2026-07-22T06:07:19Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9529 2024-03-06T10:43:58Z ddc:004 open_access

On Learning Linear Functions from Subset and Its Applications in Quantum Computing Ivanyos, Gábor Prakash, Anupam Santha, Miklos Learning from subset hidden shift problem quantum algorithms linearization Let F_{q} be the finite field of size q and let l: F_{q}^{n} -> F_{q} be a linear function. We introduce the Learning From Subset problem LFS(q,n,d) of learning l, given samples u in F_{q}^{n} from a special distribution depending on l: the probability of sampling u is a function of l(u) and is non zero for at most d values of l(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 [Friedl et al., 2014; Gábor Ivanyos, 2008] 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 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: Z_{q}^{n} -> {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 [Andrew M. Childs and Wim van Dam, 2007; Friedl et al., 2014] for HMS(q,n,r) with these parameters require exponential time. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gábor Ivanyos and Anupam Prakash and Miklos Santha 2018 Is Part Of LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2018.66 urn:nbn:de:0030-drops-95299 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.66 eng https://creativecommons.org/licenses/by/3.0/legalcode