Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $\Omega(d^2)$ queries to entries of $A$, and hence $\Omega(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-$q$ eigenvectors in time $qd^{1.5+o(1)}$. We also prove a nearly-optimal lower bound of $\tilde{\Omega}(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure.

翻译：寻找给定 $d\times d$ 矩阵 $A$ 的顶部特征向量的良好近似，是一个基础且重要的计算问题，具有广泛的应用。我们提出了两种不同的量子算法，在给定埃尔米特矩阵 $A$ 的条目查询访问权限并假设存在常数特征值间隙的条件下，可输出顶部特征向量良好近似的经典描述：一种算法的时间复杂度为 $\mathcal{\tilde{O}}(d^{1.75})$，另一种为 $d^{1.5+o(1)}$（第一种算法对近似向量 $\ell_2$ 误差的依赖略优于第二种，且采用了具有独立价值的不同技术）。我们的两种量子算法均相对于最佳经典算法实现了多项式加速，经典算法需要 $\Omega(d^2)$ 次矩阵 $A$ 条目查询，因此需要 $\Omega(d^2)$ 时间。我们将此推广至一个量子算法，可在 $qd^{1.5+o(1)}$ 时间内输出由前 $q$ 个特征向量张成子空间的经典描述。我们还证明了近似顶部特征向量的量子查询复杂度下界为近乎最优的 $\tilde{\Omega}(d^{1.5})$。我们的量子算法运行了一种对特定良性误差具有鲁棒性的经典幂法变体，其中我们通过两种不同方式在量子计算机上以微小且良性的误差实现每次矩阵-向量乘法。第一种算法采用新型“高斯相位估计”程序逐项估计矩阵-向量乘积。第二种算法利用块编码技术将矩阵-向量乘积计算为量子态，并通过新型时间高效的无偏纯态层析程序从中获得经典描述。