pith. sign in

arxiv: 2605.27191 · v1 · pith:T4IIX4BVnew · submitted 2026-05-26 · 🪐 quant-ph · eess.SP· math.OC

Statistical and Algorithmic Foundations of Probing Quantum Systems with Compressive Measurements: A Review

Zhen Qin , Michael B. Wakin , Zhihui Zhu This is my paper

Pith reviewed 2026-06-29 16:36 UTC · model grok-4.3

classification 🪐 quant-ph eess.SPmath.OC
keywords quantum state tomographycompressive sensingstructured recoverymeasurement designoptimization algorithmssample complexity
0
0 comments X

The pith

Structured quantum states enable scalable tomography by reducing degrees of freedom through compressive measurements and algorithms.

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

Quantum state tomography reconstructs unknown states but is hindered by exponential Hilbert space growth. This review examines how prior structures such as low-rankness, tensor-network representations, shallow circuits, and neural quantum states cut the effective degrees of freedom. It surveys measurement designs that preserve geometry for efficient sampling and optimization algorithms for recovery. Connecting these to compressive sensing and matrix sensing reveals shared foundations for sample complexity and scalable reconstruction.

Core claim

The survey provides a unified perspective on structured quantum state tomography through compact state representations, geometric preservation in measurement frameworks, and optimization algorithms, showing that these enable scalable recovery by drawing on principles from compressive sensing and structured inverse problems.

What carries the argument

The three interconnected themes of compact state representations, measurement design ranging from informationally complete POVMs to randomized measurements, and computational algorithms for reconstruction from empirical data.

If this is right

  • Sample complexity bounds depend on the specific structure of the quantum state.
  • Geometric properties of measurements determine the efficiency of information extraction.
  • Optimization methods allow practical reconstruction of structured states.
  • Common theoretical foundations apply across compressive sensing and quantum tomography.

Where Pith is reading between the lines

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

  • Experimental setups could prioritize measurements that exploit known structures in quantum systems.
  • Similar approaches might apply to other high-dimensional inverse problems in physics.
  • Future work could test these methods on larger systems to verify scalability claims.

Load-bearing premise

The models of structured quantum states and the geometric preservation properties of the measurements accurately reflect the conditions needed for scalable recovery.

What would settle it

An experiment or calculation showing that a structured quantum state requires as many measurements as a general state for accurate reconstruction despite the structure.

Figures

Figures reproduced from arXiv: 2605.27191 by Michael B. Wakin, Zhen Qin, Zhihui Zhu.

Figure 1
Figure 1. Figure 1: Illustration of the MPO in (30) from two perspectives: (a) each entry of the density matrix can be [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the PEPO from each element of the density matrix is illustrated in a diagrammatic [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.

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 / 1 minor

Summary. The manuscript is a review surveying quantum state tomography (QST) for structured quantum states. It organizes the literature around three themes—compact state representations (low-rank, tensor-network, shallow-circuit, and neural quantum states), measurement design frameworks ranging from informationally complete POVMs to randomized measurements and their geometric preservation properties, and optimization algorithms for reconstruction—while connecting these to principles from compressive sensing, matrix sensing, and structured inverse problems to highlight shared foundations for sample complexity, measurement efficiency, and scalable recovery.

Significance. If the synthesis holds, the review offers a useful consolidation of results on how prior structure reduces effective degrees of freedom in QST. By framing disparate QST results under compressive-sensing concepts, it could help readers identify cross-cutting theoretical tools for sample-complexity bounds and recovery guarantees, serving as a reference point for the quantum information community.

minor comments (1)
  1. [Abstract] Abstract: the phrase 'prior structure... can substantially reduce the effective degrees of freedom' is repeated in the abstract and introduction; a single consolidated statement would improve conciseness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript, for recognizing its potential significance as a consolidation of results on structured quantum state tomography under compressive-sensing principles, and for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity: survey with no internal derivations

full rationale

The paper is a review surveying prior literature on structured quantum state tomography, compressive sensing, and related inverse problems. It presents no original derivations, equations, or predictions that could reduce to fitted inputs or self-citations by construction. All content consists of summaries of external results, with the abstract and structure confirming its role as a unifying perspective rather than a self-contained proof chain. No load-bearing steps match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; no new free parameters, axioms, or invented entities are introduced in the abstract. The survey relies on standard assumptions from quantum information and compressive sensing literature.

