AI 中文总结
研究量子位稳定器状态的无偏学习问题，成功将稳定器自举框架推广到量子位系统，提出高效量子算法，能以高概率输出接近未知态保真度的稳定器状态，还给出高保真度简化算法。
AI 中文摘要
学习量子态的经典描述是量子计算中的一项基本任务。稳定器状态是最重要的量子态类别之一，在量子纠错和容错计算中起着核心作用。为减轻实际噪声的影响，无偏学习稳定器状态成为一个自然且动机充分的问题。最近，Chen等人通过使用稳定器自举框架解决了量子比特系统的这个问题。然而，量子位稳定器状态的无偏学习在很大程度上仍未被探索，因为量子位设置引入了基本结构差异，阻碍了现有量子比特技术的直接推广。本文成功将稳定器自举框架推广到量子位系统，并提出了第一个用于无偏学习量子位稳定器状态的高效量子算法。具体而言，给定一个与某个稳定器状态保真度为τ的未知n量子位纯态|ψ⟩的副本，我们的算法输出一个稳定器状态|φ⟩，使得|⟨φ|ψ⟩|²≥τ - ε的概率很高。该算法仅使用单副本和四副本测量，其样本和时间复杂度为(d/τ)^(O(d²log(1/τ)))·poly(n, 1/ε)，其中维度d是奇质数。作为直接推论，我们的算法能够有效地估计量子态的魔力，用其稳定器保真度来量化。此外，我们还为高保真度 regime τ > cos²(π/8) 提出了一种简化算法，建立了先前量子比特工作中基于阈值方法的量子位类似物。
英文摘要
Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown $n$-qudit pure state $|ψ\rangle$ that has fidelity $τ$ with some stabilizer state, our algorithm outputs a stabilizer state $|ϕ\rangle$ such that $\left| \braket{ϕ|ψ} \right|^2 \geq τ- \varepsilon$ with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as $(d/τ)^{O(d^2 \log(1/τ))} \cdot \mathrm{poly}(n, 1/\varepsilon)$, where the dimension $d$ is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime $τ> \cos^2(π/8)$, establishing a qudit analogue of the threshold-based approach in prior qubit work.
