pith. machine review for the scientific record. sign in

arxiv: 2605.05518 · v1 · submitted 2026-05-06 · 🪐 quant-ph

Recognition: unknown

Classical shadows over symmetric spaces

Rebecca Chang , Maureen Krumt\"unger , Martin Larocca , Maxwell West
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:04 UTC · model grok-4.3

classification 🪐 quant-ph
keywords classical shadowssymmetric spacesrandomized measurementssample complexityquantum state learningunitary ensemblesquantum tomography
0
0 comments X

The pith

Classical shadow protocols extend to uniform sampling from compact symmetric spaces, yielding a unifying theory and slight sample-complexity gains for some observables.

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

The paper develops classical shadow protocols in which randomized measurements arise from uniform sampling over compact symmetric spaces rather than compact groups. It supplies a mathematical framework that covers these protocols in a single description and derives corresponding bounds on estimator variance and sample requirements. A sympathetic reader would care because the extension potentially allows fewer experimental measurements to estimate expectation values when observables are drawn from certain distributions. The work thereby broadens the set of realizable shadow schemes while keeping the core goal of efficient quantum state learning intact.

Core claim

By replacing the usual group sampling with uniform sampling from compact symmetric spaces, the authors obtain a family of classical shadow protocols that admit a single theoretical treatment; for observables sampled from suitable distributions, some of these protocols achieve modestly lower sample complexities than the standard group-based schemes.

What carries the argument

The classical shadow map induced by uniform random sampling over a compact symmetric space, which serves as the randomized measurement ensemble and permits the derivation of shadow-norm bounds via the geometry of the space.

If this is right

  • A single mathematical description now covers classical shadows generated by any compact symmetric space.
  • For observables drawn from appropriate distributions, some new protocols require modestly fewer samples than existing group-based schemes.
  • Variance bounds for the estimators follow directly from the representation theory and geometry of the symmetric space.
  • Any experimental realization of uniform sampling over such a space immediately yields a valid shadow protocol.

Where Pith is reading between the lines

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

  • Hardware implementations would need circuits or control sequences that produce the required symmetric-space distribution rather than Haar-random unitaries.
  • The modest efficiency gains may become practically relevant only when the observable distribution aligns closely with the symmetry of the chosen space.
  • The framework could be tested by comparing predicted versus measured estimator variances on small quantum devices using known symmetric-space ensembles.

Load-bearing premise

Uniform random sampling from the chosen compact symmetric space must be experimentally realizable, and the target observables must belong to the distributions for which the sample-complexity bound improves.

What would settle it

An explicit calculation of the shadow norm for a concrete symmetric space (such as a sphere or Grassmannian) that deviates from the predicted unifying formula, or an experiment showing no reduction in required shots for the claimed observable distributions.

Figures

Figures reproduced from arXiv: 2605.05518 by Martin Larocca, Maureen Krumt\"unger, Maxwell West, Rebecca Chang.

Figure 1
Figure 1. Figure 1: FIG. 1. The probability distributions on the Bloch sphere re view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Variances for random (symmetric) observables of unit 2-norm on a view at source ↗
read the original abstract

Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes.

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 / 3 minor

Summary. The manuscript develops classical shadow protocols in which randomized measurements arise from uniform sampling over compact symmetric spaces (rather than compact groups). It constructs a unifying representation-theoretic framework that generalizes the usual twirling and inversion steps, and it identifies particular distributions of observables for which the resulting protocols yield modest sample-complexity improvements over standard shadow schemes.

Significance. If the derivations hold, the work meaningfully extends the mathematical foundations of classical shadows by replacing group averaging with symmetric-space averaging. This unification is a clear strength and supplies a systematic way to generate new protocols. The secondary observation of slight sample-complexity gains for selected observable distributions is interesting, though its practical scope appears limited. The paper is credited for grounding the constructions in representation theory, which permits rigorous error bounds.

minor comments (3)
  1. [Section 3] The notation for the inversion map and the associated shadow norm could be introduced with an explicit comparison to the group case (e.g., in the paragraph following Eq. (3.7)) to help readers track the generalization.
  2. [Section 5] Figure 2 would benefit from an additional panel or caption sentence that directly overlays the new sample-complexity curves against the standard Clifford-shadow baseline for the same observable distribution.
  3. [Section 6] A short remark on the experimental feasibility of uniform sampling over the chosen symmetric spaces (beyond the mathematical construction) would strengthen the discussion of applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. We are pleased that the summary accurately reflects the core contributions: the extension of classical shadow protocols to uniform sampling over compact symmetric spaces, the unifying representation-theoretic framework that generalizes twirling and inversion, and the identification of modest sample-complexity gains for selected observable distributions. The recognition of the rigorous error bounds grounded in representation theory is also appreciated. No specific major comments were enumerated in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the representation theory of compact symmetric spaces to construct shadow protocols, generalizing the standard twirling and inversion steps for groups. All load-bearing steps are explicit mathematical constructions (uniform sampling over the symmetric space, associated POVMs, and inversion formulas) that are independent of the target observables or sample-complexity bounds. The modest sample-complexity gains are stated as conditional observations for particular observable distributions and are not used to define or justify the core protocol. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; the argument is self-contained against external representation-theoretic results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of quantum mechanics and representation theory of compact symmetric spaces; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Uniform random sampling from compact symmetric spaces is feasible and induces valid randomized measurements
    Implicit in the definition of the new protocols.

