timescales short compared to measurement intervals.
    Invoked to derive the quantized-step decay of stabilizer nullity.
  • standard math Projective measurements in a fixed basis commute with the stabilizer formalism in the computational case.
    Used to obtain the analytical model for nullity dynamics.

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Forward citations

Cited by 3 Pith papers

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  2. Operational interpretation of the Stabilizer Entropy

    quant-ph 2025-07 unverdicted novelty 7.0

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Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages · cited by 3 Pith papers · 6 internal anchors

  1. [1]

    ManyQLowD

    Thus, we need∼Nnumber of measurements to inject extensive magic into the initial stabilizer state, and only a constant number of measurements to remove a constant amount of magic for Haar random states. Now, we get a more accurate estimate of the steady state value of νby solving for the steady-state of the Markov chain approximately. To this end, let us ...

    work page 2022

  2. [2]

    Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D

    D. Gottesman,Stabilizer codes and quantum error correc- tion. Caltech Ph. D, Ph.D. thesis, Thesis, eprint: quant- ph/9705052 (1997)

    work page arXiv 1997

  3. [3]

    The Heisenberg Representation of Quantum Computers

    D. Gottesman, The heisenberg representation of quan- tum computers, arXiv quant-ph/9807006 (1998)

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  4. [4]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

    work page 2004

  5. [5]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2011)

    work page 2011

  6. [6]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Learning t-doped stabilizer states, Quantum8, 1361 (2024)

    work page 2024

  7. [7]

    Howard, J

    M. Howard, J. Wallman, V. Veitch, and J. Emerson, Con- textuality supplies the ‘magic’for quantum computation, Nature510, 351 (2014)

    work page 2014

  8. [8]

    Veitch, S

    V. Veitch, S. H. Mousavian, D. Gottesman, and J. Emer- son, The resource theory of stabilizer quantum computa- tion, New J. Phys.16, 013009 (2014)

    work page 2014

  9. [9]

    E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature549, 172 (2017)

    work page 2017

  10. [10]

    Bravyi, D

    S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low- rank stabilizer decompositions, Quantum3, 181 (2019)

    work page 2019

  11. [11]

    Bu and D

    K. Bu and D. E. Koh, Efficient classical simulation of clif- ford circuits with nonstabilizer input states, Phys. Rev. Lett.123, 170502 (2019)

    work page 2019

  12. [12]

    Leone, S

    L. Leone, S. F. E. Oliviero, Y. Zhou, and A. Hamma, Quantum chaos is quantum, Quantum5, 453 (2021)

    work page 2021

  13. [13]

    A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys.303, 2 (2003)

    work page 2003

  14. [14]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Phys. Rev. A71, 022316 (2005)

    work page 2005

  15. [15]

    Chitambar and G

    E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys.91, 025001 (2019)

    work page 2019

  16. [16]

    Liu and A

    Z.-W. Liu and A. Winter, Many-body quantum magic, PRX Quantum3, 020333 (2022)

    work page 2022

  17. [17]

    Howard and E

    M. Howard and E. Campbell, Application of a resource theory for magic states to fault-tolerant quantum com- puting, Phys. Rev. Lett.118, 090501 (2017)

    work page 2017

  18. [18]

    Horodecki and J

    M. Horodecki and J. Oppenheim, (quantumness in the context of) resource theories, International Jour- nal of Modern Physics B27, 1345019 (2013), https://doi.org/10.1142/S0217979213450197

    work page doi:10.1142/s0217979213450197 2013

  19. [19]

    Veitch, C

    V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computation, New Journal of Physics14, 113011 (2012)

    work page 2012

  20. [20]

    Heinrich and D

    M. Heinrich and D. Gross, Robustness of Magic and Symmetries of the Stabiliser Polytope, Quantum3, 132 (2019)

    work page 2019

  21. [21]

    J. R. Seddon, B. Regula, H. Pashayan, Y. Ouyang, and E. T. Campbell, Quantifying quantum speedups: Im- proved classical simulation from tighter magic mono- tones, PRX Quantum2, 010345 (2021)

    work page 2021

  22. [22]

