Improved quantum lower and upper bounds for matrix scaling

TL;DR

This paper establishes fundamental limits on quantum speedups for classical matrix scaling algorithms, showing near-optimal lower bounds in high-precision regimes and providing improved quantum algorithms for low-precision cases.

Contribution

It proves that quantum algorithms cannot significantly outperform classical algorithms in high-precision matrix scaling, and introduces enhanced quantum methods for low-precision scenarios.

Findings

Quantum algorithms require   queries in high-precision regimes.

Quantum algorithms need   queries for approximate row-sum computations.

New quantum algorithms outperform previous bounds in low-precision matrix scaling.

Abstract

Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for $n \times n$ matrices with at most $m$ non-zero entries and with $\ell_2$-error $\varepsilon=\widetilde\Theta(1/m)$ must make $\widetilde\Omega(m)$ queries to the matrix, even when the success probability is exponentially small in $n$. Additionally, we show that for $\varepsilon\in[1/n,1/2]$, any quantum algorithm capable of producing $\frac{\varepsilon}{100}$-$\ell_1$-approximations of the row-sum vector of a (dense) normalized matrix uses $\Omega(n/\varepsilon)$ queries, and that there exists a constant $\varepsilon_0>0$ for which this problem takes $\Omega(n^{1.5})$ queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-$\varepsilon$ regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an $\varepsilon$-$\ell_1$-scaling in time $\widetilde O(n^{1.5}/\varepsilon^2)$, whereas the best previously known bounds were $\widetilde O(n^2\mathrm{polylog}(1/\varepsilon))$ (classical) and $\widetilde O(n^{1.5}/\varepsilon^3)$ (quantum).