Quantum Speedups for Approximating the John Ellipsoid
Xiaoyu Li, Zhao Song, Junwei Yu
TL;DR
This work provides the first quantum algorithm for approximating the John ellipsoid of a symmetric convex polytope defined by a tall matrix A, achieving a sublinear dependence on the ambient dimension n in the tall-matrix regime. The approach harnesses quantum spectral approximation and quantum-leverage-score techniques within a fixed-point iteration framework to compute the Lewis weights that certify the John ellipsoid, yielding a runtime of ṫilde{O}(ε^{−2} √n d^{1.5} + ε^{−3} d^ω) and ṫilde{O}(ε^{−2} √{nd}) queries to A, with high success probability. A key contribution is the quantum fixed-point iteration, plus a telescoping analysis that guarantees convergence to a (1+ε)-approximate solution; the method also extends to a tensor generalization via Kronecker products, which decomposes into two ordinary John ellipsoid problems. The results highlight the potential of quantum sketching and spectral-approximation techniques to accelerate high-dimensional geometric optimization, with practical implications for high-dimensional sampling, linear programming, and related machine-learning tasks.
Abstract
In 1948, Fritz John proposed a theorem stating that every convex body has a unique maximal volume inscribed ellipsoid, known as the John ellipsoid. The John ellipsoid has become fundamental in mathematics, with extensive applications in high-dimensional sampling, linear programming, and machine learning. Designing faster algorithms to compute the John ellipsoid is therefore an important and emerging problem. In [Cohen, Cousins, Lee, Yang COLT 2019], they established an algorithm for approximating the John ellipsoid for a symmetric convex polytope defined by a matrix $A \in \mathbb{R}^{n \times d}$ with a time complexity of $O(nd^2)$. This was later improved to $O(\text{nnz}(A) + d^ω)$ by [Song, Yang, Yang, Zhou 2022], where $\text{nnz}(A)$ is the number of nonzero entries of $A$ and $ω$ is the matrix multiplication exponent. Currently $ω\approx 2.371$ [Alman, Duan, Williams, Xu, Xu, Zhou 2024]. In this work, we present the first quantum algorithm that computes the John ellipsoid utilizing recent advances in quantum algorithms for spectral approximation and leverage score approximation, running in $O(\sqrt{n}d^{1.5} + d^ω)$ time. In the tall matrix regime, our algorithm achieves quadratic speedup, resulting in a sublinear running time and significantly outperforming the current best classical algorithms.