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Efficient Distributed Inner-Product Estimation via Pauli Sampling

M. Hinsche1,*,†, M. Ioannou1,*,‡, S. Jerbi1, L. Leone1, J. Eisert1,2, and J. Carrasco1,*,§

  • *These authors contributed equally to this work.
  • †Contact author: m.hinsche@fu-berlin.de
  • ‡Contact author: marios.ionannou@fu-berlin.de
  • §Contact author: jose.carrasco@fu-berlin.de
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PRX Quantum 6, 030354 – Published 18 September, 2025

DOI: https://doi.org/10.1103/g53f-z8cr

Abstract

Cross-platform verification is the task of comparing the output states produced by different physical platforms using solely local quantum operations and classical communication. While protocols have previously been suggested for this task, their exponential sample complexity renders them impractical even for intermediate-scale quantum systems. In this work, we propose a novel protocol for this task based on Pauli sampling, a subroutine that generates Paulis distributed according to their weight in the expansion of a quantum state in the Pauli basis. We show that our protocols for both Pauli sampling and cross-platform verification are efficient for pure states with low magic and entanglement [i.e., of the order O(logn)]. Conversely, we show superpolynomial lower bounds on the complexity of both tasks for states with ω(log(n)) magic and entanglement. Interestingly, when considering states with real amplitudes, the requirements of our protocol for cross-platform verification can be significantly weakened.

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Popular Summary

Progress in quantum technologies has been rapid in recent years, with quantum simulations and computers being built that seemed inconceivable not very long ago. That said, such devices have predictive power only if it is clear that they function precisely as anticipated. This basic yet profound insight focuses attention on the tasks of verification and benchmarking. While a number of techniques have been developed to characterize the quality of components of quantum devices, often, by resorting to random measurements as in randomized benchmarking, it is much less clear how to certify the correctness of the functioning of entire devices.

In fact, it is in the very nature of quantum devices to go beyond classical computational means, so one cannot simply classically and efficiently simulate a quantum device and then check whether the outcome has been right. But if this is not possible, how can one ever be sure that a device has been prepared in precisely the right quantum state? To benchmark quantum devices with quantum devices may at first seem like an insurmountable chicken-and-egg problem, giving rise to a substantial bottleneck.

In this work, we make substantial progress toward resolving this bottleneck. We ask what properties of pairs of quantum states render cross-platform verification, i.e., the computation of their inner product, efficient. This has to be done within the separate laboratory paradigm—as one preparation may have been done with some architecture in one laboratory and the other preparation in another one. In our work, we propose novel algorithms for the distributed estimation of tr(ρσ) and demonstrate regimes in which these algorithms run efficiently. At the same time, we identify some limitations: we show that the task remains hard even if we severely restrict the class of input states. Interestingly, both our hardness and easiness results connect to commonly studied properties of states, namely, their magic and their multipartite entanglement. We are able to formulate our results by bringing the technique of Pauli sampling to a new level.

Our work provides a framework that allows us to assess when one should perform cross-platform verification and to literally verify the performance of one intractable quantum system with another one.

Article Text

References (57)

