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.

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)