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Resolve the Quantum PCP (QPCP) conjecture

Resolve the quantum PCP conjecture by proving or refuting whether every QMA language admits a quantum PCP with constant query complexity and polynomial overhead, thereby establishing (or disproving) the quantum analogue of the classical PCP theorem.

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Background

The paper highlights a conceptual connection between zero‑knowledge PCPs and quantum PCPs via quantum codes: good code distance corresponds to hiding information (akin to zero knowledge). While there is evidence for good quantum LTCs, local decodability is blocked by no‑cloning, and QPCP remains a central open problem in quantum complexity.

By drawing parallels, the authors suggest that studying perfect zero‑knowledge PCPs could shed light on QPCP, but they do not provide a specific formulation beyond pointing to the conjecture’s status.

References

Finally, we highlight a connection between zero-knowledge PCPs and one of the most imporant open problems in quantum complexity theory: the quantum PCP (QPCP) conjecture.

— A Zero-Knowledge PCP Theorem (2411.07972 - Gur et al., 12 Nov 2024) in Subsection “Open problems”, Section 1 (Introduction), paragraph “The quantum PCP conjecture.”