pith. sign in

arxiv: 2601.07824 · v3 · submitted 2026-01-12 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.stat-mech

Computing quantum magic of state vectors

Piotr Sierant , Jofre Vall\`es-Muns , Artur Garcia-Saez This is my paper

Pith reviewed 2026-05-16 14:45 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.stat-mech
keywords quantum magicstabilizer Rényi entropymanafast Hadamard transformstate vectorsnon-stabilizernessmany-body systemscomputational algorithms
0
0 comments X

The pith

Fast Hadamard transforms compute SRE and mana for qubit and qutrit state vectors at O(N d^{2N}) cost.

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

The paper develops algorithms that use the fast Hadamard transform to calculate non-stabilizerness measures exactly for pure states supplied as vectors. For qubits the target is the stabilizer Rényi entropy; for qutrits it is the mana. The new scaling replaces the direct triple-exponential cost with a quadratic-exponential cost multiplied by a linear factor in the number of qudits. The same transform can be paired with Monte Carlo sampling to estimate SRE and can be extended to compute mana for mixed states. An open-source implementation supplies built-in support for multithreading, distributed parallelism and GPU acceleration.

Core claim

The fast Hadamard transform can be applied directly to the sums that define the stabilizer Rényi entropy for qubits and the mana for qutrits, yielding numerically exact values for any pure state vector at cost O(N d^{2N}).

What carries the argument

Fast Hadamard transform applied to the character sums that define SRE and mana.

If this is right

  • SRE and mana become accessible for system sizes that were previously out of reach.
  • The algorithms admit straightforward parallelization across CPU cores and GPUs.
  • Monte Carlo sampling combined with the transform yields practical estimates of SRE for large vectors.
  • Mana can be evaluated for mixed states as well as pure states.

Where Pith is reading between the lines

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

  • The reduced scaling opens the possibility of mapping how magic accumulates across quantum phase transitions.
  • The same transform technique may be adaptable to other resource measures that are expressed as sums over characters.
  • Integration with tensor-network or variational methods could push the reachable system sizes even farther.

Load-bearing premise

The quantum state must be supplied exactly as a full state vector and the defining sums must admit direct application of the fast Hadamard transform without approximation.

What would settle it

For any small N where both methods fit in memory, the numerical value of SRE or mana produced by the fast-transform algorithm must agree with the value from the naive triple sum to machine precision.

Figures

Figures reproduced from arXiv: 2601.07824 by Artur Garcia-Saez, Jofre Vall\`es-Muns, Piotr Sierant.

Figure 1
Figure 1. Figure 1: Left: Variance σf of the “energy” f(Xa) at β = 1, which controls the statistical uncertainty of the SRE in the sampling Algorithm 3, for states evolved under quantum circuits (46) of depth t, plotted as a function of system size N. Right: Absolute error of the SRE M2 at circuit depth t = 4, comparing the numerically exact result from Algorithm 2 with its sampling estimate Mˆ2, as a function of the number o… view at source ↗
read the original abstract

Non-stabilizerness, also known as ``magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer R\'enyi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number $N$ of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits ($d=2$) and the mana for qutrits ($d=3$) for pure states given as state vectors. Our methods compute SRE and mana at cost $O(N d^{2N})$, providing an exponential improvement over the naive $O(d^{3N})$ scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG ( https://github.com/bsc-quantic/HadaMAG.jl/ ), which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.

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

0 major / 2 minor

Summary. The paper introduces numerically exact algorithms that use the fast Hadamard transform to evaluate the stabilizer Rényi entropy (SRE) for pure qubit states and the mana for pure qutrit states supplied as full state vectors. It claims an exact complexity of O(N d^{2N}) (versus the naive O(d^{3N})), with extensions to Monte Carlo sampling for SRE estimation and to mixed-state mana, all implemented in the open-source Julia package HadaMAG with multithreading, MPI, and GPU support.

