Supersimulators

We prove that every randomized Boolean function admits a supersimulator: a randomized polynomial-size circuit whose output on random inputs cannot be efficiently distinguished from reality with constant advantage, even by polynomially larger distinguishers. Our result builds on the landmark complexity-theoretic regularity lemma of Trevisan, Tulsiani and Vadhan (2009), which, in contrast, provides a simulator that fools smaller distinguishers. We circumvent lower bounds for the simulator size by letting the distinguisher size bound vary with the target function, while remaining below an absolute upper bound independent of the target function. This dependence on the target function arises naturally from our use of an iteration technique originating in the graph regularity literature.