We revisit the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries. Given black-box query access to an unknown FLO, previous proposals required $\widetilde{\mathcal{O}}(n^5 / \varepsilon^2)$ queries, where $n$ is the system size and $\varepsilon$ is the error in diamond distance. These algorithms also use unphysical operations (i.e., violating fermionic superselection rules) and/or $n$ auxiliary modes to prepare Choi states of the FLO. In this work, we establish efficient and experimentally friendly protocols that obey superselection, use minimal ancilla (at most $1$ extra mode), and exhibit improved dependence on both parameters $n$ and $\varepsilon$. For arbitrary (active) FLOs this algorithm makes at most $\widetilde{\mathcal{O}}(n^4 / \varepsilon)$ queries, while for number-conserving (passive) FLOs we show that $\mathcal{O}(n^3 / \varepsilon)$ queries suffice. The complexity of the active case can be further reduced to $\widetilde{\mathcal{O}}(n^3 / \varepsilon)$ at the cost of using $n$ ancilla. This marks the first FLO learning algorithm that attains Heisenberg scaling in precision. As a side result, we also demonstrate an improved copy complexity of $\widetilde{\mathcal{O}}(n η^2 / \varepsilon^2)$ for time-efficient state tomography of $η$-particle Slater determinants in $\varepsilon$ trace distance, which may be of independent interest.

翻译：本文重新审视了学习费米子线性光学（FLO），即费米子高斯幺正变换的问题。给定对一个未知FLO的黑盒查询访问，先前方案需要$\widetilde{\mathcal{O}}(n^5 / \varepsilon^2)$次查询，其中$n$为系统规模，$\varepsilon$为钻石范数误差。这些算法还使用了非物理操作（即违反费米子超选择定则）和/或$n$个辅助模式来制备该FLO的Choi态。本工作中，我们建立了高效且实验友好的协议，这些协议遵守超选择定则，使用最少的辅助系统（至多$1$个额外模式），并展现出对参数$n$和$\varepsilon$的改进依赖关系。对于任意（主动型）FLO，该算法至多进行$\widetilde{\mathcal{O}}(n^4 / \varepsilon)$次查询；而对于粒子数守恒（被动型）FLO，我们证明$\mathcal{O}(n^3 / \varepsilon)$次查询即足够。主动型情况的复杂度可进一步降至$\widetilde{\mathcal{O}}(n^3 / \varepsilon)$，但代价是使用$n$个辅助模式。这标志着首个在精度上达到海森堡标度的FLO学习算法。作为附带结果，我们还展示了在$\varepsilon$迹距离下，对$η$粒子Slater行列式进行时间高效态层析的拷贝复杂度改进为$\widetilde{\mathcal{O}}(n η^2 / \varepsilon^2)$，这可能具有独立的研究价值。