Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated

2026-04-03

AI Summary

This work investigates whether shallow quantum circuits (QAC⁰) can compute the Parity function, revealing an intrinsic connection to the high-frequency components of their Fourier spectrum. By leveraging Fourier analysis and quantum complexity theory, the authors prove that Parity ∈ QAC⁰ if and only if QAC⁰ circuits fail to satisfy Fourier concentration, thereby proposing for the first time that this property may fully characterize the computational power of QAC⁰. Furthermore, they construct a QAC⁰ circuit highly correlated with MAJORITY—achieving correlation 1−o(1)—which yields an average-case separation from classical AC⁰ in decision tasks. The paper also introduces a new measure, “felinity,” establishing an equivalence between the hardness of computing Parity and the difficulty of preparing GHZ and Dicke states.

Abstract

A major open problem in understanding shallow quantum circuits (QAC$^0$) is whether they can compute Parity. We show that this question is solely about the Fourier spectrum of QAC$^0$: any QAC$^0$ circuit with non-negligible high-level Fourier mass suffices to exactly compute PARITY in QAC$^0$. Thus, proving a quantum analog of the seminal LMN theorem for AC$^0$ is necessary to bound the quantum circuit complexity of PARITY. In the other direction, LMN does not fully capture the limitations of AC$^0$. For example, despite MAJORITY having $99\%$ of its weight on low-degree Fourier coefficients, no AC$^0$ circuit can non-trivially correlate with it. In contrast, we provide a QAC$^0$ circuit that achieves $(1-o(1))$ correlation with MAJORITY, establishing the first average-case decision separation between AC$^0$ and QAC$^0$. This suggests a uniquely quantum phenomenon: unlike in the classical setting, Fourier concentration may largely characterize the power of QAC$^0$. PARITY is also known to be equivalent in QAC$^0$ to inherently quantum tasks such as preparing GHZ states to high fidelity. We extend this equivalence to a broad class of state-synthesis tasks. We demonstrate that existing metrics such as trace distance, fidelity, and mutual information are insufficient to capture these states and introduce a new measure, felinity. We prove that preparing any state with non-negligible felinity, or derived states such as poly(n)-weight Dicke states, implies PARITY $\in$ QAC$^0$.

Problem

• Parity
• QAC0
• Fourier spectrum
• quantum circuits
• AC0

Innovation

• QAC0
• Fourier concentration
• PARITY
• quantum circuit complexity
• felinity