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HD_1 in BPP but not in UPP (conjecture)

Prove that the 1-Hamming Distance problem HD_1 belongs to BPP and does not belong to UPP.

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Background

This conjecture, cited from prior work, would separate UPP from BPP and clarify the landscape of problems that admit constant-cost protocols under different error models. It is connected to the broader question of characterizing UPP ∩ BPP and has implications for the existence of BPP problems with new flavors of randomness.

References

This would imply UPP ⊄ BPP, and in particular the conjecture of [HHPTZ22] that HD_1 ∈ BPP\setminusUPP.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section: Constant-Cost Communication: the Story so Far and Farther (Sign-rank)