Classical Verification of Quantum Learning

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

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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


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.

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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)


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
  • computational learning theory
  • quantum learning theory
  • interactive proofs
  • quantum oracles
  • agnostic learning


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