高效学习Clifford解纠缠器和具有指数级更多$T$门的典型$t$-掺杂酉算子
Efficient learning of Clifford disentanglers and typical $t$-doped unitaries with exponentially more $T$ gates
- Freie Universität Berlin（柏林自由大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出高效算法，利用Bell分布的二次对称性恢复Clifford酉算子隐藏的张量积结构，实现典型$t$-掺杂酉算子的正确学习，支持指数级更多$T$门，并用于哈密顿量压缩和容错基准测试。
AI中文摘要：
高度纠缠且高度非稳定子的量子态不一定难以学习。我们给出了高效算法，用于测试和恢复未知纯态矢量中隐藏的张量积结构，这些态具有形式$\lvertψ\rangle = U_C \bigotimes_i \lvertψ_i\rangle$，其中$U_C$是任意未知的Clifford酉算子。尽管Clifford可以彻底打乱可见的乘积结构，我们证明了Bell分布保留了一族特征性的二次对称性。通过同时对这些对称性进行块对角化，我们的算法将问题线性化，并能够以多项式样本和计算复杂度恢复解纠缠Clifford和隐藏的划分。这可以被视为阿贝尔StateHSP范式的扩展，其中经典后处理暴露了真正的二次结构。应用于Choi态时，该方法为典型的$t$-掺杂Clifford酉算子提供了高效的正确学习算法，在包含指数级更多$T$门的参数范围内，比以前可访问的范围更广：当$t\sim n$时，所需条件仅对指数小部分的电路失效，并且即使当$t=2n$时，对于恒定比例的电路仍然成立。我们的框架还为压缩结构化多体哈密顿量提供了工具，并提出了早期容错时代编码逻辑乘积态的基准测试协议。
英文摘要：
Highly entangled and highly non-stabilizer quantum states need not be hard to learn. We give efficient algorithms for testing and recovering hidden tensor-product structure in unknown pure state vectors of the form $\lvertψ\rangle = U_C \bigotimes_i \lvertψ_i\rangle$, where $U_C$ is an arbitrary unknown Clifford unitary. Although the Clifford can thoroughly scramble the visible product structure, we prove that the Bell distribution retains a characteristic family of quadratic symmetries. By simultaneously block-diagonalising these symmetries, our algorithms linearize the problem and manage to recover both a disentangling Clifford and the hidden partitions with polynomial sample and computational complexity. This may be viewed as an extension of the abelian StateHSP paradigm in which classical post-processing exposes genuinely quadratic structure. Applied to Choi states, the method yields efficient proper learning algorithms for typical $t$-doped Clifford unitaries in regimes containing exponentially more $T$ gates than previously accessible: the required condition fails only for an exponentially small fraction of circuits when $t\sim n$, and continues to hold for a constant fraction even when $t=2n$. Our framework also provides tools for compressing structured many-body Hamiltonians and suggests benchmarking protocols for encoded logical product states in the early fault-tolerant regime.
