论文标题
用于解决量子线性系统问题的有效量子算法
Efficient quantum algorithms for solving quantum linear system problems
论文作者
论文摘要
我们将求解方程式的线性系统$ a \ mathbf {x} = \ mathbf {b} $转换为找到具有增强矩阵$ c $的单数值零的零值的问题，并呈现两个用于解决此问题的量子算法的问题。第一种算法通过应用$ o \ left的查询复杂性（sκ\ log \ left（1/ε\ right）\右）的查询复杂性的量子本本特征过滤算法直接解决了问题，以$ s $ s $ -s $ -s-s $ s-s $ -ssparse $ c $ c $，其中$κ$是$κ$的条件，而$κ$是$ s $ $ a $ a $ a $ a $ a $ a $ a $ a $ ^ is $ ^。第二个算法使用量子共振过渡方法，查询复杂度缩放为$ o \ left [sκ+ \ log \ left（1/ε\ right）/\ log \ log \ log \ log \ left（1/ε\ right）\ right] $。两种算法都符合$κ$的最佳查询复杂性，并且比以前的算法更简单。
We transform the problem of solving linear system of equations $A\mathbf{x}=\mathbf{b}$ to a problem of finding the right singular vector with singular value zero of an augmented matrix $C$, and present two quantum algorithms for solving this problem. The first algorithm solves the problem directly by applying the quantum eigenstate filtering algorithm with query complexity of $O\left( sκ\log \left( 1/ε\right) \right) $ for a $s$-sparse matrix $C$, where $κ$ is the condition number of the matrix $A$, and $ε$ is the desired precision. The second algorithm uses the quantum resonant transition approach, the query complexity scales as $O\left[sκ+ \log\left( 1/ε\right)/\log \log \left( 1/ε\right) \right] $. Both algorithms meet the optimal query complexity in $κ$, and are simpler than previous algorithms.
