Before revision:  In this paper, we primarily study thetolerant versions of the classicuniformity andidentity testing problems, providing improved query complexity bounds in the conditional sampling model. In this paper , we prove that tolerant uniformity testing in the conditional sampling model can be solved using O(\varepsilon^{-2}) queries, which is known to be optimal and improves upon the O(\varepsilon^{-20})-query algorithm of  [CRS15].
After revision:  In this paper, we primarily study thetolerant versions of the classicuniformity andidentity testing problems, providing improved query complexity bounds for several classic distribution testing problems in this model. First , we prove that tolerant uniformity testing in the conditional sampling model can be solved using O(\varepsilon^{-2}) queries, which is known to be optimal and improves upon the O(\varepsilon^{-20})-query algorithm of [CRS15].