aiqbench analysis
This research resolves a fundamental open problem in quantum complexity theory by establishing that the ability of shallow quantum circuits (QAC$^0$) to compute Parity is entirely determined by their Fourier spectrum. The authors introduce "felinity" as a novel metric to characterize inherently quantum state-synthesis tasks, such as preparing GHZ states, which existing classical measures fail to capture. These findings clarify the unique computational power of shallow quantum circuits and provide necessary conditions for proving quantum analogs of classical lower bound theorems.
Parity $\notin$ QAC0 $\iff$ QAC0 is Fourier-Concentrated
arXiv:2604.02793v2 Announce Type: replace 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$^0$.
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$^0$.
Quantum Physics (quant-ph); Computational Complexity (cs.CC)
arXiv:2604.02793 [quant-ph]
(or arXiv:2604.02793v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2604.02793 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 316 words
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