Improved Circuit Lower Bounds and Quantum-Classical Separations

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).

• The paper demonstrates that classical lower bound techniques extend to GC^0[p] circuits by approximating G(k) gates with low-degree probabilistic polynomials, matching known AC^0[p] bounds.
• It establishes the strongest known unconditional quantum-classical separations, showing that constant-depth quantum circuits solve problems that are intractable for GC^0 and GC^0[p] circuits.
• The work leverages an extended multi-output switching lemma and natural properties to derive quasipolynomial-time learning algorithms, revealing new limitations and frontiers in circuit complexity.

Improved Circuit Lower Bounds and Quantum-Classical Separations

Introduction and Context

The paper "Improved Circuit Lower Bounds and Quantum-Classical Separations" (2408.16406) investigates the limitations of existing lower bound techniques in circuit complexity by extending their applicability to a novel and more expressive circuit class, $GC^0$ and its modular extensions $GC^0[p]$. The class $GC^0$ generalizes $AC^0$ by permitting unbounded-fan-in gates that compute arbitrary functions on small Hamming balls and behave constantly outside. This architecture encapsulates both conventional gates and arbitrary bounded-fan-in gates, connecting the study of locality in circuits with previously established switching lemma and polynomial method frameworks.

The work is motivated by two major research questions:

1. Determining the expressive limits for which current lower bound techniques remain effective;
2. Unifying the structural underpinnings that enable extensions of disparate methods—combinatorial (switching lemma) and algebraic (probabilistic polynomial method)—to more general circuit classes, to thereby better understand the frontiers of complexity theory and guidance for new breakthroughs.

Main Contributions

This manuscript delivers several core results, extending classical lower bounds, analyzing quantum-classical separations, and presenting implications in learning theory and complexity.

Extension of the Polynomial Method to $GC^0[p]$

The authors prove that any lower bound against $AC^0[p]$ established via the polynomial method applies identically to $GC^0[p]$, for appropriate settings of the locality parameter $k = O(n^{1/2d})$. This is shown by constructing low-degree probabilistic polynomials that closely approximate $G(k)$ gates: any $G(k)$ can be approximated on the Boolean cube by a polynomial of degree $O(k + \log(1/\epsilon))$. The bound is shown to be tight.

Key result: For the majority function $\mathrm{MAJ}$ (and $MOD_q$, for $q \neq p$), any depth-$d$ $GC^0[p]$ circuit of size less than $2^{\Omega(n^{1/2(d-1)})}$ fails to compute the function, matching the best-known bounds for $AC^0[p]$.

Crucially, this demonstrates that both the switching lemma and the polynomial method are fundamentally insensitive to the specific gate type differences between $AC^0$ and $GC^0[p]$, provided locality (in the Hamming ball sense) is preserved.

Extension to Algorithmic Methods

The framework for extending lower bounds holds partially for Williams' algorithmic method. The authors exhibit that $E^{NP}$ does not have exponential-size $GCC^0$ circuits for locality $k \le O(n^{\delta/\log n})$, generalizing Williams's classical $E^{NP} \not\subset ACC^0$ result. This is achieved by derandomizing $GC^0$-CircuitSAT using efficiently constructed probabilistic circuits with bounded fan-in that simulate $G(k)$ gates. However, the lower bound incurs a $1/k$ loss in the exponent compared to $ACC^0$, and the possibility of completely lifting the algorithmic method (without degradation) remains open.

Quantum-Classical Separations

By leveraging the refined classical lower bounds, the paper secures the strongest known unconditional quantum-classical separations for shallow circuits:

• $BQLOGTIME$ is strictly stronger than $GC^0$, with oracle separations established that extend and strengthen previous relativized separations between $BQP$ and $PH$.
• The 2D Hidden Linear Function (2D HLF) problem and related relational problems are shown to be solvable by constant-depth quantum circuits ($QNC^0$), but are hard for $GC^0$ and $GC^0[p]$—even against exponential-size classical circuits—on suitable distributions. This separation survives various extensions such as the presence of quantum advice and noise.
• For every prime $p$, there are search and interactive problems where $QNC^0$ or $QNC^0/\mathsf{qpoly}$ circuits succeed with high probability, but $GC^0[k][p]$ (even with random advice, and for $k = O(n^{1/2d})$) fail with negligible probability.

Structural and Technical Advances

• The multi-output multi-switching lemma is extended to $GC^0$, generalizing prior results for $AC^0$ and serving as the foundation for many average-case lower bounds.
• The inherent barrier character of the results is explicit: if any function is in $GC^0[p]$ for moderate $k$, neither the polynomial method nor switching lemma will yield improved bounds over $AC^0[p]$.

Learning Algorithms

Through the connection between natural proofs and PAC learning, the paper establishes quasipolynomial-time algorithms for learning $GC^0[k][p]$ in the uniform PAC model with membership queries, generalizing earlier results for $AC^0[p]$. This leverages the naturalness of the polynomial method lower bounds for these enriched classes.

Implications

Circuit Complexity

These results precisely delineate the limits of two of the principal lower bound techniques in circuit complexity. By showing neither the probabilistic polynomial method nor the switching lemma can distinguish $AC^0[p]$ from the much larger $GC^0[p]$ class, the work exposes a significant flexibility—and corresponding weakness—of these classical tools. The tight translation of lower bounds identifies locality (computation restricted to small Hamming balls) as the unifying structure exploited by current techniques.

This observation suggests that further progress on superpolynomial or exponential lower bounds for more potent circuit classes (e.g., $TC^0$ or $ACC^0$ with larger locality) will require fundamentally new ideas or paradigms beyond random restrictions and probabilistic polynomials.

Quantum Complexity and Separations

On the quantum-classical front, the strengthened separations have direct implications for complexity theory, providing robust evidence (relative to various oracles and in diverse setting—including noisy computation and circuits with advice) that quantum models outperform even highly expressive classical shallow circuits.

The new structural results integrate with, and significantly extend, oracle results such as $BQP \not\subset PH$ and hardness of simulating certain quantum tasks, indicating the enduring power of quantum computation even when classical models are granted gates of high local expressiveness.

Learning Theory

The extension of natural property lower bounds to learning guarantees for $GC^0[p]$ suggests that, for a large swath of constant-depth, locally powerful classical circuits, efficient learning remains possible under standard distributional and query models—a fact that may inform both theoretical study and applications in secure computation and pseudorandomness.

Open Problems and Future Directions

Several key questions are isolated for further research:

• Whether $G(k)$ gates precisely characterize the limitations of the switching lemma and polynomial method, or whether even broader notions of locality might further unify and/or expose the barrier.
• The extent to which the algorithmic method can be lifted to $GCC^0$ without quantitative loss.
• Deeper connections and relative power between $GC^0$ classes and other traditionally studied classes such as $NC^1$ and $TC^0$ as $k$ scales.
• The prospect of quantum-classical separations in even more general settings, especially without quantum advice or with genuinely explicit functions.

Conclusion

This work systematically extends the core lower bound techniques in circuit complexity to a broad new circuit class that simultaneously captures the limitations and reach of these methods. It establishes new quantum-classical separations, matching and surpassing previous state-of-the-art results, and clarifies the structural features (notably, Hamming ball locality) exploited by these techniques. These results define both the horizon and limitations of current approaches and illuminate new directions for research in both classical and quantum complexity theory.