Improved quantum lower and upper bounds for matrix scaling

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 × n matrices with at most m non-zero entries and with ℓ2-error varepsilon=widetildeTheta(1/m) must make widetildeOmega(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for varepsilon∈[1/n,1/2], any quantum algorithm capable of producing fracvarepsilon100-ℓ1-approximations of the row-sum vector of a (dense) normalized matrix uses Omega(n/varepsilon) queries, and that there exists a constant varepsilon0>0 for which this problem takes Omega(n1.5) 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-ℓ1-scaling in time widetilde O(n1.5/varepsilon2), whereas the best previously known bounds were widetilde O(n2polylog(1/varepsilon)) (classical) and widetilde O(n1.5/varepsilon3) (quantum).