提出可高效学习弱相互作用费米子幺正算符的算法，突破量子化学与多体物理中的关键难题。
Mildly-Interacting Fermionic Unitaries are Efficiently Learnable
- 基于费米子高斯维度概念，设计能处理近高斯幺正算符的可学习框架。
- 在钻石距离下以多项式时间学习至误差ε，依赖参数为n、2^t和1/ε，其中t为非高斯门数。
- 适用于量子化学模拟中复杂但结构简单的费米子系统，适合量子算法研究者参考。
近期工作表明可高效学习费米子高斯幺正算符（又称最近邻匹配电路或非相互作用费米子幺正算符）。然而，人们自然会问：能否高效学习接近高斯的幺正算符？例如由少量非高斯门构成的算符。这类算符在量子化学和多体物理中具有重要意义，但此前尚无有效学习算法。本文首次给出此类结果：设计一个算法，对最多含O(t)个非高斯门的n模费米子幺正算符U进行查询，可在时间poly(n, 2^t, 1/ε)内返回一个在钻石距离下逼近U至ε的电路。这解决了Mele和Herasymenko提出的中心开放问题，并在最强距离度量下实现。实际上，该算法更通用：定义了幺正高斯维度这一性质，证明可学习任意满足高斯维度至少为2n - O(t)的n模幺正算符，且该类包含虽需高达2^{O(t)}个非高斯门构造但仍可学习的算符。此外，我们还给出一个poly(n, 1/ε)时间算法，用于区分一算符是否具有至少为k的高斯维度，或在弗罗贝尼乌斯距离下与所有此类算符相距ε以上，前提二者必居其一。过程中我们还获得了关于近高斯费米子幺正算符的结构性成果，可能具独立研究价值。
原文摘要 · Abstract (English)
Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that are near Gaussian: for example, unitaries prepared with a small number of non-Gaussian circuit elements. These operators find significance in quantum chemistry and many-body physics, yet no algorithm exists to learn them. We give the first such result by devising an algorithm which makes queries to an $n$-mode fermionic unitary $U$ prepared by at most $O(t)$ non-Gaussian gates and returns a circuit approximating $U$ to diamond distance $\varepsilon$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. This resolves a central open question of Mele and Herasymenko under the strongest distance metric. In fact, our algorithm is much more general: we define a property of unitary Gaussianity known as unitary Gaussian dimension and show that our algorithm can learn $n$-mode unitaries of Gaussian dimension at least $2n - O(t)$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. Indeed, this class subsumes unitaries prepared by at most $O(t)$ non-Gaussian gates but also includes several unitaries that require up to $2^{O(t)}$ non-Gaussian gates to construct. In addition, we give a $\textrm{poly}(n,1/\varepsilon)$-time algorithm to distinguish whether an $n$-mode unitary is of Gaussian dimension at least $k$ or $\varepsilon$-far from all such unitaries in Frobenius distance, promised that one is the case. Along the way, we prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.
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