  1. J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).
  2. R. O’Donnell and J. Wright, in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’16 (Association for Computing Machinery, New York, 2016), p. 899.
  3. M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Practical characterization of quantum devices without tomography, Phys. Rev. Lett. 107, 210404 (2011).
  4. S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few Pauli measurements, Phys. Rev. Lett. 106, 230501 (2011).
  5. A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. Joshi, R. Blatt, C. F. Roos, and P. Zoller, Cross-platform verification of intermediate scale quantum devices, Phys. Rev. Lett. 124, 010504 (2020).
  6. A. Anshu, Z. Landau, and Y. Liu, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2022), p. 44.
  7. J. Carrasco, A. Elben, C. Kokail, B. Kraus, and P. Zoller, Theoretical and experimental perspectives of quantum verification, PRX Quantum 2, 010102 (2021).
  8. Y. Qian, Y. Du, Z. He, M.-h. Hsieh, and D. Tao, Multimodal deep representation learning for quantum cross-platform verification, Phys. Rev. Lett. 133, 130601 (2024).
  9. C. Zheng, X. Yu, and K. Wang, Cross-platform comparison of arbitrary quantum processes, npj Quantum Inf. 10, 4 (2024).
  10. D. Zhu, Z.-P. Cian, C. Noel, A. Risinger, D. Biswas, L. Egan, Y. Zhu, A. M. Green, C. H. Alderete, N. H. Nguyen, Q. Wang, A. Maksymov, Y. Nam, M. Cetina, N. M. Linke, M. Hafezi, and C. Monroe, Cross-platform comparison of arbitrary quantum computations, Nat. Commun. 13, 6620 (2022).
  11. A. Montanaro, Learning stabilizer states by Bell sampling, ArXiv:1707.04012.
  12. D. Gross, S. Nezami, and M. Walter, Schur-Weyl duality for the Clifford group with applications: Property testing, a robust Hudson theorem, and de Finetti representations, Commun. Math. Phys. 385, 1325 (2021).
  13. H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Quantum advantage in learning from experiments, Science 376, 1182 (2022).
  14. D. Hangleiter and M. J. Gullans, Bell sampling from quantum circuits, Phys. Rev. Lett. 133, 020601 (2024).
  15. S. Grewal, V. Iyer, W. Kretschmer, and D. Liang, Efficient learning of quantum states prepared with few non-Clifford gates, ArXiv:2305.13409.
  16. S. Grewal, V. Iyer, W. Kretschmer, and D. Liang, in Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024 (Association for Computing Machinery, New York, 2024), p. 1352.
  17. F. E. Gutiérrez, Simple algorithms to test and learn local Hamiltonians, ArXiv:2404.06282.
  18. S. Bravyi, D. Gosset, and Y. Liu, How to simulate quantum measurement without computing marginals, Phys. Rev. Lett. 128, 220503 (2022).
  19. M. J. Bremner, A. Montanaro, and D. J. Shepherd, Achieving quantum supremacy with sparse and noisy commuting quantum computations, Quantum 1, 8 (2017).
  20. P. Clifford and R. Clifford, in Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’18 (Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2018), p. 146.
  21. L. Leone, S. F. E. Oliviero, and A. Hamma, Learning t-doped stabilizer states, Quantum 8, 1361 (2024).
  22. N.-H. Chia, C.-Y. Lai, and H.-H. Lin, Efficient learning of t-doped stabilizer states with single-copy measurements, Quantum 8, 1250 (2024).
  23. L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer Rényi entropy, Phys. Rev. Lett. 128, 050402 (2022).
  24. L. Leone, S. F. E. Oliviero, and A. Hamma, Nonstabilizerness determining the hardness of direct fidelity estimation, Phys. Rev. A 107, 022429 (2023).
  25. G. Lami and M. Collura, Nonstabilizerness via perfect Pauli sampling of matrix product states, Phys. Rev. Lett. 131, 180401 (2023).
  26. T. Haug and L. Piroli, Stabilizer entropies and nonstabilizerness monotones, Quantum 7, 1092 (2023).
  27. P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dalmonte, Many-body magic via Pauli-Markov chains—from criticality to gauge theories, PRX Quantum 4, 040317 (2023).
  28. Z. Ji, Y.-K. Liu, and F. Song, in Advances in Cryptology—CRYPTO 2018: 38th Annual International Cryptology Conference, Santa Barbara, California, USA, August 19–23, 2018, Proceedings, Part III (Springer-Verlag, Berlin, Heidelberg, 2018), p. 126.