pith-pipeline@v0.9.1-grok · 5738 in / 1017 out tokens · 25732 ms · 2026-06-29T16:36:41.609159+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

174 extracted references · 20 canonical work pages · 7 internal anchors

  1. [1]

    Cambridgeuniversity press, 2010

    M.A.NielsenandI.L.Chuang,Quantumcomputationandquantuminformation. Cambridgeuniversity press, 2010. 29

    2010

  2. [2]

    Quantum comput- ers,

    T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum comput- ers,”nature, vol. 464, no. 7285, pp. 45–53, 2010

    2010

  3. [3]

    Quantum communication,

    N. Gisin and R. Thew, “Quantum communication,”Nature photonics, vol. 1, no. 3, pp. 165–171, 2007

    2007

  4. [4]

    Quantum sensing,

    C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,”Reviews of modern physics, vol. 89, no. 3, p. 035002, 2017

    2017

  5. [5]

    Quantum computing in the nisq era and beyond,

    J. Preskill, “Quantum computing in the nisq era and beyond,”Quantum, vol. 2, p. 79, 2018

    2018

  6. [6]

    Ibmquantumbreaksthe100-qubitprocessorbarrier,

    J.Chow,O.Dial,andJ.Gambetta,“Ibmquantumbreaksthe100-qubitprocessorbarrier,”IBMResearch Blog, 2021

    2021

  7. [7]

    Quantum simulations with ultracold quantum gases,

    I. Bloch, J. Dalibard, and S. Nascimbene, “Quantum simulations with ultracold quantum gases,”Na- ture Physics, vol. 8, no. 4, pp. 267–276, 2012

    2012

  8. [8]

    Measurement of qubits,

    D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,”Physical Review A, vol. 64, no. 5, p. 052312, 2001

    2001

  9. [9]

    A tomographic approach to wigner’s function,

    J. Bertrand and P. Bertrand, “A tomographic approach to wigner’s function,”Foundations of Physics, vol. 17, no. 4, pp. 397–405, 1987

    1987

  10. [10]

    Determination of quasiprobability distributions interms of probability dis- tributions for the rotated quadrature phase,

    K. Vogel andH. Risken, “Determination of quasiprobability distributions interms of probability dis- tributions for the rotated quadrature phase,”Physical Review A, vol. 40, no. 5, p. 2847, 1989

    1989

  11. [11]

    Quantum-state tomography and discrete wigner function,

    U. Leonhardt, “Quantum-state tomography and discrete wigner function,”Physical review letters, vol. 74, no. 21, p. 4101, 1995

    1995

  12. [12]

    Quantum-state estimation,

    Z. Hradil, “Quantum-state estimation,”Physical Review A, vol. 55, no. 3, p. R1561, 1997

    1997

  13. [13]

    Quantumtomographyviacompressedsensing: error bounds, sample complexity and efficient estimators,

    S.T.Flammia,D.Gross,Y.-K.Liu,andJ.Eisert,“Quantumtomographyviacompressedsensing: error bounds, sample complexity and efficient estimators,”New Journal of Physics, vol. 14, no. 9, p. 095022, 2012

    2012

  14. [14]

    Matrix product state representations,

    D. Perez-García, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix product state representations,” Quantum Information and Computation, vol. 7, no. 5-6, pp. 401–430, 2007

    2007

  15. [15]

    Matrixproductoperatorrepresentations,

    B.Pirvu,V.Murg,J.I.Cirac,andF.Verstraete,“Matrixproductoperatorrepresentations,”NewJournal of Physics, vol. 12, no. 2, p. 025012, 2010

    2010

  16. [16]

    Matrix product states and projected entan- gledpairstates: Concepts,symmetries,theorems,

    J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, “Matrix product states and projected entan- gledpairstates: Concepts,symmetries,theorems,”ReviewsofModernPhysics,vol.93,no.4,p.045003, 2021

    2021

  17. [17]

    Learningshallow quantumcircuits,

    H.-Y.Huang,Y.Liu,M.Broughton,I.Kim,A.Anshu,Z.Landau,andJ.R.McClean,“Learningshallow quantumcircuits,”inProceedingsofthe56thAnnualACMSymposiumonTheoryofComputing,pp.1343– 1351, 2024

