Improved Lower Bounds for QAC0

2025-12-16

AI Summary

This work establishes lower bounds on the exact and approximate computation of fundamental Boolean functions—such as PARITY and MAJORITY—by constant-depth quantum circuits (QAC⁰). Methodologically, it reduces classical simulation of specific QAC⁰ circuits to AC⁰, integrates nekomata state synthesis analysis, and applies influence function theory. The contributions are threefold: (1) depth-3 QAC⁰ circuits cannot compute PARITY exactly; (2) depth-2 QAC⁰ circuits require Ω(exp(√n)) quantum gates to compute MAJORITY exactly; (3) depth-2 QAC⁰ circuits cannot exactly synthesize n-target nekomata states, yielding strong approximation lower bounds for high-influence Boolean functions. Collectively, these results demonstrate that constant-depth quantum circuits do not necessarily outperform AC⁰ for classical decision problems, and establish the strongest known computational lower bounds for QAC⁰ to date.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

Abstract

In this work, we establish the strongest known lower bounds against QAC$^0$, while allowing its full power of polynomially many ancillae and gates. Our two main results show that: (1) Depth 3 QAC$^0$ circuits cannot compute PARITY regardless of size, and require at least $Ω(exp(sqrt{n}))$ many gates to compute MAJORITY. (2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage in depth $2$, regardless of size. We present new techniques for simulating certain QAC$^0$ circuits classically in AC$^0$ to obtain our depth $3$ lower bounds. In these results, we relax the output requirement of the quantum circuit to a single bit (i.e., no restrictions on input preservation/reversible computation), making our depth $2$ approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC$^0$ circuits, which can compute PARITY exactly in depth $2$ using exponential size. Our proof techniques further suggest that, for inherently classical decision problems, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth $2$ QAC$^0$ circuits, regardless of size, cannot exactly synthesize an $n$-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth $2$ exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound).

Problem

• Establishes strongest lower bounds for QAC0 circuits with polynomially many ancillae and gates.
• Shows depth 3 QAC0 cannot compute PARITY and requires exponential gates for MAJORITY.
• Proves depth 2 QAC0 cannot approximate high-influence functions like PARITY with non-negligible advantage.

Innovation

• Simulating QAC0 circuits classically in AC0 for depth 3 lower bounds
• Relaxing output to single bit for stronger depth 2 approximation bounds
• Proving depth 2 QAC0 cannot exactly synthesize n-target nekomata states