Quantum Speedups for Approximating the John Ellipsoid

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 ∈ Rn × d with a time complexity of O(nd²). This was later improved to O(nnz(A) + d^ω) by [Song, Yang, Yang, Zhou 2022], where nnz(A) is the number of nonzero entries of A and ω is the matrix multiplication exponent. Currently ω ≈ 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(√nd1.5 + 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.