    2024

  18. [18]

    From architectures to applications: A review of neural quantum states,

    H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, “From architectures to applications: A review of neural quantum states,”arXiv preprint arXiv:2402.09402, 2024

    work page arXiv 2024

  19. [19]

    Distinguishability of quantum states under restricted fam- ilies of measurements with an application to quantum data hiding,

    W. Matthews, S. Wehner, and A. Winter, “Distinguishability of quantum states under restricted fam- ilies of measurements with an application to quantum data hiding,”Communications in Mathematical Physics, vol. 291, pp. 813–843, 2009

    2009

  20. [20]

    Low rank matrix recovery from rank one measurements,

    R. Kueng, H. Rauhut, and U. Terstiege, “Low rank matrix recovery from rank one measurements,” Applied and Computational Harmonic Analysis, vol. 42, no. 1, pp. 88–116, 2017

    2017

  21. [21]

    Fast state tomography with optimal error bounds,

    M. Guţă, J. Kahn, R. Kueng, and J. A. Tropp, “Fast state tomography with optimal error bounds,” Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 20, p. 204001, 2020. 30

    2020

  22. [22]

    Evenly distributed unitaries: On the structure of unitary de- signs,

    D. Gross, K. Audenaert, and J. Eisert, “Evenly distributed unitaries: On the structure of unitary de- signs,”Journal of mathematical physics, vol. 48, no. 5, 2007

    2007

  23. [23]

    Exact and approximate unitary 2-designs and their application to fidelity estimation,

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, “Exact and approximate unitary 2-designs and their application to fidelity estimation,”Physical Review A—Atomic, Molecular, and Optical Physics, vol. 80, no. 1, p. 012304, 2009

    2009

  24. [24]

    Unitarydesignsandcodes,

    A.RoyandA.J.Scott,“Unitarydesignsandcodes,”Designs,codesandcryptography,vol.53,pp.13–31, 2009

    2009

  25. [25]

    Quantum Tomography From Few Full-Rank Observables

    V. Voroninski, “Quantum tomography from few full-rank observables,”arXiv preprint arXiv:1309.7669, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  26. [26]

    Distinguishing multi-partite states by local measurements,

    C. Lancien and A. Winter, “Distinguishing multi-partite states by local measurements,”Communica- tions in Mathematical Physics, vol. 323, pp. 555–573, 2013

    2013

  27. [27]

    Fastandrobustquantumstatetomographyfromfewbasis measurements,

    F.G.Brandão,R.Kueng,andD.S.França,“Fastandrobustquantumstatetomographyfromfewbasis measurements,”arXiv preprint arXiv:2009.08216, 2020

    work page arXiv 2009

  28. [28]

    Optimal quantum state tomogra- phy with local informationally complete measurements,

    C. Jameson, Z. Qin, A. Goldar, M. B. Wakin, Z. Zhu, and Z. Gong, “Optimal quantum state tomogra- phy with local informationally complete measurements,”arXiv preprint arXiv:2408.07115, 2024

    work page arXiv 2024

  29. [29]

    When does adaptivity help for quantum state learn- ing?,

    S. Chen, B. Huang, J. Li, A. Liu, and M. Sellke, “When does adaptivity help for quantum state learn- ing?,” in2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pp. 391–404, IEEE, 2023

    2023

  30. [30]

    Adaptivity can help exponentially for shadow tomography,

    S. Chen, W. Gong, and Z. Zhang, “Adaptivity can help exponentially for shadow tomography,”arXiv preprint arXiv:2412.19022, 2024

    work page arXiv 2024

  31. [31]

    Compressive sensing [lecture notes],

    R. G. Baraniuk, “Compressive sensing [lecture notes],”IEEE signal processing magazine, vol. 24, no. 4, pp. 118–121, 2007

    2007

  32. [32]

    Non-square matrix sensing without spurious localminimaviatheburer-monteiroapproach,

    D. Park, A. Kyrillidis, C. Carmanis, and S. Sanghavi, “Non-square matrix sensing without spurious localminimaviatheburer-monteiroapproach,”inArtificialintelligenceandstatistics,pp.65–74,PMLR, 2017

    2017

  33. [33]

