AI 中文总结
本文定义了费米子熵这一可高效测量的非高斯性强单调量，推导其操作意义、样本复杂度界，并应用于Matchgate电路生成酉设计的研究，明确了非高斯门的最优用量及生成类Haar动力学的非高斯性成本。
AI 中文摘要
费米子高斯态是一类经典可处理的核心量子态，而费米子非高斯性是超越自由费米子动力学所需的资源。关键挑战在于通过兼具数学严谨性与实验可及性的单调量来量化该资源。本文证明，由关联矩阵平方的弗罗贝尼乌斯范数定义的费米子熵，是一种强纯态高斯单调量。其简洁的闭式表达式也使其可直接测量：我们证明，利用O(ε⁻²)次双副本测量，可对关联的费米子纯度进行加性误差ε下的无偏估计，且该复杂度与系统大小无关。此外，我们证明费米子熵满足渐近连续性，并由此直接确立其作为非高斯性蒸馏渐近速率上界的操作意义。我们进一步推导了费米子高斯态容错测试的线性样本复杂度界，较现有技术实现了二次改进。作为本文结果的进一步应用，我们研究由Matchgate电路补充Majorana局域非高斯门生成的酉设计，证明仅要达到误差低于0.4%的近似态2-设计，就需要线性数量的此类门。结合相对误差设计的已知近线性上界，这确定了所有相关设计概念下的最优掺杂水平（对数因子除外），并揭示了在该架构中生成类Haar量子动力学所需的大量非高斯性成本。
英文摘要
Fermionic Gaussian states form a central class of classically tractable quantum states, while fermionic non-Gaussianity provides the resource required to go beyond free-fermion dynamics. A key challenge is to quantify this resource through monotones that are both mathematically rigorous and experimentally accessible. Here, we show that the fermionic entropy, defined through the squared Frobenius norm of the correlation matrix, is a strong pure-state Gaussian monotone. Its simple closed-form expression also makes it directly measurable: we show that the associated fermionic purity can be unbiasedly estimated up to additive error $\varepsilon$ using $O(\varepsilon^{-2})$ two-copy measurements, independently of the system size. Moreover, we prove that the fermionic entropy obeys asymptotic continuity and, as a direct consequence, establish its operational meaning as the upper bound to the asymptotic rate of non-Gaussianity distillation. We further derive a linear sample complexity bound for tolerant testing of fermionic Gaussian states, providing a quadratic improvement over the state of the art. As a further application of our results, we study unitary designs generated by Matchgate circuits supplemented with Majorana-local non-Gaussian gates. We prove that a linear number of such gates is necessary even to achieve an approximate state $2$-design with error below $0.4\%$. Combined with known nearly linear upper bounds for relative-error designs, this determines the optimal doping level, up to logarithmic factors, across all relevant design notions and reveals the extensive non-Gaussianity cost required to generate Haar-like quantum dynamics in this architecture.
