Improved Circuit Lower Bounds and Quantum-Classical Separations

TL;DR

The paper introduces GC^0(k), a shallow circuit class built from G(k) gates that operate locally on small Hamming balls, and proves that the polynomial-method lower bounds for AC^0[p] lift losslessly to GC^0(k)[p], yielding exponential-size lower bounds for MAJ and MOD_q that match the best AC^0[p] bounds. It also extends Williams’ algorithmic method to GCC^0, obtaining unconditional E^NP lower bounds in the GCC^0 setting, and establishes a quasipolynomial-time PAC-learning algorithm for GC^0(k)[p]. A central conceptual contribution is the identification of locality in the G(k) model as a unifying property that underpins both switching lemmas and the polynomial method, revealing a barrier between AC^0[p] and GC^0(k)[p]. Leveraging these improved classical bounds, the work delivers the strongest unconditional quantum-classical separations to date, including BQLOGTIME ⊄ GC^0, QNC^0 ⊄ GC^0(k), and related QNC^0/qpoly versus GC^0(k)[p] separations, with oracle constructions extending Raz–Tal-type results. The results collectively deepen our understanding of how limited locality in circuit models constrains classical computation even when quantum models exhibit significant advantages, and they open multiple avenues for further refining lower-bound techniques in the GC^0 framework.

Abstract

We continue the study of the circuit class GC^0, which augments AC^0 with unbounded-fan-in gates that compute arbitrary functions inside a sufficiently small Hamming ball but must be constant outside it. While GC^0 can compute functions requiring exponential-size circuits, Kumar (CCC 2023) showed that switching-lemma lower bounds for AC^0 extend to GC^0 with no loss in parameters. We prove a parallel result for the polynomial method: any lower bound for AC^0[p] obtained via the polynomial method extends to GC^0[p] without loss in parameters. As a consequence, we show that the majority function MAJ requires depth-$d$ GC^0[p] circuits of size $2^{Ω(n^{1/2(d-1)})}$, matching the best-known lower bounds for AC^0[p]. This yields the most expressive class of non-monotone circuits for which exponential-size lower bounds are known for an explicit function. We also prove a similar result for the algorithmic method, showing that E^NP requires exponential-size GCC^0 circuits, extending a result of Williams (JACM 2014). Finally, leveraging our improved classical lower bounds, we establish the strongest known unconditional separations between quantum and classical circuit classes. We separate QNC^0 from GC^0 and GC^0[p] in various settings and show that BQLOGTIME is not contained in GC^0. As a consequence, we construct an oracle relative to which BQP lies outside uniform GC^0, extending the Raz-Tal oracle separation between BQP and PH (STOC 2019).

Theorems & Definitions (121)

• Theorem 1.3: Main result of kumar2023tight, Informal
• Theorem 1.4: Improved circuit lower bounds, Informal
• Theorem 1.5: $\mathsf{GC}^0(k)[p]$ lower bound, Restatement of \ref{['thm:gc0-lower-bound']}
• Lemma 1.6: Restatement of \ref{['lem:gateapprox']}
• Lemma 1.7: Restatement of \ref{['lem:prob-poly-lower-bound']}
• Theorem 1.8: Restatement of \ref{['thm:probpoly']}
• Remark 1.9
• Theorem 1.10: $\mathsf{E}^\mathsf{NP} \not\subseteq \mathsf{GCC}^0(k)$, Restatement of \ref{['thm:main-gcc0']}
• Theorem 1.11: $\mathsf{GCC}^0(k)$-CircuitSAT algorithm, Restatement of \ref{['thm:gccfasteval']}
• Theorem 1.12: Restatement of \ref{['thm:gk-depth-2']}