Fast and Practical Quantum-Inspired Classical Algorithms for Solving Linear Systems

• URL: http://arxiv.org/abs/2307.06627v2
• Date: Thu, 30 Nov 2023 12:33:24 GMT
• Title: Fast and Practical Quantum-Inspired Classical Algorithms for Solving Linear Systems
• Authors: Qian Zuo and Tongyang Li
• Abstract summary: We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems.
• Score: 11.929584800629673
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical algorithms that produce a data structure for the solution $x\in\mathbb{R}^{n}$ of the linear system $Ax=b$ with the ability to sample and query its entries. The resulting $x$ satisfies $\|x-A^{+}b\|\leq\epsilon\|A^{+}b\|$, where $\|\cdot\|$ is the spectral norm and $A^+$ is the Moore-Penrose inverse of $A$. Our algorithm has time complexity $\widetilde{O}(\kappa_F^4/\kappa\epsilon^2)$ in the general case, where $\kappa_{F} =\|A\|_F\|A^+\|$ and $\kappa=\|A\|\|A^+\|$ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When $A$ is $s$-sparse, our algorithm has complexity $\widetilde{O}(s \kappa\log(1/\epsilon))$, matching the quantum lower bound for solving linear systems in $\kappa$ and $1/\epsilon$ up to poly-logarithmic factors [Harrow and Kothari]. When $A$ is $s$-sparse and symmetric positive-definite, our algorithm has complexity $\widetilde{O}(s\sqrt{\kappa}\log(1/\epsilon))$. Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.

Related papers

• Quantum spectral method for gradient and Hessian estimation [4.193480001271463]
  Gradient descent is one of the most basic algorithms for solving continuous optimization problems. We propose a quantum algorithm that returns an $varepsilon$-approximation of its gradient with query complexity $widetildeO (1/varepsilon)$. We also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method.
  arXiv  Detail & Related papers  (2024-07-04T11:03:48Z)
• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Quantum machine learning with subspace states [8.22379888383833]
  We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum sampling algorithm that samples from the distribution $Pr[S]= det(X_SX_ST)$ for $|S|=d$ using $O(nd)$ gates and with circuit depth $O(dlog n)$. The second is a quantum singular value estimation algorithm for compound matrices $mathcalAk$, the speedup for this algorithm is potentially exponential.
  arXiv  Detail & Related papers  (2022-01-31T19:34:47Z)
• Faster quantum-inspired algorithms for solving linear systems [0.05076419064097732]
  We show that there is a classical algorithm that outputs a data structure for $x$ allowing sampling and querying to the entries. This output can be viewed as a classical analogue to the output of quantum linear solvers.
  arXiv  Detail & Related papers  (2021-03-18T15:12:44Z)
• 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)
• An improved quantum-inspired algorithm for linear regression [15.090593955414137]
  We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm. We show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting.
  arXiv  Detail & Related papers  (2020-09-15T17:58:25Z)
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation [0.8057006406834467]
  We present a quantum algorithm to solve systems of linear equations of the form $Amathbfx=mathbfb$. The algorithm achieves an exponential improvement with respect to $N$ over classical methods.
  arXiv  Detail & Related papers  (2020-09-09T18:00:09Z)
• Learning nonlinear dynamical systems from a single trajectory [102.60042167341956]
  We introduce algorithms for learning nonlinear dynamical systems of the form $x_t+1=sigma(Thetastarx_t)+varepsilon_t$. We give an algorithm that recovers the weight matrix $Thetastar$ from a single trajectory with optimal sample complexity and linear running time.
  arXiv  Detail & Related papers  (2020-04-30T10:42:48Z)

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.