Interactive proofs for verifying (quantum) learning and testing

Matthias C. Caro; Jens Eisert; Marcel Hinsche; Marios Ioannou; Alexander Nietner; Ryan Sweke

doi:10.22331/q-2026-10-20-2239

Interactive proofs for verifying (quantum) learning and testing

Matthias C. Caro^1,2, Jens Eisert^2,3, Marcel Hinsche², Marios Ioannou², Alexander Nietner², and Ryan Sweke^4,5,6,7

¹Department of Computer Science, University of Warwick, UK
²Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany
³Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany
⁴IBM Quantum, Almaden Research Center, San Jose, CA, USA
⁵African Institute for Mathematical Sciences (AIMS), South Africa
⁶Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7600, South Africa
⁷National Institute for Theoretical and Computational Sciences (NITheCS), South Africa

Published:	2026-10-20, volume 10, page 2239
Editor:	Yu Tong
Eprint:	arXiv:2410.23969v3
Doi:	https://doi.org/10.22331/q-2026-10-20-2239
Citation:	Quantum 10, 2239 (2026).

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Abstract

We consider the problem of testing and learning from data in the presence of resource constraints, such as limited memory or weak data access, which place limitations on the efficiency and feasibility of testing or learning. In particular, we ask the following question: Could a resource-constrained learner/tester use interaction with a resource-unconstrained but untrusted party to solve a learning or testing problem more efficiently than they could without such an interaction? In this work, we answer this question both abstractly and for concrete problems, in two complementary ways: For a wide variety of scenarios, we prove that a resource-constrained learner cannot gain any advantage through classical interaction with an untrusted prover. As a special case, we show that for the vast majority of testing and learning problems in which quantum memory is a meaningful resource, a memory-constrained quantum algorithm cannot overcome its limitations via classical communication with a memory-unconstrained quantum prover. In contrast, when quantum communication is allowed, we construct a variety of interactive proof protocols, for specific learning and testing problems, which allow memory-constrained quantum verifiers to gain significant advantages through delegation to untrusted provers. These results highlight both the limitations and potential of delegating learning and testing problems to resource-rich but untrusted third parties.

Featured image: A schematic of the interactive proofs setting analyzed in this work.

Popular summary

In today's computing landscape, more and more heavy computation runs in a handful of giant data centres. Quantum computing is widely expected to follow the same pattern: A few centres will host fault-tolerant quantum hardware that users access remotely. This raises a question of trust. How can the user be sure that the provider has actually executed the desired quantum computation and not some cheaper alternative? The fragile nature of quantum computation makes this kind of verification especially challenging in the quantum era. Over the last two decades, a rich toolbox has been developed to address this challenge, enabling users of varying computational power to delegate quantum computations to untrusted service providers over different types of communication channels.

However, not all tasks of interest are computations on explicitly specified inputs. With the growing importance of machine learning and artificial intelligence in classical as well as quantum computing, learning and testing problems are gaining prominence. In quantum learning and testing, algorithms are designed to extract information about an unknown quantum state or process from black-box quantum data access. Existing tools for verified quantum computing are not designed to cover scenarios where the service provider uses high-quality data that the client does not have access to; in fact, verification is subtle even for classical algorithms for data processing.

We begin to map out a theory of verification for learning and testing with quantum data. Our first contribution is an impossibility result: For a broad class of learning and testing problems, when the user is strictly weaker than the provider, classical communication alone does not suffice for the user to verify the provider's answer. We complement this with positive results showing that, for a variety of quantum learning and testing tasks of interest, quantum communication allows a client limited to single-copy quantum processing to reap the copy complexity benefits of delegating tasks to a provider capable of multi-copy quantum processing. Our work shows that if quantum computing power does end up concentrated in few centres, quantum communication across a quantum internet is in many cases necessary and in several cases sufficient to allow a user with modest quantum capabilities to verifiably and efficiently delegate quantum learning and testing tasks to powerful providers.

► BibTeX data

@article{Caro2026interactiveproofs,
  doi = {10.22331/q-2026-10-20-2239},
  url = {https://doi.org/10.22331/q-2026-10-20-2239},
  title = {Interactive proofs for verifying (quantum) learning and testing},
  author = {Caro, Matthias C. and Eisert, Jens and Hinsche, Marcel and Ioannou, Marios and Nietner, Alexander and Sweke, Ryan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2239},
  month = oct,
  year = {2026}
}

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[2] Abbas Poshtvan, Oleksandra Lapiha, Mina Doosti, Dominik Leichtle, Luka Music, and Elham Kashefi, "Selectively Blind Quantum Computation", arXiv:2504.17612, (2025).

[3] Fabian Wiesner, Jens Eisert, and Anna Pappa, "Unifying communication paradigms in measurement-based delegated quantum computing", arXiv:2506.21988, (2025).

[4] Miranda Christ, Daniel Reichman, and Jonathan Shafer, "Protocols for Verifying Smooth Strategies in Bandits and Games", arXiv:2507.10567, (2025).

[5] Ari Biswas, Mark Bun, Clément Canonne, and Satchit Sivakumar, "Interactive Proofs For Distribution Testing With Conditional Oracles", arXiv:2511.22122, (2025).

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