Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated
This addresses a major open problem in quantum circuit complexity, with implications for understanding the power of shallow quantum circuits compared to classical ones.
The paper tackles the problem of whether shallow quantum circuits (QAC0) can compute Parity, showing that this question is equivalent to the Fourier spectrum of QAC0, and provides a QAC0 circuit achieving high correlation with MAJORITY, establishing the first average-case decision separation between AC0 and QAC0.
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$^0$.