*Joran van Apeldoorn, Sander Gribling, Yinan Li, Harold Nieuwboer, Michael Walter, Ronald de Wolf*

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 ~O(sqrt(mn)/eps^4) for scaling or balancing an nx n matrix (given by an oracle) with m non-zero entries to within ell_1-error eps. Their classical analogs use ~O(m/eps^2) time, and every classical algorithm for scaling or balancing with small constant eps 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 eps . We emphasize that even for constant eps these problems are already non-trivial (and relevant in applications).

**On arXiv**