  29. S. Aaronson, A. Bouland, B. Fefferman, S. Ghosh, U. Vazirani, C. Zhang, and Z. Zhou, in 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2024).
  30. A. Gu, L. Leone, S. Ghosh, J. Eisert, S. F. Yelin, and Y. Quek, Pseudomagic quantum states, Phys. Rev. Lett. 132, 210602 (2024).
  31. T. Haug, K. Bharti, and D. E. Koh, Pseudorandom unitaries are neither real nor sparse nor noise-robust, Quantum 9, 1759 (2025).
  32. M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Inf. Comput. 7, 1 (2007).
  33. Z.-W. Liu and A. Winter, Many-body quantum magic, PRX Quantum 3, 020333 (2022).
  34. L. Leone and L. Bittel, Stabilizer entropies are monotones for magic-state resource theory, Phys. Rev. A 110, L040403 (2024).
  35. A. Gu, S. F. E. Oliviero, and L. Leone, Doped stabilizer states in many-body physics and where to find them, Phys. Rev. A 110, 062427 (2024).
  36. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011), 2nd ed.
  37. S. Bravyi, D. P. Divincenzo, R. Oliveira, and B. M. Terhal, The complexity of stoquastic local Hamiltonian problems, Quantum Inf. Comput. 8, 361 (2008).
  38. Y. Teng, R. Samajdar, K. Van Kirk, F. Wilde, S. Sachdev, J. Eisert, R. Sweke, and K. Najafi, Learning topological states from randomized measurements using variational tensor network tomography, ArXiv:2406.00193.
  39. M. Cramer, M. B. Plenio, S. T. Flammia, D. Gross, S. D. Bartlett, R. Somma, O. Landon-Cardinal, Y.-K. Liu, and D. Poulin, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  40. J. Denzler, S. Varona, T. Guaita, and J. Carrasco, Highly-entangled, highly-doped states that are efficiently cross-device verifiable, Phys. Rev. Res. 7, 023320 (2025).
  41. T. Haug, S. Lee, and M. S. Kim, Efficient quantum algorithms for stabilizer entropies, Phys. Rev. Lett. 132, 240602 (2024).
  42. W. Gong, J. Haferkamp, Q. Ye, and Z. Zhang, On the sample complexity of purity and inner product estimation, ArXiv:2410.12712.
  43. S. Arunachalam and L. Schatzki, in 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025), Leibniz International Proceedings in Informatics (LIPIcs) (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2025), Vol. 327, p. 11:1.
  44. T. Haug, L. Aolita, and M. S. Kim, Probing quantum complexity via universal saturation of stabilizer entropies, Quantum 9, 1801 (2025).
  45. G. Lami and M. Collura, Unveiling the stabilizer group of a matrix product state, Phys. Rev. Lett. 133, 010602 (2024).
  46. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, United Kingdom, 2018).
  47. The bound Sα(ψf,S)=ω(logn) applies only to cuts where the subsystem size is also at least ω(logn).
  48. S. Chen, J. Cotler, H.-Y. Huang, and J. Li, in 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE Computer Society, Los Alamitos, CA, USA, 2022), p. 574.
  49. H.-Y. Huang, R. Kueng, and J. Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021).
  50. H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett. 86, 910 (2001).
  51. O. Crawford, B. van Straaten, D. Wang, T. Parks, E. Campbell, and S. Brierley, Efficient quantum measurement of Pauli operators in the presence of finite sampling error, Quantum 5, 385 (2021).
  52. B. Wu, J. Sun, Q. Huang, and X. Yuan, Overlapped grouping measurement: A unified framework for measuring quantum states, Quantum 7, 896 (2023).
  53. C. Hadfield, S. Bravyi, R. Raymond, and A. Mezzacapo, Measurements of quantum Hamiltonians with locally-biased classical shadows, Commun. Math. Phys. 391, 951 (2022).
  54. A median-of-means estimator could be used to reduce the complexity of N1 to 2(1+Δ/λ2)/γ2ln(3/δ) (or O(log(δ−1)/ϵ2)), but for ease of exposition, we opt for a simpler empirical mean estimator.
  55. We now assume that ϵ2<1/16, so that ϵ2<1/4 and 1−ϵ2>3/4>1/2.
  56. R. F. Werner, All teleportation and dense coding schemes, J. Phys. A: Math. Gen. 34, 7081 (2001).
  57. D. Gross, K. Audenaert, and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48, 052104 (2007).

Outline

Information

PRX Quantum 6, 030354– Published 18 September, 2025
  • Received 4 August 2024
  • Revised 15 April 2025
  • Accepted 7 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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