    Nonconvex optimization meets low-rank matrix factorization: An overview,

    Y. Chi, Y. M. Lu, and Y. Chen, “Nonconvex optimization meets low-rank matrix factorization: An overview,”IEEE Transactions on Signal Processing, vol. 67, no. 20, pp. 5239–5269, 2019

    2019

  34. [34]

    Compressedsensing,

    D.L.Donoho, “Compressedsensing,”IEEETransactionsoninformationtheory,vol.52, no.4, pp.1289– 1306, 2006

    2006

  35. [35]

    Near-optimalsignalrecoveryfromrandomprojections: Universalencoding strategies?,

    E.J.CandesandT.Tao, “Near-optimalsignalrecoveryfromrandomprojections: Universalencoding strategies?,”IEEE transactions on information theory, vol. 52, no. 12, pp. 5406–5425, 2006

    2006

  36. [36]

    Tightoracleinequalitiesforlow-rankmatrixrecoveryfromaminimalnum- ber of noisy random measurements,

    E.J.CandesandY.Plan,“Tightoracleinequalitiesforlow-rankmatrixrecoveryfromaminimalnum- ber of noisy random measurements,”IEEE Transactions on Information Theory, vol. 57, no. 4, pp. 2342– 2359, 2011

    2011

  37. [37]

    Boumal,An introduction to optimization on smooth manifolds

    N. Boumal,An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

    2023

  38. [38]

    Therestrictedisometrypropertyanditsimplicationsforcompressedsensing,

    E.J.Candes,“Therestrictedisometrypropertyanditsimplicationsforcompressedsensing,”Comptes rendus mathematique, vol. 346, no. 9-10, pp. 589–592, 2008

    2008

  39. [39]

    Convexrecoveryofastructuredsignalfromindependentrandomlinearmeasurements,

    J.A.Tropp,“Convexrecoveryofastructuredsignalfromindependentrandomlinearmeasurements,” inSampling Theory, a Renaissance, pp. 67–101, Springer, 2015

    2015

  40. [40]

    Newanalysisofmanifoldembeddingsandsignalrecoveryfromcom- pressivemeasurements,

    A.EftekhariandM.B.Wakin,“Newanalysisofmanifoldembeddingsandsignalrecoveryfromcom- pressivemeasurements,”AppliedandComputationalHarmonicAnalysis,vol.39,no.1,pp.67–109,2015. 31

    2015

  41. [41]

    Randomprojectionsofsmoothmanifolds,

    R.G.BaraniukandM.B.Wakin, “Randomprojectionsofsmoothmanifolds,”Foundationsofcomputa- tional mathematics, vol. 9, no. 1, pp. 51–77, 2009

    2009

  42. [42]

    Predicting many properties of a quantum system from very few measurements,

    H.-Y. Huang, R. Kueng, and J. Preskill, “Predicting many properties of a quantum system from very few measurements,”Nature Physics, vol. 16, no. 10, pp. 1050–1057, 2020

    2020

  43. [43]

    A simple proof of the restricted isometry property for random matrices,

    R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,”Constructive Approximation, vol. 28, no. 3, pp. 253–263, 2008

    2008

  44. [44]

    Atomicdecompositionbybasispursuit,

    S.S.Chen,D.L.Donoho,andM.A.Saunders,“Atomicdecompositionbybasispursuit,”SIAMreview, vol. 43, no. 1, pp. 129–159, 2001

    2001

  45. [45]

    PhDthesis,PhDthesis,StanfordUniversity,2002

    M.Fazel,Matrixrankminimizationwithapplications. PhDthesis,PhDthesis,StanfordUniversity,2002

    2002

  46. [46]

    Robust uncertainty principles: Exact signal reconstruction fromhighlyincompletefrequencyinformation,

    E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction fromhighlyincompletefrequencyinformation,”IEEETransactionsoninformationtheory,vol.52,no.2, pp. 489–509, 2006

    2006

  47. [47]

    Guaranteedminimum-ranksolutionsoflinearmatrixequations via nuclear norm minimization,

    B.Recht,M.Fazel,andP.A.Parrilo,“Guaranteedminimum-ranksolutionsoflinearmatrixequations via nuclear norm minimization,”SIAM review, vol. 52, no. 3, pp. 471–501, 2010

    2010

  48. [48]

    Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,

    R. Aharoni and Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,”Linear Algebra and Its Applications, vol. 120, pp. 165–175, 1989

    1989

  49. [49]

