On zero error algorithms having oracle access to one query

It is known that Sp2 ⊆ ZPP NP [3]. The reverse direction of whether ZPP is contained in Sp2 remains open. We show that if the zero-error algorithm is allowed to ask only one query to the NP oracle (for any input and random string), then it can be simulated in Sp2. That is, we prove that ZPP ⊆ Sp2. Next we consider whether the above result can be improved as ZPP ⊆ P and point out a difficulty in doing so. Via a simple proof, we observe that BPP ⊆ ZPP (a result implicitly proven in some prior work). Thus, achieving the above improvement would imply BPP ⊆ P, settling a long standing open problem. We then argue that the above mentioned improvement can be obtained for the next level of the polynomial time hierarchy. Namely, we prove that ZPP p 2 [1] ⊆ P p 2 . On the other hand, by adapting our proof of our main result it can be shown that ZPP p 2 [1] ⊆ S NP[1] 2 . For the purpose of comparing these two results, we prove that P p 2 ⊆ S NP[1] 2 . We conclude by observing that the above claims extend to the higher levels of the hierarchy: for k ≥ 2, ZPP p k [1] ⊆ P p k [2] and P p k ⊆ S Σ k−1 [1] 2 .