Improved quantum lower and upper bounds for matrix scaling

• URL: http://arxiv.org/abs/2109.15282v1
• Date: Thu, 30 Sep 2021 17:29:23 GMT
• Title: Improved quantum lower and upper bounds for matrix scaling
• Authors: Sander Gribling, Harold Nieuwboer
• Abstract summary: We investigate the possibilities for quantum speedups for classical second-order algorithms. We show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime. We give improved quantum algorithms in the low-precision regime.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

Related papers

• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix [2.7050250604223693]
  Finding a good approximation of the top eigenvector of a given $dtimes d$ matrix $A$ is a basic and important computational problem. We give two different quantum algorithms that output a classical description of a good approximation of the top eigenvector.
  arXiv  Detail & Related papers  (2024-05-23T16:33:13Z)
• Quantum Time-Space Tradeoffs for Matrix Problems [0.5524804393257919]
  We consider the time and space required for quantum computers to solve a range of problems involving matrices. For almost all matrices $A$, we prove that quantum circuits with at most $T$ input queries and $S$ qubits of memory require $T=Omega(n2/S)$. Because many of our lower bounds match deterministic algorithms with the same time and space complexity, we show that quantum computers cannot provide any advantage for these problems with any space bound.
  arXiv  Detail & Related papers  (2024-01-10T18:38:43Z)
• A square-root speedup for finding the smallest eigenvalue [0.6597195879147555]
  We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix's low-energy subspace.
  arXiv  Detail & Related papers  (2023-11-07T22:52:56Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Robustness of Quantum Algorithms for Nonconvex Optimization [7.191453718557392]
  We show that quantum algorithms can find an $epsilon$-SOSP with poly-logarithmic,, or exponential number of queries in $d. We also characterize the domains where quantum algorithms can find an $epsilon$-SOSP with poly-logarithmic,, or exponential number of queries in $d.
  arXiv  Detail & Related papers  (2022-12-05T19:10:32Z)
• Optimal Query Complexities for Dynamic Trace Estimation [59.032228008383484]
  We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly. We provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $epsilon$ error with $delta$ failure probability. Our lower bounds (1) give the first tight bounds for Hutchinson's estimator in the matrix-vector product model with Frobenius norm error even in the static setting, and (2) are the first unconditional lower bounds for dynamic trace estimation.
  arXiv  Detail & Related papers  (2022-09-30T04:15:44Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Quantum algorithms for matrix scaling and matrix balancing [9.010461408997646]
  Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical methods: Sinkhorn's algorithm for matrix scaling and Osborne's algorithm for matrix balancing.
  arXiv  Detail & Related papers  (2020-11-25T15:26:59Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.