On learning linear functions from subset and its applications in quantum computing

TL;DR

This paper introduces a new algorithm for learning linear functions from subset samples over finite fields, with applications to quantum computing problems like the Hidden Multiple Shift, improving efficiency over previous methods.

Contribution

The paper presents a generalized, efficient randomized algorithm for the Learning From Subset problem, extending prior work and applying it to quantum algorithms for the Hidden Multiple Shift problem.

Findings

Algorithm with sample complexity (n+d)^{O(d)}

Polynomial-time solution for certain quantum shift problems

Improved over previous exponential-time algorithms

Abstract

Let $\mathbb{F}_q$ be the finite field of size $q$ and let $\ell: \mathbb{F}_q^n \to \mathbb{F}_q$ be a linear function. We introduce the {\em Learning From Subset} problem LFS$(q,n,d)$ of learning $\ell$, given samples $u \in \mathbb{F}_q^n$ from a special distribution depending on $\ell$: the probability of sampling $u$ is a function of $\ell(u)$ and is non zero for at most $d$ values of $\ell(u)$. We provide a randomized algorithm for LFS$(q,n,d)$ with sample complexity $(n+d)^{O(d)}$ and running time polynomial in $\log q$ and $(n+d)^{O(d)}$. Our algorithm generalizes and improves upon previous results \cite{Friedl, Ivanyos} that had provided algorithms for LFS$(q,n,q-1)$ with running time $(n+q)^{O(q)}$. We further present applications of our result to the {\em Hidden Multiple Shift} problem HMS$(q,n,r)$ in quantum computation where the goal is to determine the hidden shift $s$ given oracle access to $r$ shifted copies of an injective function $f: \mathbb{Z}_q^n \to \{0, 1\}^l$, that is we can make queries of the form $f_s(x,h) = f(x-hs)$ where $h$ can assume $r$ possible values. We reduce HMS$(q,n,r)$ to LFS$(q,n, q-r+1)$ to obtain a polynomial time algorithm for HMS$(q,n,r)$ when $q=n^{O(1)}$ is prime and $q-r=O(1)$. The best known algorithms \cite{CD07, Friedl} for HMS$(q,n,r)$ with these parameters require exponential time.