    E. T. Campbell and D. E. Browne, Bound states for magic state distillation in fault-tolerant quantum com- putation, Phys. Rev. Lett.104, 030503 (2010)

    work page 2010

  23. [23]

    Beverland, E

    M. Beverland, E. Campbell, M. Howard, and V. Kli- uchnikov, Lower bounds on the non-clifford resources for quantum computations, Quantum Science Tech.5, 035009 (2020)

    work page 2020

  24. [24]

    Jiang and X

    J. Jiang and X. Wang, Lower bound for the t count via unitary stabilizer nullity, Physical Review Applied19, 034052 (2023)

    work page 2023

  25. [25]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer r´ enyi entropy, Phys. Rev. Lett.128, 050402 (2022)

    work page 2022

  26. [26]

    J. A. Monta˜ n` a L´ opez and P. Kos, Exact solution of long- range stabilizer r´ enyi entropy in the dual-unitary xxz model*, Journal of Physics A: Mathematical and The- oretical57, 475301 (2024)

    work page 2024

  27. [27]

    Chatziioannou, K

    X. Turkeshi, A. Dymarsky, and P. Sierant, Pauli spec- trum and nonstabilizerness of typical quantum many- body states, Physical Review B111, 10.1103/phys- revb.111.054301 (2025)

    work page doi:10.1103/phys- 2025

  28. [28]

    Haug and L

    T. Haug and L. Piroli, Stabilizer entropies and nonstabi- lizerness monotones, Quantum7, 1092 (2023)

    work page 2023

  29. [29]

    Haug and L

    T. Haug and L. Piroli, Quantifying nonstabilizerness of matrix product states, Phys. Rev. B107, 035148 (2023)

    work page 2023

  30. [30]

    P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dal- monte, Many-body magic via pauli-markov chains—from criticality to gauge theories, PRX Quantum4, 040317 (2023)

    work page 2023

  31. [31]

    Lami and M

    G. Lami and M. Collura, Nonstabilizerness via perfect pauli sampling of matrix product states, Phys. Rev. Lett. 131, 180401 (2023)

    work page 2023

  32. [32]

    Passarelli, R

    G. Passarelli, R. Fazio, and P. Lucignano, Nonstabilizer- ness of permutationally invariant systems, Phys. Rev. A 110, 022436 (2024)

    work page 2024

  33. [33]

    P. S. Tarabunga and T. Haug, Efficient mutual magic and magic capacity with matrix product states, arXiv:2504.07230 (2025)

    work page arXiv 2025

  34. [34]

    M. Frau, P. S. Tarabunga, M. Collura, M. Dalmonte, and E. Tirrito, Nonstabilizerness versus entanglement in matrix product states, Phys. Rev. B110, 045101 (2024)

    work page 2024

  35. [35]

    Haug and M

    T. Haug and M. Kim, Scalable measures of magic re- source for quantum computers, PRX Quantum4, 010301 (2023)

    work page 2023

  36. [36]

    S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, Measuring magic on a quantum processor, npj Quantum Information8, 148 (2022)

    work page 2022

  37. [37]

    T. Haug, S. Lee, and M. S. Kim, Efficient quantum al- gorithms for stabilizer entropies, Phys. Rev. Lett.132, 240602 (2024)

    work page 2024

  38. [38]

    Efficient witnessing and testing of magic in mixed quantum states

    T. Haug and P. S. Tarabunga, Efficient witnessing and testing of magic in mixed quantum states, arXiv preprint arXiv:2504.18098 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  39. [39]

    A. Gu, S. F. E. Oliviero, and L. Leone, Doped stabilizer states in many-body physics and where to find them, Phys. Rev. A110, 062427 (2024)

    work page 2024

  40. [40]

    Rattacaso, L

    D. Rattacaso, L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer entropy dynamics after a quantum quench, Phys. Rev. A108, 042407 (2023)

    work page 2023

  41. [41]

