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Is A0PP PP-low?

Determine whether the complexity class A_0PP (also known as SBQP) is PP-low; specifically, prove or refute that PP with an A_0PP oracle does not exceed PP in power, i.e., establish whether PP^{A_0PP} = PP holds.

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Background

The paper contrasts APP with the related counting class A_0PP (also known as SBQP). Being PP-low means that granting PP oracle access to a class does not increase its computational power, i.e., PPC = PP. While APP is known to be PP-low, the lowness of A_0PP remains unsettled.

Clarifying whether A_0PP is PP-low would sharpen the landscape of low-for-PP classes and further situate A_0PP relative to APP, PP, and QMA. The authors explicitly note this as an outstanding uncertainty.

References

Compared with the class ${A_0PP}={SBQP} \subseteq PP$ , ${A_0PP}$ contains $QMA$ and is not known to be $PP$-low, while $APP$ is not known to contain even $NP$ but is $PP$-low.

— Even quantum advice is unlikely to solve PP (2403.09994 - Yirka, 15 Mar 2024) in Section 2 (Preliminaries), APP discussion following Definition 'APP'