Classical Verification of Quantum Learning

Authors Matthias C. Caro , Marcel Hinsche , Marios Ioannou, Alexander Nietner , Ryan Sweke



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Author Details

Matthias C. Caro
  • Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
  • Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA
Marcel Hinsche
  • Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
Marios Ioannou
  • Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
Alexander Nietner
  • Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
Ryan Sweke
  • IBM Quantum, Almaden Research Center, San Jose, CA, USA
  • Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany

Acknowledgements

We thank Jens Eisert for his valuable input to discussions on this project and for suggestions on improving the draft. We thank Srinivasan Arunachalam, Jack O'Connor, Yihui Quek, Jonathan Shafer, Thomas Vidick, and the ITCS reviewers for insightful comments and discussions.

Cite AsGet BibTex

Matthias C. Caro, Marcel Hinsche, Marios Ioannou, Alexander Nietner, and Ryan Sweke. Classical Verification of Quantum Learning. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.24

Abstract

Quantum data access and quantum processing can make certain classically intractable learning tasks feasible. However, quantum capabilities will only be available to a select few in the near future. Thus, reliable schemes that allow classical clients to delegate learning to untrusted quantum servers are required to facilitate widespread access to quantum learning advantages. Building on a recently introduced framework of interactive proof systems for classical machine learning, we develop a framework for classical verification of quantum learning. We exhibit learning problems that a classical learner cannot efficiently solve on their own, but that they can efficiently and reliably solve when interacting with an untrusted quantum prover. Concretely, we consider the problems of agnostic learning parities and Fourier-sparse functions with respect to distributions with uniform input marginal. We propose a new quantum data access model that we call "mixture-of-superpositions" quantum examples, based on which we give efficient quantum learning algorithms for these tasks. Moreover, we prove that agnostic quantum parity and Fourier-sparse learning can be efficiently verified by a classical verifier with only random example or statistical query access. Finally, we showcase two general scenarios in learning and verification in which quantum mixture-of-superpositions examples do not lead to sample complexity improvements over classical data. Our results demonstrate that the potential power of quantum data for learning tasks, while not unlimited, can be utilized by classical agents through interaction with untrusted quantum entities.

Subject Classification

ACM Subject Classification
  • Theory of computation → Machine learning theory
  • Theory of computation → Quantum computation theory
  • Theory of computation → Interactive proof systems
Keywords
  • computational learning theory
  • quantum learning theory
  • interactive proofs
  • quantum oracles
  • agnostic learning

