Quantum Approximation I. Embeddings of Finite Dimensional L_p Spaces

TL;DR

This paper investigates the quantum complexity of embedding finite-dimensional L_p spaces, revealing regions where quantum computation outperforms classical methods and establishing bounds for approximation rates.

Contribution

It provides tight bounds on quantum query complexity for embeddings of finite-dimensional L_p spaces, highlighting where quantum advantages exist.

Findings

Quantum computation can significantly improve approximation rates in certain parameter regions.

Matching upper and lower bounds are established for quantum query complexity.

Results serve as foundational for future analysis of function space approximations.

Abstract

We study approximation of embeddings between finite dimensional L_p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper.