论文标题
量子低级近似问题
The quantum low-rank approximation problem
论文作者
论文摘要
我们考虑了著名的低级近似问题的量子版本。具体而言，我们考虑两个归一化量子状态，$ρ$和$σ$之间的距离$ d（ρ，σ）$，其中$σ$的排名最多为$ r $。对于痕量距离和Hilbert-Schmidt距离，我们通过分析求解最佳状态$σ$，从而最小化了这一距离。对于Hilbert-schmidt距离，独特的最佳状态为$σ=τ_r +n_r $，其中$τ_r=π_r=π_rρπ_r$是通过将$ρ$投影到其$ r $ r $的主要组件上，带有projector $π_r$，而$ n_r $是$ n_r $是$ n_r $是$ N_R = \ n_r = \ frac = \ frac = \ frac {1- \ frac {1- \ text {tr}（τ_r）} {r}π_r$。对于痕量距离，该状态也是最佳的，但不是独特的最佳选择，我们提供了最佳状态。我们简要讨论我们的结果如何通过量子计算机上的各种优化来执行主成分分析（PCA）。
We consider a quantum version of the famous low-rank approximation problem. Specifically, we consider the distance $D(ρ,σ)$ between two normalized quantum states, $ρ$ and $σ$, where the rank of $σ$ is constrained to be at most $R$. For both the trace distance and Hilbert-Schmidt distance, we analytically solve for the optimal state $σ$ that minimizes this distance. For the Hilbert-Schmidt distance, the unique optimal state is $σ= τ_R +N_R$, where $τ_R = Π_R ρΠ_R$ is given by projecting $ρ$ onto its $R$ principal components with projector $Π_R$, and $N_R$ is a normalization factor given by $N_R = \frac{1- \text{Tr}(τ_R)}{R}Π_R$. For the trace distance, this state is also optimal but not uniquely optimal, and we provide the full set of states that are optimal. We briefly discuss how our results have application for performing principal component analysis (PCA) via variational optimization on quantum computers.
