AI 中文总结
针对量子1-PCA问题，提出一种使用非自适应单量子比特泡利测量的算法，在混合态和谱间隙条件下，以近线性时间恢复主导特征向量，并推广了纯态情形的结果。
AI 中文摘要
我们考虑量子1-PCA问题：给定一个未知的$n$量子比特混合态的副本，恢复其主导特征向量的经典描述。我们的目标是使用非自适应和单量子比特测量来实现这一目标。对于具有最大特征值$\lambda$和谱间隙至少为$\Delta > 0$的$n$量子比特态，我们给出一个算法，该算法以高概率恢复主导特征向量至保真度至少$1 - \varepsilon$，使用$\tilde{O}\left( {2^n \cdot \eta^2} / {\Delta^3 \varepsilon^3}\right)$个副本和$\tilde{O}((2^n/\Delta \varepsilon) \cdot \operatorname{poly}(\eta/\Delta\varepsilon))$时间，其中$\eta = \max (1 - \lambda, \varepsilon)$。我们所有的测量都是非自适应选择的，并在单量子比特泡利基下进行。当谱间隙为常数且所需精度与噪声水平相当时，即$\varepsilon = \Omega (\eta)$，我们的运行时间和副本复杂度变为$\tilde{O} (2^n / \varepsilon)$。这推广了Grewal等人[ arXiv:2601.04444 ]的保证，他们实现了类似的速率，但假设$\eta = 0$，即状态是纯的。我们的结果表明，在状态错误指定的情况下，相同的速率仍然成立，最多相差多对数因子。从技术角度来看，我们的算法通过递归构造近似保留目标特征向量的低维子空间来工作。为了实现运行时间对希尔伯特空间维度的近线性依赖，我们开发了一种新颖的结构化泡利采样方案，该方案能够快速批量计算指数多个投影泡利矩阵。
英文摘要
We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $λ$ and spectral gap at least $Δ> 0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \varepsilon$ with high probability using $\tilde{O}\left( {2^n \cdot η^2} / {Δ^3 \varepsilon^3}\right)$ copies and $\tilde{O}((2^n/Δ\varepsilon) \cdot \operatorname{poly}(η/Δ\varepsilon))$ time, where $η= \max (1 - λ, \varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\varepsilon = Ω(η)$, our runtime and copy complexity become $\tilde{O} (2^n / \varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $η= 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.