Significance. If the exactness and complexity claims hold, the work removes a major computational bottleneck for quantifying non-stabilizerness, enabling routine calculations on systems an order of magnitude larger than before. The open-source, parallelized implementation is a concrete strength that directly supports reproducible large-scale studies of magic in many-body systems.

minor comments (2)
  1. [§3] The complexity analysis in the main text would benefit from an explicit step-by-step count of operations in the FHT application to the defining sums (currently summarized in the abstract).
  2. [Fig. 2] Figure captions should state the precise system sizes and hardware used for the reported timings to allow direct reproduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our contributions, and recommendation to accept. No major comments were raised that require point-by-point response.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives an O(N d^{2N}) algorithm for exact SRE and mana computation by applying the standard fast Hadamard transform directly to the sums that define these quantities for pure state vectors. This mapping is presented as a straightforward consequence of linear-algebra properties of the FHT and does not rely on self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The complexity improvement over the naive O(d^{3N}) scaling follows immediately from the known O(d^N log d^N) cost of the FHT without circular reduction. The open-source package provides an independent verification route, confirming the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical properties of the fast Hadamard transform applied to sums over stabilizer states; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math The fast Hadamard transform efficiently computes the required sums defining SRE and mana for pure states given as vectors
    This follows from established properties of the Walsh-Hadamard transform in quantum information and signal processing.

pith-pipeline@v0.9.0 · 5589 in / 1273 out tokens · 84186 ms · 2026-05-16T14:45:40.200922+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Non-Local Magic Resources for Fermionic Gaussian States

    quant-ph 2026-04 unverdicted novelty 6.0

    Closed-form formula computes non-local magic for fermionic Gaussian states from two-point correlations in polynomial time.

  2. Quantum Complexity and New Directions in Nuclear Physics and High-Energy Physics Phenomenology

    quant-ph 2026-04 unverdicted novelty 2.0

    A review of how quantum information science is expected to provide new tools and insights for nuclear and high-energy physics phenomenology and quantum simulations.

Reference graph

Works this paper leans on

127 extracted references · 127 canonical work pages · cited by 2 Pith papers · 6 internal anchors

  1. [1]

    Quantum computing and the entanglement frontier

    J. Preskill, “Quantum computing and the entanglement frontier,” (2012), arXiv:1203.5813

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  2. [2]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022)

    work page 2022

  3. [3]

    Vidal, Phys

    G. Vidal, Phys. Rev. Lett.91, 147902 (2003)

    work page 2003

  4. [4]

    Orús, Ann

    R. Orús, Ann. Phys.349, 117 (2014)

    work page 2014

  5. [5]

    Orús, Nat

    R. Orús, Nat. Rev. Phys.1, 538–550 (2019)

    work page 2019

  6. [6]

    S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein,Tensor network contractions: methods and applications to quantum many-body systems(Springer Nature, 2020)

    work page 2020

  7. [7]

    M. C. Bañuls, Annu. Rev. Condens. Matter Phys.14, 173–191 (2023)

    work page 2023

  8. [8]

    Verstraete, V

    F. Verstraete, V. Murg, and J. Cirac, Adv. Phys.57, 143 (2008)

    work page 2008

  9. [9]

    Schuch, M

    N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett.100, 030504 (2008)

    work page 2008

  10. [10]

    Schollwöck, Ann

    U. Schollwöck, Ann. Phys.326, 96 (2011)

    work page 2011

  11. [11]

    L. G. Valiant, inProceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC ’01 (Association for Computing Machinery, New York, NY, USA, 2001) p. 114–123

    work page 2001

  12. [12]

    B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A65, 032325 (2002)

    work page 2002

  13. [13]

    Bravyi, Quantum Info

    S. Bravyi, Quantum Info. Comput.5, 216–238 (2005)

    work page 2005

  14. [14]

    The Heisenberg representation of quantum computers

    D. Gottesman, “The heisenberg representation of quantum computers,” (1998), arXiv:quant- ph/9807006

    work page arXiv 1998

  15. [15]

    Aaronson and D

    S. Aaronson and D. Gottesman, Phys. Rev. A70, 052328 (2004)

    work page 2004

  16. [16]

    Sierant, P

    P. Sierant, P. Stornati, and X. Turkeshi, PRX Quantum7, 010302 (2026)

    work page 2026

  17. [17]

    Liu and A

    Z.-W. Liu and A. Winter, PRX Quantum3, 020333 (2022)

    work page 2022

  18. [18]

    Veitch, S

    V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New J. Phys.16, 013009 (2014)

    work page 2014

  19. [19]

    Heimendahl, M

    A. Heimendahl, M. Heinrich, and D. Gross, Journal of Mathematical Physics63, 112201 (2022)

    work page 2022

  20. [20]

    Haug and L

    T. Haug and L. Piroli, Quantum7, 1092 (2023)

    work page 2023

  21. [21]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(2000)

    work page 2000

  22. [22]

    Bravyi and A

    S. Bravyi and A. Kitaev, Phys. Rev. A71, 022316 (2005)

    work page 2005

  23. [23]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A52, 3457 (1995)

    work page 1995

  24. [24]

