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Interactive Oracle Arguments in the QROM and Applications to Succinct Verification of Quantum Computation

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Topics in Cryptology – CT-RSA 2024 (CT-RSA 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14643))

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Abstract

This work is motivated by the following question: can an untrusted quantum server convince a classical verifier of the answer to an efficient quantum computation using only polylogarithmic communication? We show how to achieve this in the quantum random oracle model (QROM), after a non-succinct instance-independent setup phase.

We introduce and formalize the notion of post-quantum interactive oracle arguments for languages in QMA, a generalization of interactive oracle proofs (Ben-Sasson–Chiesa–Spooner). We then show how to compile any non-adaptive public-coin interactive oracle argument (with private setup) into a succinct argument (with setup) in the QROM.

To conditionally answer our motivating question via this framework under the post-quantum hardness assumption of LWE, we show that the ZX local Hamiltonian problem with at least inverse-polylogarithmic relative promise gap has an interactive oracle argument with instance-independent setup, which we can then compile.

Assuming a variant of the quantum PCP conjecture that we introduce called the weak ZX quantum PCP conjecture, we obtain a succinct argument for QMA (and consequently the verification of quantum computation) in the QROM (with non-succinct instance-independent setup) which makes only black-box use of the underlying cryptographic primitives.

The author was partially supported by the National Science Foundation (NSF) grants CCF-1947889 and CNS-1414119. Thanks to Mark Bun and Nicholas Spooner for their mentorship and help with refining and revising this project.

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Notes

  1. 1.

    The Pauli X, Z matrices are \(X = \begin{pmatrix} 0 &{} 1\\ 1 &{} 0 \end{pmatrix}, Z = \begin{pmatrix} 1 &{} 0\\ 0 &{} -1 \end{pmatrix}\) and they are used frequently in physics and quantum computation.

  2. 2.

    Consider, for example, the universal gate set \(G = \{H, X, \textsc {CCNOT} \}\). Note that \(H = \frac{1}{\sqrt{2}} (X + Z)\) and \(\textsc {CCNOT} = I - \frac{1}{4}(I - Z_1)(I - Z_2)(I - X_3)\). G is a universal gate set with real matrices and can be used to obtain propagation Hamiltonians whose Pauli decomposition has the real Pauli matrices X and Z.

  3. 3.

    It is known how to efficiently prepare such history state for an efficient quantum computation, but we do not include the details here.

  4. 4.

    We conjecture that it is possible to reduce the number of messages to 3 in our work. More details are provided in the extended version of this paper [21].

  5. 5.

    In this paper we will only work with classical Merkle trees where the data are classical strings and the (honest) algorithms are executed on classical devices. However, their security is established against a cheating quantum device in the quantum random oracle model.

  6. 6.

    This is equivalent to sending each overlapping intermediate node once instead of sending it multiple times inside possibly overlapping paths for each leaf. However, for easier notation and exposition, we send the authentication paths for each leaf and require this consistency condition when verifying a batch of authentication paths.

  7. 7.

    It is an open question whether they are equivalent under classical reductions. In fact, the proof checking formulation itself could end up being more specific than that provided in [1] which was the reason why it was not straightforward to prove the equivalence under classical reductions. For the details of the quantum reduction, we refer the reader to the proof of Theorem 5.5. in [24].

  8. 8.

    Later, we will use the term “public-coin protocols with private setup” to highlight this again.

  9. 9.

    Please refer to the extended version of this paper [21] for full details on the choice of these thresholds as well as the Test and Hadamard round of the Mahadev protocol.

  10. 10.

    In most useful interactive oracle arguments including the argument system for the local Hamiltonian problem discussed in this paper, we do not have to know the input length exactly, but it suffices to know an upper bound.

  11. 11.

    Keeping the randomness used in the setup enables the verifier to store information such as secret keys and/or trapdoors without revealing them to the prover.

  12. 12.

    or its oracle messages.

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Acknowledgements

The author was partially supported by NSF grants CCF-1947889 and CNS-1414119. Thanks to Mark Bun and Nicholas Spooner for their mentorship and help with refining and revising this project. Thanks - in alphabetical order - to Adam Smith, Alex Bredariol Grilo, Alex Lombardi, Anand Natarajan, Andrea Coladangelo, Anurag Anshu, Assaf Kfoury, Azer Bestavros, Chris Laumann, Fermi Ma, Ibrahim Faisal, James Bartusek, Jiayu Zhang, Leo Reyzin, Ludmila Glinskih, Luowen Qian, Mayank Varia, Muhammad Faisal, Nadya Voronova, Nathan Ju, Ran Canetti, Sam Gunn, Steve Homer, Thomas Vidick, Urmila Mahadev, and others for their time listening to ideas, helpful discussions, and explanations. I specially thank Thomas Vidick for his online lectures that spurred my interest in this research topic as well as his continued support. Thanks to [3, 20] for making the  sources of their papers accessible which helped in typesetting this paper. Thanks to UCLA’s Institute for Pure and Applied Mathematics (IPAM) for the support received to participate in the Graduate Summer School on Post-quantum and Quantum Cryptography where I discussed this work with other participants. Thanks to the reviewers of this paper for helping with iterating and refining it.

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    Islam Faisal

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Faisal, I. (2024). Interactive Oracle Arguments in the QROM and Applications to Succinct Verification of Quantum Computation. In: Oswald, E. (eds) Topics in Cryptology – CT-RSA 2024. CT-RSA 2024. Lecture Notes in Computer Science, vol 14643. Springer, Cham. https://doi.org/10.1007/978-3-031-58868-6_16

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