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References

  1. Scott Aaronson. The learnability of quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088):3089-3114, 2007. URL: https://doi.org/10.1098/rspa.2007.0113.
  2. Scott Aaronson. Shadow tomography of quantum states. SIAM Journal on Computing, 49(5):STOC18-368, 2019. URL: https://doi.org/10.1137/18M120275X.
  3. Amira Abbas, David Sutter, Christa Zoufal, Aurélien Lucchi, Alessio Figalli, and Stefan Woerner. The power of quantum neural networks. Nature Computational Science, 1(6):403-409, 2021. URL: https://doi.org/10.1038/s43588-021-00084-1.
  4. Dorit Aharonov, Jordan Cotler, and Xiao-Liang Qi. Quantum algorithmic measurement. Nature Communications, 13(1):1-9, 2022. URL: https://doi.org/10.1038/s41467-021-27922-0.
  5. Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf. Two new results about quantum exact learning. Quantum, 5:587, November 2021. URL: https://doi.org/10.22331/q-2021-11-24-587.
  6. Srinivasan Arunachalam and Ronald de Wolf. Guest column: A survey of quantum learning theory. SIGACT News, 48, 2017. URL: https://doi.org/10.1145/3106700.3106710.
  7. Srinivasan Arunachalam and Ronald de Wolf. Optimal quantum sample complexity of learning algorithms. Journal of Machine Learning Research, 19(71):1-36, 2018. URL: http://jmlr.org/papers/v19/18-195.html.
  8. Srinivasan Arunachalam, Alex B. Grilo, and Henry Yuen. Quantum statistical query learning, 2020. URL: https://arxiv.org/abs/2002.08240.
  9. Alp Atıcı and Rocco A. Servedio. Improved bounds on quantum learning algorithms. Quantum Information Processing, 4(5):355-386, 2005. URL: https://doi.org/10.1007/s11128-005-0001-2.
  10. Alp Atıcı and Rocco A. Servedio. Quantum algorithms for learning and testing juntas. Quantum Information Processing, 6(5):323-348, 2007. URL: https://doi.org/10.1007/s11128-007-0061-6.
  11. Leonardo Banchi, Jason Pereira, and Stefano Pirandola. Generalization in quantum machine learning: A quantum information standpoint. PRX Quantum, 2(4):040321, 2021. URL: https://doi.org/10.1103/PRXQuantum.2.040321.
  12. Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. On the Power of Nonstandard Quantum Oracles. In Omar Fawzi and Michael Walter, editors, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023), volume 266 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:25, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2023.11.
  13. Debajyoti Bera and Sagnik Chatterjee. Efficient quantum agnostic improper learning of decision trees, 2022. URL: https://arxiv.org/abs/2210.00212.
  14. Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. SIAM Journal on Computing, 26(5):1411-1473, 1997. URL: https://doi.org/10.1137/S0097539796300921.
  15. Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the vapnik-chervonenkis dimension. Journal of the ACM (JACM), 36(4):929-965, 1989. URL: https://doi.org/10.1145/76359.76371.
  16. Anne Broadbent, Joseph Fitzsimons, and Elham Kashefi. Universal blind quantum computation. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science. IEEE, October 2009. URL: https://doi.org/10.1109/focs.2009.36.
  17. Nader H. Bshouty and Jeffrey C. Jackson. Learning DNF over the uniform distribution using a quantum example oracle. SIAM Journal on Computing, 28(3):1136-1153, 1998. URL: https://doi.org/10.1137/S0097539795293123.
  18. Ran Canetti and Ari Karchmer. Covert learning: How to learn with an untrusted intermediary. In Theory of Cryptography Conference, pages 1-31. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-90456-2_1.
  19. Matthias C. Caro. Quantum learning boolean linear functions w.r.t. product distributions. Quantum Information Processing, 19, 2020. URL: https://doi.org/10.1007/s11128-020-02661-1.
  20. Matthias C. Caro. Binary classification with classical instances and quantum labels. Quantum Machine Intelligence, 3, 2021. URL: https://doi.org/10.1007/s42484-021-00043-z.
  21. Matthias C. Caro. Learning quantum processes and hamiltonians via the pauli transfer matrix, 2022. URL: https://arxiv.org/abs/2212.04471.
  22. Matthias C. Caro and Ishaun Datta. Pseudo-dimension of quantum circuits. Quantum Machine Intelligence, 2:14, 2020. URL: https://doi.org/10.1007/s42484-020-00027-5.
  23. Matthias C. Caro, Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, and Ryan Sweke. Encoding-dependent generalization bounds for parametrized quantum circuits. Quantum, 5:582, 2021. URL: https://doi.org/10.22331/q-2021-11-17-582.
  24. Matthias C. Caro, Marcel Hinsche, Marios Ioannou, Alexander Nietner, and Ryan Sweke. Classical verification of quantum learning, 2023. URL: https://arxiv.org/abs/2306.04843.
  25. Matthias C Caro, Hsin-Yuan Huang, Marco Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J Coles. Generalization in quantum machine learning from few training data. Nature Communications, 13, 2022. URL: https://doi.org/10.1038/s41467-022-32550-3.
  26. Matthias C. Caro, Hsin-Yuan Huang, Nicholas Ezzell, Joe Gibbs, Andrew T. Sornborger, Lukasz Cincio, Patrick J. Coles, and Zoë Holmes. Out-of-distribution generalization for learning quantum dynamics. Nature Communications, 14, 2023. URL: https://doi.org/10.1038/s41467-023-39381-w.
  27. Kean Chen, Qisheng Wang, Peixun Long, and Mingsheng Ying. Unitarity estimation for quantum channels. IEEE Transactions on Information Theory, 69(8):5116-5134, 2023. URL: https://doi.org/10.1109/TIT.2023.3263645.