    Bravyi, G

    S. Bravyi, G. Smith, and J. A. Smolin, Phys. Rev. X6, 021043 (2016)

    work page 2016

  25. [25]

    Bravyi and D

    S. Bravyi and D. Gosset, Phys. Rev. Lett.116, 250501 (2016)

    work page 2016

  26. [26]

    Bravyi, D

    S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Quantum3, 181 (2019)

    work page 2019

  27. [27]

    Howard and E

    M. Howard and E. Campbell, Phys. Rev. Lett.118, 090501 (2017)

    work page 2017

  28. [28]

    Heinrich and D

    M. Heinrich and D. Gross, Quantum3, 132 (2019)

    work page 2019

  29. [29]

    Sarkar, C

    S. Sarkar, C. Mukhopadhyay, and A. Bayat, New Journal of Physics22, 083077 (2020)

    work page 2020

  30. [30]

    Hamaguchi, K

    H. Hamaguchi, K. Hamada, and N. Yoshioka, Quantum8, 1461 (2024)

    work page 2024

  31. [31]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. Lett.128, 050402 (2022)

    work page 2022

  32. [32]

    Leone and L

    L. Leone and L. Bittel, Phys. Rev. A110, L040403 (2024)

    work page 2024

  33. [33]

    Haug and L

    T. Haug and L. Piroli, Phys. Rev. B107, 035148 (2023)

    work page 2023

  34. [34]

    P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dalmonte, PRX Quantum4, 040317 (2023)

    work page 2023

  35. [35]

    Odavić, T

    J. Odavić, T. Haug, G. Torre, A. Hamma, F. Franchini, and S. M. Giampaolo, SciPost Phys. 15, 131 (2023)

    work page 2023

  36. [36]

    Passarelli, R

    G. Passarelli, R. Fazio, and P. Lucignano, Phys. Rev. A110, 022436 (2024)

    work page 2024

  37. [37]

    P. S. Tarabunga and C. Castelnovo, Quantum8, 1347 (2024)

    work page 2024

  38. [38]

    P. R. N. Falcão, P. S. Tarabunga, M. Frau, E. Tirrito, J. Zakrzewski, and M. Dalmonte, Phys. Rev. B111, L081102 (2025)

    work page 2025

  39. [39]

    Jasser, J

    B. Jasser, J. Odavić, and A. Hamma, Phys. Rev. B112, 174204 (2025)

    work page 2025

  40. [40]

    Bera and M

    S. Bera and M. Schirò, SciPost Phys.19, 159 (2025)

    work page 2025

  41. [41]

    Y.-M. Ding, Z. Wang, and Z. Yan, PRX Quantum6, 030328 (2025)

    work page 2025

  42. [42]

    Hoshino, M

    M. Hoshino, M. Oshikawa, and Y. Ashida, Phys. Rev. X16, 011037 (2026). Accepted in Quantum 2026-04-05, click title to verify. Published under CC-BY 4.0.22

    work page 2026

  43. [43]

    Hoshino and Y

    M. Hoshino and Y. Ashida, Phys. Rev. Lett.136, 080402 (2026)

    work page 2026

  44. [44]

    Rattacaso, L

    D. Rattacaso, L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. A108, 042407 (2023)

    work page 2023

  45. [45]

    Turkeshi, A

    X. Turkeshi, A. Dymarsky, and P. Sierant, Phys. Rev. B111, 054301 (2025)

    work page 2025

  46. [46]

    Turkeshi, E

    X. Turkeshi, E. Tirrito, and P. Sierant, Nature Communications16, 2575 (2025)

    work page 2025

  47. [47]

    Tirrito, X

    E. Tirrito, X. Turkeshi, and P. Sierant, Phys. Rev. Lett.135, 220401 (2025)

    work page 2025

  48. [48]

    Odavić, M

    J. Odavić, M. Viscardi, and A. Hamma, Phys. Rev. B112, 104301 (2025)

    work page 2025

  49. [49]

    T. Haug, L. Aolita, and M. Kim, Quantum9, 1801 (2025)

    work page 2025

  50. [50]

    J. A. Montañà López and P. Kos, J. Phys. A: Math. and Theor.57, 475301 (2024)

    work page 2024

  51. [51]

    P. S. Tarabunga and E. Tirrito, npj Quantum Information11, 166 (2025)

    work page 2025

  52. [52]

    Santra, A

    G. C. Santra, A. Windey, S. Bandyopadhyay, A. Legramandi, and P. Hauke, “Complexity transitions in chaotic quantum systems,” (2025), arXiv:2505.09707 [quant-ph]

    work page arXiv 2025

  53. [53]

