Kumar used a switching lemma to prove exponential-size lower bounds for a circuit class GC^0 that not only contains AC^0 but can--with a single gate--compute functions that require exponential-size TC^0 circuits. His main result was that switching-lemma lower bounds for AC^0 lift to GC^0 with no loss in parameters, even though GC^0 requires exponential-size TC^0 circuits. Informally, GC^0 is AC^0 with unbounded-fan-in gates that behave arbitrarily inside a sufficiently small Hamming ball but must be constant outside it. We show that polynomial-method lower bounds for AC^0[p] lift to GC^0[p] with no loss in parameters, complementing Kumar's result for GC^0 and the switching lemma. As an application, we prove Majority requires depth-d GC^0[p] circuits of size $2^{\Omega(n^{1/2(d-1)})}$, matching the state-of-the-art lower bounds for AC^0[p]. We also show that E^NP requires exponential-size GCC^0 circuits (the union of GC^0[m] for all m), extending the result of Williams. It is striking that the switching lemma, polynomial method, and algorithmic method all generalize to GC^0-related classes, with the first two methods doing so without any loss. We also establish the strongest known unconditional separations between quantum and classical circuits: 1. There's an oracle relative to which BQP is not contained in the class of languages decidable by uniform families of size-$2^{n^{O(1)}}$ GC^0 circuits, generalizing Raz and Tal's relativized separation of BQP from the polynomial hierarchy. 2. There's a search problem that QNC^0 circuits can solve but average-case hard for exponential-size GC^0 circuits. 3. There's a search problem that QNC^0/qpoly circuits can solve but average-case hard for exponential-size GC^0[p] circuits. 4. There's an interactive problem that QNC^0 circuits can solve but exponential-size GC^0[p] circuits cannot.

翻译：Kumar 使用切换引理证明了电路类 GC^0 的指数规模下界，该类不仅包含 AC^0，而且能够通过单个门计算需要指数规模 TC^0 电路的函数。他的主要结果是，针对 AC^0 的切换引理下界可以无损地提升到 GC^0，尽管 GC^0 需要指数规模的 TC^0 电路。非正式地说，GC^0 是 AC^0 的扩展，其无扇入限制的门在足够小的汉明球内行为任意，但在球外必须为常数。我们证明了针对 AC^0[p] 的多项式方法下界可以无损地提升到 GC^0[p]，从而补充了 Kumar 关于 GC^0 和切换引理的结果。作为一个应用，我们证明了多数函数需要规模为 $2^{\Omega(n^{1/2(d-1)})}$ 的深度-d GC^0[p] 电路，这与 AC^0[p] 的最新下界相匹配。我们还证明了 E^NP 需要指数规模的 GCC^0 电路（所有 m 对应的 GC^0[m] 的并集），从而扩展了 Williams 的结果。值得注意的是，切换引理、多项式方法和算法方法都可以推广到 GC^0 相关类，且前两种方法在推广过程中没有任何损失。我们还建立了已知最强的量子与经典电路之间的无条件分离：1. 存在一个预言机，使得 BQP 不包含在由规模为 $2^{n^{O(1)}}$ 的均匀 GC^0 电路族可判定的语言类中，这推广了 Raz 和 Tal 关于 BQP 与多项式层次结构的相对化分离。2. 存在一个搜索问题，QNC^0 电路可以解决，但对于指数规模的 GC^0 电路是平均情况困难的。3. 存在一个搜索问题，QNC^0/qpoly 电路可以解决，但对于指数规模的 GC^0[p] 电路是平均情况困难的。4. 存在一个交互问题，QNC^0 电路可以解决，但指数规模的 GC^0[p] 电路无法解决。