Quantum Speedup for Spectral Approximation of Kronecker Products
Yeqi Gao, Zhao Song, Ruizhe Zhang
TL;DR
The paper addresses efficient spectral approximation of Kronecker products $A_1\otimes A_2$, a task in tensor-based optimization with high classical costs. It introduces a quantum sketching framework building on quantum repeated halving, generalized leverage scores, and 2D (Kronecker) sampling to produce a classical diagonal sketch $D$ with $\|D\|_0=O(\epsilon^{-2}d^2\log d)$ such that $(1-\epsilon)A^T A\preceq A^T D^T D A\preceq(1+\epsilon)A^T A$, where $A=A_1\otimes A_2$ and $D\sim\mathsf{LS}(A)$. The main result achieves $\widetilde{O}(\sqrt{nd}/\epsilon)$ row queries to the input and runtime $\widetilde{O}( r\sqrt{nd}/\epsilon + d^{\omega})$, offering a polynomial quantum speedup in the tall-matrix regime and delivering a fully classical output suitable for downstream sketching tasks. By generalizing leverage-score sampling to the Kronecker/tensor setting and adapting the halving framework to 2D, the approach enables scalable spectral approximation for tensor-structured problems with potential impact on tensor regression and related optimization pipelines.
Abstract
Given its widespread application in machine learning and optimization, the Kronecker product emerges as a pivotal linear algebra operator. However, its computational demands render it an expensive operation, leading to heightened costs in spectral approximation of it through traditional computation algorithms. Existing classical methods for spectral approximation exhibit a linear dependency on the matrix dimension denoted by $n$, considering matrices of size $A_1 \in \mathbb{R}^{n \times d}$ and $A_2 \in \mathbb{R}^{n \times d}$. Our work introduces an innovative approach to efficiently address the spectral approximation of the Kronecker product $A_1 \otimes A_2$ using quantum methods. By treating matrices as quantum states, our proposed method significantly reduces the time complexity of spectral approximation to $O_{d,ε}(\sqrt{n})$.