    Alternating projections on manifolds,

    A. S. Lewis and J. Malick, “Alternating projections on manifolds,”Mathematics of Operations Research, vol. 33, no. 1, pp. 216–234, 2008

    2008

  50. [50]

    Hard thresholding pursuit: an algorithm for compressive sensing,

    S. Foucart, “Hard thresholding pursuit: an algorithm for compressive sensing,”SIAM Journal on nu- merical analysis, vol. 49, no. 6, pp. 2543–2563, 2011

    2011

  51. [51]

    Iterative hard thresholding for compressed sensing,

    T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,”Applied and computational harmonic analysis, vol. 27, no. 3, pp. 265–274, 2009

    2009

  52. [52]

    A singular value thresholding algorithm for matrix completion,

    J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM Journal on optimization, vol. 20, no. 4, pp. 1956–1982, 2010

    1956

  53. [53]

    Lowranktensorrecoveryviaiterativehardthresholding,

    H.Rauhut,R.Schneider,andŽ.Stojanac,“Lowranktensorrecoveryviaiterativehardthresholding,” Linear Algebra and its Applications, vol. 523, pp. 220–262, 2017

    2017

  54. [54]

    Findinglow-ranksolutionsvianonconvexma- trix factorization, efficiently and provably,

    D.Park,A.Kyrillidis,C.Caramanis,andS.Sanghavi,“Findinglow-ranksolutionsvianonconvexma- trix factorization, efficiently and provably,”SIAM Journal on Imaging Sciences, vol. 11, no. 4, pp. 2165– 2204, 2018

    2018

  55. [55]

    Structuredlow-rankmatrixfactorization: Globaloptimality,algorithms, andapplications,

    B.D.HaeffeleandR.Vidal,“Structuredlow-rankmatrixfactorization: Globaloptimality,algorithms, andapplications,”IEEEtransactionsonpatternanalysisandmachineintelligence,vol.42,no.6,pp.1468– 1482, 2019

    2019

  56. [56]

    An optimal statistical and computational framework for gener- alized tensor estimation,

    R. Han, R. Willett, and A. R. Zhang, “An optimal statistical and computational framework for gener- alized tensor estimation,”The Annals of Statistics, vol. 50, no. 1, pp. 1–29, 2022

    2022

  57. [57]

    A scalable factorization approach for high-order structured tensor recovery,

    Z. Qin, M. B. Wakin, and Z. Zhu, “A scalable factorization approach for high-order structured tensor recovery,”arXiv preprint arXiv:2506.16032, 2025

    work page arXiv 2025

  58. [58]

    Therandom- ized measurement toolbox,

    A.Elben,S.T.Flammia,H.-Y.Huang,R.Kueng,J.Preskill,B.Vermersch,andP.Zoller,“Therandom- ized measurement toolbox,”Nature Reviews Physics, vol. 5, no. 1, pp. 9–24, 2023

    2023

  59. [59]

    Quantumstatetomographyformatrixprod- uct density operators,

    Z.Qin,C.Jameson,Z.Gong,M.B.Wakin,andZ.Zhu,“Quantumstatetomographyformatrixprod- uct density operators,”IEEE Transactions on Information Theory, vol. 70, no. 7, pp. 5030–5056, 2024

    2024

  60. [60]

    Sample-efficient quantum state tomography for structured quantum states in one dimension,

    Z. Qin, C. Jameson, A. Goldar, M. B. Wakin, Z. Gong, and Z. Zhu, “Sample-efficient quantum state tomography for structured quantum states in one dimension,”arXiv preprint arXiv:2410.02583, 2024. 32

    work page arXiv 2024

  61. [61]

    Quantum state tomography for tensor networks in two dimensions,

    Z. Qin and Z. Zhu, “Quantum state tomography for tensor networks in two dimensions,”Physical Review A, vol. 113, no. 2, p. 022414, 2026

    2026

  62. [63]

    Precision-guaranteed quantum tomography,

    T. Sugiyama, P. S. Turner, and M. Murao, “Precision-guaranteed quantum tomography,”Physical Re- view Letters, vol. 111, no. 16, p. 160406, 2013

    2013

  63. [64]

    Efficient quantum state tomography,

    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,”Nature communications, vol. 1, no. 1, pp. 1–7, 2010

    2010

  64. [65]