    S. F. E. Oliviero, L. Leone, and A. Hamma, Magic-state resource theory for the ground state of the transverse- field ising model, Phys. Rev. A106, 042426 (2022)

    work page 2022

  42. [42]

    Odavi´ c, T

    J. Odavi´ c, T. Haug, G. Torre, A. Hamma, F. Franchini, and S. M. Giampaolo, Complexity of frustration: A new source of non-local non-stabilizerness, SciPost Phys.15, 131 (2023)

    work page 2023

  43. [43]

    Passarelli, P

    G. Passarelli, P. Lucignano, D. Rossini, and A. Rus- somanno, Chaos and magic in the dissipative quantum 12 kicked top, Quantum9, 1653 (2025)

    work page 2025

  44. [44]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Nonstabiliz- erness determining the hardness of direct fidelity estima- tion, Phys. Rev. A107, 022429 (2023)

    work page 2023

  45. [45]

    Magni and X

    B. Magni and X. Turkeshi, Quantum complexity and chaos in many-qudit doped clifford circuits (2025), arXiv:2506.02127 [quant-ph]

    work page arXiv 2025

  46. [46]

    Jasser, J

    B. Jasser, J. Odavic, and A. Hamma, Stabilizer entropy and entanglement complexity in the sachdev-ye-kitaev model (2025), arXiv:2502.03093 [quant-ph]

    work page arXiv 2025

  47. [47]

    Tirrito, P

    E. Tirrito, P. S. Tarabunga, G. Lami, T. Chanda, L. Leone, S. F. Oliviero, M. Dalmonte, M. Collura, and A. Hamma, Quantifying nonstabilizerness through en- tanglement spectrum flatness, Physical Review A109, L040401 (2024)

    work page 2024

  48. [48]

    T. Haug, L. Aolita, and M. Kim, Probing quantum com- plexity via universal saturation of stabilizer entropies, arXiv:2406.04190 (2024)

    work page arXiv 2024

  49. [49]

    A. Gu, S. F. Oliviero, and L. Leone, Magic-induced computational separation in entanglement theory, PRX Quantum6, 020324 (2025)

    work page 2025

  50. [50]

    Ben-Zion, J

    D. Ben-Zion, J. McGreevy, and T. Grover, Disentangling quantum matter with measurements, Phys. Rev. B101, 115131 (2020)

    work page 2020

  51. [51]

    S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum er- ror correction in scrambling dynamics and measurement- induced phase transition, Physical Review Letters125, 030505 (2020)

    work page 2020

  52. [52]

    Fidkowski, J

    L. Fidkowski, J. Haah, and M. B. Hastings, How dynam- ical quantum memories forget, Quantum5, 382 (2021)

    work page 2021

  53. [53]

    Brown and O

    W. Brown and O. Fawzi, Short random circuits define good quantum error correcting codes, in2013 IEEE In- ternational Symposium on Information Theory(2013) pp. 346–350

    work page 2013

  54. [54]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Phys. Rev. X9, 031009 (2019)

    work page 2019

  55. [55]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    work page 2019

  56. [56]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

    work page 2018

  57. [57]

    P. S. Tarabunga and E. Tirrito, Magic transition in measurement-only circuits (2024), arXiv:2407.15939 [quant-ph]

    work page arXiv 2024

  58. [58]

    Bejan, C

    M. Bejan, C. McLauchlan, and B. B´ eri, Dynamical magic transitions in monitored clifford+tcircuits, PRX Quan- tum5, 030332 (2024)

    work page 2024

  59. [59]

    G. E. Fux, E. Tirrito, M. Dalmonte, and R. Fazio, Entan- glement – nonstabilizerness separation in hybrid quan- tum circuits, Phys. Rev. Res.6, L042030 (2024)

    work page 2024

  60. [60]

    Russomanno, G

    A. Russomanno, G. Passarelli, D. Rossini, and P. Lucig- nano, Efficient evaluation of the nonstabilizerness in uni- tary and monitored quantum many-body systems (2025), arXiv:2502.01431 [quant-ph]

    work page arXiv 2025

  61. [61]

