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On Learning Linear Functions from Subset and Its Applications in Quantum Computing

Authors Gábor Ivanyos, Anupam Prakash, Miklos Santha

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Author Details

Gábor Ivanyos
  • Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary
Anupam Prakash
  • CNRS, IRIF, Université Paris Diderot 75205 Paris, France
Miklos Santha
  • CNRS, IRIF, Université Paris Diderot 75205 Paris, France
  • and , Centre for Quantum Technologies, National University of Singapore, Singapore 117543

Funding

A part of the research was accomplished while the first two authors were visiting the Centre for Quantum Technologies (CQT), National University of Singapore. The research at CQT was partially funded by the Singapore Ministry of Education and the National Research Foundation under grant R-710-000-012-135. This research was supported in part by the QuantERA ERA-NET Cofund project QuantAlgo and by the Hungarian National Research, Development and Innovation Office - NKFIH, Grant K115288.

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Gábor Ivanyos, Anupam Prakash, and Miklos Santha. On Learning Linear Functions from Subset and Its Applications in Quantum Computing. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.66

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Learning from subset
  • hidden shift problem
  • quantum algorithms
  • linearization

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