On the Cryptographic Structure Required for Verifying Qubits

2026-06-03

AI Summary

This work investigates the cryptographic underpinnings of anti-commuting operator tests in classically verifiable quantum computation and introduces a framework termed the Test of Non-Commutativity (ToNC). By establishing rigorous reductions between ToNC and classical key agreement (KA) as well as oblivious transfer (OT) based on one-way functions, the paper demonstrates for the first time that ToNC can be used to construct KA and implement OT. Furthermore, it introduces a post-quantum hardcore measure theorem and an interactive XOR lemma, enabling hardness amplification for KA and OT in post-quantum settings. These results uncover a deep connection between quantum verification protocols and classical cryptographic primitives, thereby laying a new theoretical foundation for classically verifiable quantum computation.

Abstract

Classically testing for the presence of anti-commuting operators on a quantum device is a critical tool underpinning recent progress in classical verification of quantum computation. While such tests can be based on cryptographic assumptions, known constructions rely on highly structured assumptions, e.g. trapdoor claw-free functions. In this work, we seek to explain this state of affairs by constructing strong cryptography from (certain forms of) classical tests of anti-commutation. In particular, we formulate the notion of a test of non-commutation (ToNC), an interactive protocol between a quantum prover and classical verifier in which the prover's final-round response is obtained by measuring one of two binary observables $P_0,P_1$ depending on the verifier's challenge bit $c$. We prove that, for a broad range of parameters, ToNC implies classical-communication key agreement (KA), and ToNC combined with one-way functions implies oblivious transfer (OT). Along the way, we develop tools for and provide the first known results on hardness amplification for post-quantum KA and OT, where communication is classical but adversaries may be quantum. In particular, we prove the following results of independent interest. - Post-quantum hard-core measure theorem: For any efficiently sampleable high-min-entropy distribution $D$ over pairs $(x,b)$ such that quantum circuits have advantage at most $δ$ in predicting $b$ from $x$, there exists a sub-distribution $M\preceq D$ of density $(1-δ)$ on which $b$ is nearly optimally quantum-hard to predict. - Post-quantum interactive XOR lemma: Given any classically-interactive protocol, if quantum adversaries have advantage at most $δ$ in guessing a private challenger bit $b$, then two sequential repetitions reduce the advantage for predicting the XOR of the challenger bits $b_1\oplus b_2$ to at most $δ^2+\rm{negl}(λ)$.

Problem

• anti-commutation
• classical verification of quantum computation
• cryptography
• key agreement
• oblivious transfer

Innovation

• Test of Non-Commutation
• Post-quantum Cryptography
• Classical Verification of Quantum Computation
• Hardness Amplification
• Oblivious Transfer