    Enhancingquantumstatereconstructionwithstructured classical shadows,

    Z.Qin,J.M.Lukens,B.T.Kirby,andZ.Zhu,“Enhancingquantumstatereconstructionwithstructured classical shadows,”npj Quantum Information, vol. 11, no. 1, p. 147, 2025

    2025

  65. [66]

    Quan- tum certification and benchmarking,

    J.Eisert,D.Hangleiter,N.Walk,I.Roth,D.Markham,R.Parekh,U.Chabaud,andE.Kashefi,“Quan- tum certification and benchmarking,”Nature Reviews Physics, vol. 2, no. 7, pp. 382–390, 2020

    2020

  66. [67]

    A survey on the complexity of learning quantum states,

    A. Anshu and S. Arunachalam, “A survey on the complexity of learning quantum states,”Nature Reviews Physics, vol. 6, no. 1, pp. 59–69, 2024

    2024

  67. [68]

    Quantum computation and quantum information,

    M. A. Nielsen and I. Chuang, “Quantum computation and quantum information,” 2002

    2002

  68. [69]

    T. A. Severini,Elements of distribution theory, vol. 17. Cambridge University Press, 2005

    2005

  69. [70]

    Compressed sensing using generative models,

    A. Bora, A. Jalal, E. Price, and A. G. Dimakis, “Compressed sensing using generative models,” in International conference on machine learning, pp. 537–546, PMLR, 2017

    2017

  70. [71]

    A simple proof of the restricted isometry property for random matrices,

    R. G. Baraniuk, M. A. Davenport, R. A. DeVore, and M. B. Wakin, “A simple proof of the restricted isometry property for random matrices,”Constructive Approximation, 2007

    2007

  71. [72]

    Shadow tomography based on informationally complete positive operator-valued measure,

    A. Acharya, S. Saha, and A. M. Sengupta, “Shadow tomography based on informationally complete positive operator-valued measure,”Physical Review A, vol. 104, p. 052418, Nov 2021

    2021

  72. [73]

    Tight informationally complete quantum measurements,

    A. J. Scott, “Tight informationally complete quantum measurements,”Journal of Physics A: Mathemat- ical and General, vol. 39, no. 43, p. 13507, 2006

    2006

  73. [74]

    Apartialderandomizationofphaseliftusingsphericaldesigns,

    D.Gross,F.Krahmer,andR.Kueng,“Apartialderandomizationofphaseliftusingsphericaldesigns,” Journal of Fourier Analysis and Applications, vol. 21, no. 2, pp. 229–266, 2015

    2015

  74. [75]

    Accessible information and informational power of quantum 2-designs,

    M. Dall’Arno, “Accessible information and informational power of quantum 2-designs,”Physical Re- view A, vol. 90, no. 5, p. 052311, 2014

    2014

  75. [76]

    Averaging sets: a generalization of mean values and spherical de- signs,

    P. D. Seymour and T. Zaslavsky, “Averaging sets: a generalization of mean values and spherical de- signs,”Advances in Mathematics, vol. 52, no. 3, pp. 213–240, 1984

    1984

  76. [77]

    Construction of spherical t-designs,

    B. Bajnok, “Construction of spherical t-designs,”Geometriae Dedicata, vol. 43, pp. 167–179, 1992

    1992

  77. [78]

    Reexamination of optimal quantum state estimation of pure states,

    A. Hayashi, T. Hashimoto, and M. Horibe, “Reexamination of optimal quantum state estimation of pure states,”Physical Review A—Atomic, Molecular, and Optical Physics, vol. 72, no. 3, p. 032325, 2005

    2005

  78. [79]

    Mutually unbiased bases are complex projective 2-designs,

    A. Klappenecker and M. Rotteler, “Mutually unbiased bases are complex projective 2-designs,” in Proceedings.InternationalSymposiumonInformationTheory,2005.ISIT2005.,pp.1740–1744,IEEE,2005

    2005

  79. [80]

    Qubit stabilizer states are complex projective 3-designs

    R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs,”arXiv preprint arXiv:1510.02767, 2015

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  80. [81]

    Approximation by Quantum Circuits

    E. Knill, “Approximation by quantum circuits,”arXiv:quant-ph/9508006, 1995

    work page internal anchor Pith review Pith/arXiv arXiv 1995

Showing first 80 references.