利用Heisenberg缩放和物理操作学习费米子线性光学
Learning fermionic linear optics with Heisenberg scaling and physical operations
AI总结：
本文提出了一种高效学习费米子线性光学的新方法，实现了Heisenberg缩放精度，并改进了状态重构的复杂度。
摘要：
我们重新审视学习费米子线性光学（FLO）的问题，也称为费米子高斯幺正变换。给定对未知FLO的黑盒查询访问，先前的提案需要~O(n⁵/ε²)次查询，其中n是系统规模，ε是钻石距离的误差。这些算法还使用不物理的操作（即违反费米子超选择规则）和/或n个辅助模式来准备FLO的 Choi 状态。在本工作中，我们建立了高效且实验友好的协议，这些协议遵守超选择规则，使用最少的辅助模式（最多1个额外模式），并且在两个参数n和ε上表现出改进的依赖性。对于任意（主动）FLOs，该算法最多需要~O(n⁴/ε)次查询，而对于守恒数（被动）FLOs，我们显示只需O(n³/ε)次查询。主动情况的复杂度可以进一步减少到~O(n³/ε)，代价是使用n个辅助模式。这标志着首个达到Heisenberg缩放精度的FLO学习算法。作为附带结果，我们还展示了改进的复制复杂度~O(nη²/ε²)，用于在ε迹距离下对η-粒子Slater行列式进行时间高效的状态重构，这可能具有独立兴趣。
英文摘要：
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.
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