AI 中文总结
研究发现Keyl-Werner算法并非谱估计最优方法，提出的算法用更少量子态副本即可估计本征值，解决了相关问题并反驳了2016年猜想，还推导了其他更优算法。
AI 中文摘要
我们提出一种算法：给定量子态ρ的n=O(d²·(log log d / log d)²)份副本，可在总变差距离下以常数误差估计ρ的本征值。由此，学习量子态本征值所需副本数少于全态层析成像所需的Θ(d²)份。这是对有影响力的Keyl-Werner算法（需n=Θ(d²)份副本）在谱估计上的首次改进，解决了Keyl和Werner于2001年提出的问题，并反驳了Wright在2016年的猜想。我们的主要技术工具是一种新型层析成像保证：对所有方向|w⟩，特定方向|w⟩上的层析成像误差与⟨w|ρ|w⟩同步缩放。从这一更强的“相对误差”界出发，我们还得到了Bures距离下主成分分析及χ²散度下层析成像的更优算法作为推论。
英文摘要
We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $ρ$, estimates the eigenvalues of $ρ$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $Θ(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = Θ(d^2)$ copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction $|w\rangle$ scales with $\langle w | ρ|w\rangle$ for all directions simultaneously. From this stronger "relative-error" bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $χ^2$-divergence as corollaries.
Comments58 pages