  28. Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. Exponential separations between learning with and without quantum memory. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 574-585. IEEE, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00063.
  29. Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. The complexity of nisq, 2022. URL: https://arxiv.org/abs/2210.07234.
  30. Hao-Chung Cheng, Min-Hsiu Hsieh, and Ping-Cheng Yeh. The learnability of unknown quantum measurements. Quantum Info. Comput., 16(7–8):615-656, May 2016. URL: https://doi.org/10.5555/3179466.3179470.
  31. Kai-Min Chung and Han-Hsuan Lin. Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem. In Min-Hsiu Hsieh, editor, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021), volume 197 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:22, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.TQC.2021.3.
  32. Andrew W Cross, Graeme Smith, and John A Smolin. Quantum learning robust against noise. Physical Review A, 92(1):012327, 2015. URL: https://doi.org/10.1103/PhysRevA.92.012327.
  33. Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao. Efficient measure for the expressivity of variational quantum algorithms. Physical Review Letters, 128(8):080506, 2022. URL: https://doi.org/10.1103/PhysRevLett.128.080506.
  34. Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz. Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. The Annals of Mathematical Statistics, 27(3):642-669, 1956. URL: https://www.jstor.org/stable/2237374.
  35. Alexandros Eskenazis and Paata Ivanisvili. Learning low-degree functions from a logarithmic number of random queries. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 203-207, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3519981.
  36. Marco Fanizza, Yihui Quek, and Matteo Rosati. Learning quantum processes without input control, 2022. URL: https://arxiv.org/abs/2211.05005.
  37. Bill Fefferman and Shelby Kimmel. Quantum vs. Classical Proofs and Subset Verification. In Igor Potapov, Paul Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), volume 117 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:23, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.22.
  38. Vitaly Feldman. On the power of membership queries in agnostic learning. Journal of Machine Learning Research, 10(7):163-182, 2009. URL: http://jmlr.org/papers/v10/feldman09a.html.
  39. Vitaly Feldman. Distribution-specific agnostic boosting. In Proceedings Innovations in Computer Science - ICS 2010, ICS 2010, pages 241-250, Tsinghua University, Beijing, China, 2010. Tsinghua University Press. URL: https://arxiv.org/abs/0909.2927.
  40. Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. On agnostic learning of parities, monomials, and halfspaces. SIAM Journal on Computing, 39(2):606-645, 2009. URL: https://doi.org/10.1137/070684914.
  41. Joseph F Fitzsimons. Private quantum computation: an introduction to blind quantum computing and related protocols. npj Quantum Information, 3(1):23, 2017. Google Scholar
  42. Christopher A Fuchs and Jeroen Van De Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Transactions on Information Theory, 45(4):1216-1227, 1999. URL: https://doi.org/10.1109/18.761271.
  43. Alexandru Gheorghiu, Theodoros Kapourniotis, and Elham Kashefi. Verification of quantum computation: An overview of existing approaches. Theory of computing systems, 63:715-808, 2019. URL: https://doi.org/10.1007/s00224-018-9872-3.
  44. Oded Goldreich and Leonid A Levin. A hard-core predicate for all one-way functions. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 25-32, 1989. URL: https://doi.org/10.1145/73007.73010.
  45. Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive proofs for verifying machine learning. Electron. Colloquium Comput. Complex., 20(58), 2020. URL: https://eccc.weizmann.ac.il/report/2020/058.
  46. Shafi Goldwasser, Guy N Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive proofs for verifying machine learning. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.41.
  47. Parikshit Gopalan, Adam Tauman Kalai, and Adam R Klivans. Agnostically learning decision trees. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 527-536, 2008. URL: https://doi.org/10.1145/1374376.1374451.
  48. Parikshit Gopalan, Ryan O'Donnell, Rocco A Servedio, Amir Shpilka, and Karl Wimmer. Testing fourier dimensionality and sparsity. SIAM Journal on Computing, 40(4):1075-1100, 2011. URL: https://doi.org/10.1137/100785429.
  49. Alex B Grilo, Iordanis Kerenidis, and Timo Zijlstra. Learning-with-errors problem is easy with quantum samples. Physical Review A, 99(3):032314, 2019. URL: https://doi.org/10.1103/PhysRevA.99.032314.
  50. Aram W. Harrow and David J. Rosenbaum. Uselessness for an oracle model with internal randomness. Quantum Inf. Comput., 14(7-8):608-624, 2014. URL: https://doi.org/10.26421/QIC14.7-8-5.
  51. David Haussler. Decision theoretic generalizations of the pac model for neural net and other learning applications. Information and computation, 100(1):78-150, 1992. URL: https://doi.org/10.1016/0890-5401(92)90010-D.
