Revisiting Quantum Algorithms for Linear Regressions: Quadratic Speedups without Data-Dependent Parameters

• URL: http://arxiv.org/abs/2311.14823v1
• Date: Fri, 24 Nov 2023 19:41:28 GMT
• Title: Revisiting Quantum Algorithms for Linear Regressions: Quadratic Speedups without Data-Dependent Parameters
• Authors: Zhao Song, Junze Yin, Ruizhe Zhang
• Abstract summary: We develop a quantum algorithm that runs in $widetildeO(epsilon-1sqrtnd1.5) + mathrmpoly(d/epsilon)$ time. It provides a quadratic quantum speedup in $n over the classical lower bound without any dependence on data-dependent parameters.
• Score: 10.602399256297032
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b \|_2^2 $. The best classical algorithm takes $O(nd) + \mathrm{poly}(d/\epsilon)$ time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as $\kappa(A)$, the condition number of $A$. In this work, we develop a quantum algorithm that runs in $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$ time. It provides a quadratic quantum speedup in $n$ over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.

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