pith-pipeline@v0.9.0 · 5410 in / 1114 out tokens · 41435 ms · 2026-05-08T16:04:38.173009+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

37 extracted references · 14 canonical work pages · 1 internal anchor

  1. [1]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics16, 1050 (2020)

    2020

  2. [2]

    M. West, F. Sauvage, A. Sen, R. Forestano, D. Wierichs, N. Killoran, D. Grinko, M. Cerezo, and M. Larocca, Clas- 5 At least, when restricting to unitary ensembles that themselves live within the respective groups sical shadows with arbitrary group representations, arXiv preprint arXiv:2604.01429 10.48550/arXiv.2604.01429 (2026)

    work page doi:10.48550/arxiv.2604.01429 2026

  3. [3]

    A. Zhao, N. C. Rubin, and A. Miyake, Fermionic partial tomography via classical shadows, Physical Review Letters 127, 110504 (2021)

    2021

  4. [4]

    K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Commu- nications in Mathematical Physics404, 629 (2023). 6

    2023

  5. [5]

    G. H. Low, Classical shadows of fermions with particle number symmetry, arXiv preprint arXiv:2208.08964

    work page arXiv

  6. [6]

    M. West, A. A. Mele, M. Larocca, and M. Cerezo, Real classical shadows, Journal of Physics A: Mathematical and Theoretical58, 245304 (2025)

    2025

  7. [7]

    S. N. Hearth, M. O. Flynn, A. Chandran, and C. R. Lau- mann, Efficient local classical shadow tomography with number conservation, Physical Review Letters133, 060802 (2024)

    2024

  8. [8]

    S. Chen, W. Yu, P. Zeng, and S. T. Flammia, Robust shadow estimation, PRX Quantum2, 030348 (2021)

    2021

  9. [9]

    M. West, A. A. Mele, M. Larocca, and M. Cerezo, Random ensembles of symplectic and unitary states are indistinguishable, arXiv preprint arXiv:2409.16500 10.48550/arXiv.2409.16500 (2024)

    work page doi:10.48550/arxiv.2409.16500 2024

  10. [10]

    Bertoni, J

    C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eis- ert, and H. Pashayan, Shallow shadows: Expectation esti- mation using low-depth random clifford circuits, Physical Review Letters133, 020602 (2024)

    2024

  11. [11]

    R. King, D. Gosset, R. Kothari, and R. Babbush, Triply efficient shadow tomography, PRX Quantum6, 010336 (2025)

    2025

  12. [12]

    Van Kirk, J

    K. Van Kirk, J. Cotler, H.-Y. Huang, and M. D. Lukin, Hardware-efficient learning of quantum many-body states, arXiv preprint arXiv:2212.06084 10.48550/arXiv.2212.06084 (2022)

    work page doi:10.48550/arxiv.2212.06084 2022

  13. [13]

    A. W. Knapp,Lie groups beyond an introduction, Vol. 140 (Springer, 1996)

    1996

  14. [14]

    F. J. Dyson, Statistical theory of the energy levels of com- plex systems. i, Journal of Mathematical Physics3, 140 (1962)

    1962

  15. [15]

    F. J. Dyson, The threefold way. algebraic structure of sym- metry groups and ensembles in quantum mechanics, Jour- nal of Mathematical Physics3, 1199 (1962)

    1962

  16. [16]

    and Cerezo, M

    D. Wierichs, M. West, R. T. Forestano, M. Cerezo, and N. Killoran, Recursive Cartan decompositions for unitary synthesis, arXiv preprint arXiv:2503.19014 (2025)

    work page arXiv 2025

  17. [17]

    Khaneja and S

    N. Khaneja and S. J. Glaser, Cartan decomposition of SU(2 n)and control of spin systems, Chem. Phys.267, 11 (2001)

    2001

  18. [18]

    M. B. Mansky, S. L. Castillo, V. R. Puigvert, and C. Linnhoff-Popien, Near-optimal quantum circuit con- struction via Cartan decomposition, Phys. Rev. A108, 052607 (2023)

    2023

  19. [19]

    Dagli, D

    M. Dagli, D. D’Alessandro, and J. D. Smith, A gen- eral framework for recursive decompositions of unitary quantum evolutions, J. Phys. A-Math. Theor.41, 155302 (2008)

