Quantum Proofs of Proximity

Marcel Dall'Agnol1, Tom Gur1, Subhayan Roy Moulik2, and Justin Thaler3

1Department of Computer Science, University of Warwick, UK
2Department of Mathematics, UC Berkeley, USA & Department of Computer Science, University of Oxford, UK
3Department of Computer Science, Georgetown University, USA

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We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA $\textit{proofs of proximity}$ (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject inputs that are $\varepsilon$-far from $\Pi$, while only probing a minuscule portion of their input.
We investigate the complexity landscape of this model, showing that QMAPs can be $exponentially$ stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016).
Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly $decomposable$. A consequence of this framework is a QMA algorithm to verify the Parity of an $n$-bit string with $O(n^{2/3})$ queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.

Two of the most important questions in the theory of computer science are:

– Is verifying the solution to a computational problem easier than deciding if one exists?
– Can quantum computers solve problems their classical counterparts cannot?

Indeed, their formalisation in the model of polynomial-time computation yields the foundational P vs. NP and BPP vs. BQP problems. A related problem is the quantum analog of P vs. NP, namely, is QMA a strict subset of BQP? Less formally:

– Is verification easier than decision for quantum computers?

We study this question in the property-testing setting, which models super-efficient algorithms that only probe a minuscule portion of their input. In this setting, separations between verification and decision as well as quantum and classical computing are known. In other words, answers to the property-testing analogues of P vs. NP and BQP vs. BPP questions have been known for some time.

However, the property-testing analogue of QMA has not heretofore been systematically explored. Our paper fills this gap by initiating its study. We formalise super-efficient quantum verifiers, i.e., quantum proofs of proximity, and study their power and limitations. Among our results is a separation between quantum and classical proofs of proximity. This answers the QMA vs. BQP question in the property-testing setting.

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[1] Marcel Dall’Agnol, Tom Gur, and Oded Lachish, "A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Verification", SIAM Journal on Computing 52 6, 1413 (2023).

[2] Adrian She and Henry Yuen, "Unitary property testing lower bounds by polynomials", arXiv:2210.05885, (2022).

[3] Andreas Bluhm, Matthias C. Caro, and Aadil Oufkir, "Hamiltonian Property Testing", arXiv:2403.02968, (2024).

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