Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn's algorithm for matrix scaling and Osborne's algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time $\tilde O(\sqrt{mn}/\varepsilon^4)$ for scaling or balancing an $n \times n$ matrix (given by an oracle) with $m$ non-zero entries to within $\ell_1$-error $\varepsilon$. Their classical analogs use time $\tilde O(m/\varepsilon^2)$, and every classical algorithm for scaling or balancing with small constant $\varepsilon$ requires $\Omega(m)$ queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of $n$, at the expense of a worse polynomial dependence on the obtained $\ell_1$-error $\varepsilon$. We emphasize that even for constant $\varepsilon$ these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn's and Osborne's algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn's algorithm for entrywise-positive matrices to the $\ell_1$-setting, leading to an $\tilde O(n^{1.5}/\varepsilon^3)$-time quantum algorithm for $\varepsilon$-$\ell_1$-scaling in this case. We also prove a lower bound, showing that our quantum algorithm for matrix scaling is essentially optimal for constant $\varepsilon$: every quantum algorithm for matrix scaling that achieves a constant $\ell_1$-error with respect to uniform marginals needs to make at least $\Omega(\sqrt{mn})$ queries.

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用于矩阵缩放和矩阵平衡的量子算法

矩阵缩放和矩阵平衡是两个基本的线性代数问题，具有广泛的应用，例如逼近永久性和预处理线性系统以使它们在数值上更稳定。我们研究了量子算法解决这些问题的能力和局限性。我们提供两种经典方法（在两种意义上）的量子实现：用于矩阵缩放的Sinkhorn算法和用于矩阵平衡的Osborne算法。使用幅度估计作为主要工具，我们的量子实现均在时间$ \ tilde O（\ sqrt {mn} / \ varepsilon ^ 4）$中运行，以缩放或平衡$ n \ timesn $矩阵（由Oracle提供）其中$ m $个非零条目位于$ \ ell_1 $错误$ \ varepsilon $之内。他们的经典类比使用时间$ \波浪号O（m / \ varepsilon ^ 2）$，而每一个经典的用于以较小的常数$ \ varepsilon $缩放或平衡的算法都需要对输入矩阵的项进行$ \ Omega（m）$查询。因此，我们以$ n $为单位实现了多项式加速，但代价是对获得的$ \ ell_1 $-误差$ \ varepsilon $的多项式依赖性更差。我们强调，即使对于恒定的$ \ varepsilon $，这些问题也已经不容易了（与应用程序相关）。在此过程中，我们扩展了对Sinkhorn和Osborne算法的经典分析，以允许边际计算中的错误。我们还对Sinkhorn的用于入口-正矩阵的算法进行了改进的分析，以适应$ \ ell_1 $的设置，从而导致$ \代字O（n ^ {1.5} / \ varepsilon ^ 3）$时间的量子算法\在这种情况下varepsilon $-$ \ ell_1 $可缩放。我们也证明了下界