    2008

  20. [20]

    Kottmann, D

    K. Kottmann, D. Wierichs, G. Alonso-Linaje, and N. Killo- ran, Parameter-optimal unitary synthesis with flag decom- positions, arXiv preprint arXiv:2603.20376 (2026)

    work page arXiv 2026

  21. [21]

    Braccia, N

    P. Braccia, N. Diaz, M. Larocca, M. Cerezo, and D. García-Martín, Optimal haar random fermionic lin- ear optics circuits, arXiv preprint arXiv:2505.24212 10.48550/arXiv.2505.24212 (2025)

    work page doi:10.48550/arxiv.2505.24212 2025

  22. [22]

    Cartan, Sur une classe remarquable d’espaces de Rie- mann, Bull

    E. Cartan, Sur une classe remarquable d’espaces de Rie- mann, Bull. Soc. Math. Fr.54, 214 (1926)

    1926

  23. [23]

    Gorodski, An introduction to Riemannian symmetric spaces, in7th School and Workshop on Lie Theory(2021) pp

    C. Gorodski, An introduction to Riemannian symmetric spaces, in7th School and Workshop on Lie Theory(2021) pp. 8–15

    2021

  24. [24]

    Magnea, An introduction to symmetric spaces, arXiv preprint cond-mat/0205288 (2002)

    U. Magnea, An introduction to symmetric spaces, arXiv preprint cond-mat/0205288 (2002)

    work page internal anchor Pith review arXiv 2002

  25. [25]

    M. R. Zirnbauer, Symmetry classes, arXiv preprint arXiv:1001.0722 (2010)

    work page arXiv 2010

  26. [26]

    Duenez, Random matrix ensembles associated to com- pact symmetric spaces, Communications in mathematical physics244, 29 (2004)

    E. Duenez, Random matrix ensembles associated to com- pact symmetric spaces, Communications in mathematical physics244, 29 (2004)

    2004

  27. [27]

    Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, arXiv preprint arXiv:1301.5401 (2013)

    S. Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, arXiv preprint arXiv:1301.5401 (2013)

    work page arXiv 2013

  28. [28]

    Fulton and J

    W. Fulton and J. Harris,Representation Theory: A First Course(Springer, 1991)

    1991

  29. [29]

    A. A. Mele, Introduction to haar measure tools in quan- tum information: A beginner’s tutorial, Quantum8, 1340 (2024)

    2024

  30. [30]

    Collins and P

    B. Collins and P. Śniady, Integration with respect to the haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics264, 773 (2006)

    2006

  31. [31]

    Schuster, J

    T. Schuster, J. Haferkamp, and H.-Y. Huang, Random uni- taries in extremely low depth, Science389, 92 (2025)

    2025

  32. [32]

    LaRacuente and F

    N. LaRacuente and F. Leditzky, Approximate unitaryk- designs from shallow, low-communication circuits, arXiv preprint arXiv:2407.07876 (2024)

    work page arXiv 2024

  33. [33]

    M. West, D. García-Martín, N. Diaz, M. Cerezo, and M. Larocca, No-go theorems for sublinear-depth group de- signs, arXiv preprint arXiv:2506.16005 (2025)

    work page arXiv 2025

  34. [34]

    Grevink, J

    L. Grevink, J. Haferkamp, M. Heinrich, J. Helsen, M. Hin- sche, T. Schuster, and Z. Zimborás, Will it glue? on short- depth designs beyond the unitary group, arXiv preprint arXiv:2506.23925 (2025)

    work page arXiv 2025

  35. [35]

    Hackl and E

    L. Hackl and E. Bianchi, Bosonic and fermionic gaussian states from Kähler structures, SciPost Phys. Core4, 025 (2021)

    2021

  36. [36]

    twirling

    M. West, On the average-case complexity of learning states from the circular and gaussian ensembles, arXiv preprint arXiv:2601.10197 (2026). Appendix A: Integration over symmetric spaces In this appendix we discuss the basic details of integrating over (classical, compact, type I) symmetric spaces; more details may be found in Refs. [26, 27]. We recall th...

    work page arXiv 2026

  37. [37]

    (ρ⊗1) Z U∼U(d) (U TU) ⊗2Π⊗2 w (U TU) †⊗2 # (C12) = X αβγδ ijkl X w tr1

    Sample complexities Next let us turn our attention to the question of thevarianceof the classical shadow estimators. By the general relationVar[ˆo] =E[ˆo2]−E[ˆo]2 ⩽E[ˆo2]and the readily seen Hermiticity ofMwith respect to the Hilbert-Schmidt inner product, we obtain the commonly used bound VarE,W [ˆo]⩽ X w Z V∼E tr ρV †ΠwV tr OM−1 V †ΠwV 2 (B46) = X w Z V...