    Passarelli, A

    G. Passarelli, A. Russomanno, and P. Lucignano, Phys. Rev. A111, 062417 (2025)

    work page 2025

  54. [54]

    Stabilizer entanglement as a magic highway,

    Z.-Y. Hou, C. Cao, and Z.-C. Yang, “Stabilizer entanglement as a magic highway,” (2025), arXiv:2503.20873 [quant-ph]

    work page arXiv 2025

  55. [55]

    Zhang and Y

    Y. Zhang and Y. Gu, “Quantum magic dynamics in random circuits,” (2024), arXiv:2410.21128 [quant-ph]

    work page arXiv 2024

  56. [56]

    Magni, A

    B. Magni, A. Christopoulos, A. De Luca, and X. Turkeshi, Phys. Rev. X15, 031071 (2025)

    work page 2025

  57. [57]

    Magni and X

    B. Magni and X. Turkeshi, Quantum9, 1956 (2025)

    work page 1956

  58. [58]

    Zhang, S

    Y. Zhang, S. Vijay, Y. Gu, and Y. Bao, PRX Quantum7, 010344 (2026)

    work page 2026

  59. [59]

    Szombathy, A

    D. Szombathy, A. Valli, C. P. Moca, L. Farkas, and G. Zaránd, Phys. Rev. Res.7, 043072 (2025)

    work page 2025

  60. [60]

    S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, npj Quantum Info.8, 148 (2022)

    work page 2022

  61. [61]

    Haug and M

    T. Haug and M. Kim, PRX Quantum4, 010301 (2023)

    work page 2023

  62. [62]

    T. Haug, S. Lee, and M. S. Kim, Phys. Rev. Lett.132, 240602 (2024)

    work page 2024

  63. [63]

    Sarkar and T

    R. Sarkar and T. J. Yoder, Quantum8, 1307 (2024)

    work page 2024

  64. [64]

    Gross, J

    D. Gross, J. Math. Phys.47, 122107 (2006)

    work page 2006

  65. [65]

    Gross, Applied Physics B86, 367 (2007)

    D. Gross, Applied Physics B86, 367 (2007)

    work page 2007

  66. [66]

    Veitch, C

    V. Veitch, C. Ferrie, D. Gross, and J. Emerson, New Journal of Physics14, 113011 (2012)

    work page 2012

  67. [67]

    Pashayan, J

    H. Pashayan, J. J. Wallman, and S. D. Bartlett, Phys. Rev. Lett.115, 070501 (2015)

    work page 2015

  68. [68]

    C. D. White, C. Cao, and B. Swingle, Phys. Rev. B103, 075145 (2021)

    work page 2021

  69. [69]

    P. S. Tarabunga, Quantum8, 1413 (2024)

    work page 2024

  70. [70]

    Mana in haar-random states,

    C. D. White and J. H. Wilson, “Mana in haar-random states,” (2020), arXiv:2011.13937 [quant-ph]

    work page arXiv 2020

  71. [71]

    K. Goto, T. Nosaka, and M. Nozaki, Phys. Rev. D106, 126009 (2022)

    work page 2022

  72. [72]

    T. J. Sewell and C. D. White, Phys. Rev. B106, 125130 (2022)

    work page 2022

  73. [73]

    Koukoulekidis and D

    N. Koukoulekidis and D. Jennings, npj Quantum Information8, 42 (2022)

    work page 2022

  74. [74]

    R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Journal of High Energy Physics2024, 264 (2024)

    work page 2024

  75. [75]

    Ahmadi and E

    A. Ahmadi and E. Greplova, SciPost Phys.16, 043 (2024)

    work page 2024

  76. [76]

    Nyström, N

    R. Nyström, N. Pranzini, and E. Keski-Vakkuri, Phys. Rev. Res.7, 033085 (2025)

    work page 2025

  77. [77]

    C. E. P. Robin and M. J. Savage, Phys. Rev. C112, 044004 (2025)

    work page 2025

  78. [78]

    Brökemeier, S

    F. Brökemeier, S. M. Hengstenberg, J. W. T. Keeble, C. E. P. Robin, F. Rocco, and M. J. Savage, Phys. Rev. C111, 034317 (2025)

    work page 2025

  79. [79]

    C. D. White and M. J. White, Phys. Rev. D110, 116016 (2024)

    work page 2024

  80. [80]

    Chernyshev, C

    I. Chernyshev, C. E. P. Robin, and M. J. Savage, Phys. Rev. Res.7, 023228 (2025)

    work page 2025

Showing first 80 references.