    M. J. Gullans and D. A. Huse, Dynamical purifica- tion phase transition induced by quantum measurements, Phys. Rev. X10, 041020 (2020)

    work page 2020

  62. [62]

    S. F. Oliviero, L. Leone, and A. Hamma, Transitions in entanglement complexity in random quantum circuits by measurements, Physics Letters A418, 127721 (2021)

    work page 2021

  63. [63]

    True and A

    S. True and A. Hamma, Transitions in Entanglement Complexity in Random Circuits, Quantum6, 818 (2022)

    work page 2022

  64. [64]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum cir- cuits, Phys. Rev. B101, 104302 (2020)

    work page 2020

  65. [65]

    O. Lunt, M. Szyniszewski, and A. Pal, Measurement- induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits, Phys. Rev. B104, 155111 (2021)

    work page 2021

  66. [66]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B105, 094303 (2022)

    work page 2022

  67. [67]

    Le Gal, X

    Y. Le Gal, X. Turkeshi, and M. Schir` o, Entanglement dynamics in monitored systems and the role of quantum jumps, PRX Quantum5, 030329 (2024)

    work page 2024

  68. [68]

    Multipartite entanglement structure of monitored quantum circuits

    A. Lira-Solanilla, X. Turkeshi, and S. Pappalardi, Mul- tipartite entanglement structure of monitored quantum circuits (2024), arXiv:2412.16062 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  69. [69]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrish- nan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B101, 060301 (2020)

    work page 2020

  70. [70]

    Leone, S

    L. Leone, S. F. Oliviero, G. Esposito, and A. Hamma, Phase transition in stabilizer entropy and efficient purity estimation, Physical Review A109, 032403 (2024)

    work page 2024

  71. [71]

    Zhou, Z.-C

    S. Zhou, Z.-C. Yang, A. Hamma, and C. Chamon, Single t gate in a clifford circuit drives transition to universal entanglement spectrum statistics, SciPost Phys.9, 087 (2020)

    work page 2020

  72. [72]

    Turkeshi, E

    X. Turkeshi, E. Tirrito, and P. Sierant, Magic spread- ing in random quantum circuits, Nature Communications 16, 2575 (2025)

    work page 2025

  73. [73]

    Tirrito, L

    E. Tirrito, L. Lumia, A. Paviglianiti, G. Lami, A. Silva, X. Turkeshi, and M. Collura, Magic phase transitions in monitored gaussian fermions (2025), arXiv:2507.07179 [quant-ph]

    work page arXiv 2025

  74. [74]

    Paviglianiti, G

    A. Paviglianiti, G. Lami, M. Collura, and A. Silva, Es- timating non-stabilizerness dynamics without simulating it, arXiv:2405.06054 (2024)

    work page arXiv 2024

  75. [75]

    Niroula, C

    P. Niroula, C. D. White, Q. Wang, S. Johri, D. Zhu, C. Monroe, C. Noel, and M. J. Gullans, Phase transition in magic with random quantum circuits, Nature Physics 20, 1786 (2024)

    work page 2024

  76. [76]

    F. B. Trigueros and J. A. M. Guzm´ an, Nonstabilizer- ness and error resilience in noisy quantum circuits (2025), arXiv:2506.18976 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  77. [77]

    X. Wang, M. M. Wilde, and Y. Su, Quantifying the magic of quantum channels, New J. Phys.21, 103002 (2019)

    work page 2019

  78. [78]

    Leone and L

    L. Leone and L. Bittel, Stabilizer entropies are monotones for magic-state resource theory, Physical Review A110, L040403 (2024)

    work page 2024

  79. [79]

    Exact entanglement probability distribution of bi-partite randomised stabilizer states

    O. Dahlsten and M. B. Plenio, Exact entanglement prob- ability distribution of bi-partite randomised stabilizer states, arXiv preprint quant-ph/0511119 (2005)

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  80. [80]

    T´ oth and O

    G. T´ oth and O. G¨ uhne, Entanglement detection in the stabilizer formalism, Phys. Rev. A72, 022340 (2005)

    work page 2005

Showing first 80 references.