  52. Lunjia Hu and Charlotte Peale. Comparative Learning: A Sample Complexity Theory for Two Hypothesis Classes. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 72:1-72:30, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.72.
  53. Hsin-Yuan Huang, Michael Broughton, Jordan Cotler, Sitan Chen, Jerry Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John Preskill, et al. Quantum advantage in learning from experiments. Science, 376(6598):1182-1186, 2022. URL: https://doi.org/10.1126/science.abn7293.
  54. Hsin-Yuan Huang, Sitan Chen, and John Preskill. Learning to predict arbitrary quantum processes, 2023. URL: https://arxiv.org/abs/2210.14894.
  55. Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050-1057, 2020. URL: https://doi.org/10.1038/s41567-020-0932-7.
  56. Hsin-Yuan Huang, Richard Kueng, and John Preskill. Information-theoretic bounds on quantum advantage in machine learning. Physical Review Letters, 126(19):190505, 2021. URL: https://doi.org/10.1103/PhysRevLett.126.190505.
  57. Jeffrey C. Jackson, Christino Tamon, and Tomoyuki Yamakami. Quantum dnf learnability revisited. In International Computing and Combinatorics Conference, pages 595-604. Springer, 2002. URL: https://doi.org/10.1007/3-540-45655-4_63.
  58. Adam Tauman Kalai, Yishay Mansour, and Elad Verbin. On agnostic boosting and parity learning. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 629-638, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374466.
  59. Varun Kanade and Adam Kalai. Potential-based agnostic boosting. In Y. Bengio, D. Schuurmans, J. Lafferty, C. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems, volume 22. Curran Associates, Inc., 2009. URL: https://proceedings.neurips.cc/paper/2009/file/13f9896df61279c928f19721878fac41-Paper.pdf.
  60. Varun Kanade, Andrea Rocchetto, and Simone Severini. Learning dnfs under product distributions via μ-biased quantum fourier sampling. Quantum Information & Computation, 19(15&16):1261-1278, 2019. URL: https://doi.org/10.26421/QIC19.15-16.
  61. Michael Kearns. Efficient noise-tolerant learning from statistical queries. Journal of the ACM (JACM), 45(6):983-1006, 1998. URL: https://doi.org/10.1145/293347.293351.
  62. Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert E Schapire, and Linda Sellie. On the learnability of discrete distributions. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 273-282, 1994. Google Scholar
  63. Michael J. Kearns, Robert E. Schapire, and Linda M. Sellie. Toward efficient agnostic learning. Mach. Learn., 17(2–3):115-141, November 1994. URL: https://doi.org/10.1007/BF00993468.
  64. Michael R Kosorok. Introduction to empirical processes and semiparametric inference. Springer, 2008. URL: https://doi.org/10.1007/978-0-387-74978-5.
  65. Eyal Kushilevitz and Yishay Mansour. Learning decision trees using the fourier spectrum. SIAM Journal on Computing, 22(6):1331-1348, 1993. URL: https://doi.org/10.1137/0222080.
  66. Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, fourier transform, and learnability. Journal of the ACM (JACM), 40(3):607-620, 1993. URL: https://doi.org/10.1145/174130.174138.
  67. U. Mahadev. Classical verification of quantum computations, 2018. URL: https://arxiv.org/abs/1804.01082.
  68. Pascal Massart. The tight constant in the dvoretzky-kiefer-wolfowitz inequality. The Annals of Probability, 18(3):1269-1283, 1990. URL: https://www.jstor.org/stable/2244426.
  69. Ashley Montanaro. The quantum query complexity of learning multilinear polynomials. Information Processing Letters, 112(11):438-442, 2012. URL: https://doi.org/10.1016/j.ipl.2012.03.002.
  70. Saachi Mutreja and Jonathan Shafer. Pac verification of statistical algorithms. In Gergely Neu and Lorenzo Rosasco, editors, Proceedings of Thirty Sixth Conference on Learning Theory, volume 195 of Proceedings of Machine Learning Research, pages 5021-5043. PMLR, July 2023. URL: https://proceedings.mlr.press/v195/mutreja23a.html.
  71. Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In Amnon Ta-Shma, editor, 38th Computational Complexity Conference (CCC 2023), volume 264 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:27, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.22.
  72. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. Google Scholar
  73. Jack O'Connor. Delegating machine learning with succinct proofs. Master’s thesis, University of Warwick, 2021. Google Scholar
  74. John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, 2018. URL: https://doi.org/10.22331/q-2018-08-06-79.
  75. Rocco A. Servedio and Steven J. Gortler. Equivalences and separations between quantum and classical learnability. SIAM Journal on Computing, 33(5):1067-1092, 2004. URL: https://doi.org/10.1137/S0097539704412910.
  76. Michel Talagrand. Sharper bounds for gaussian and empirical processes. The Annals of Probability, pages 28-76, 1994. URL: https://doi.org/10.1214/aop/1176988847.
  77. Vladimir N. Vapnik and Alexei Ya. Chervonenkis. On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities. Theory of Probability & Its Applications, 16(2):264-280, 1971. URL: https://doi.org/10.1137/1116025.
  78. Mark M. Wilde. From classical to quantum shannon theory, 2011. URL: https://arxiv.org/abs/1106.1445.
  79. Chi Zhang. An improved lower bound on query complexity for quantum pac learning. Information Processing Letters, 111(1):40-45, 2010. URL: https://doi.org/10.1016/j.ipl.2010.10.007.