Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound

  title={Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound},
  author={Aleksandrs Belovs},
  journal={computational complexity},
  • Aleksandrs Belovs
  • Published 26 November 2013
  • Computer Science, Mathematics
  • computational complexity
In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h. The task is to identify the variables the function depends on.When h is the XOR or the OR function, this gives a restricted variant of the Bernstein–Vazirani or the combinatorial group testing problem, respectively.We analyze the general case using the… 
Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing
This tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm.
A ug 2 01 6 Exact quantum query complexity of EXACT nk , l
The minimum number of queries for an exact quantum algorithm computing the function f is denoted by QE(f), and the following n bit function with 0 ≤ k ≤ l ≤ n is considered.
Tight Quantum Lower Bound for Approximate Counting with Quantum States
The lower bounds are proven using variants of the adversary bound by Belovs and employing analysis closely related to the Johnson association scheme, giving tight trade-offs between all types of resources available to the algorithm.
Quantum algorithms for learning graphs and beyond
We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a
Quantum algorithm for learning secret strings and its experimental demonstration
It is proved that any classical deterministic algorithm needs at least n queries to the oracle fs to learn the n-bit secret string s in both the worst case and the average case, and an optimal classical Deterministic algorithm learning any s using n queries is presented.
Optimal one-shot quantum algorithm for EQUALITY and AND
It is shown that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.
Variations on Quantum Adversary
This work obtains a version of the bound for general input oracles, which is a lower bound for exact transformation and an upper bound for approximate transformation and possesses the tight composition property.
Quantum Algorithm for Monotonicity Testing on the Hypercube
A bounded-error quantum algorithm is developed that makes $\tilde O(n^{1/4}\varepsilon^{-1/2})$ queries to a Boolean function $f$, accepts a monotone function, and rejects a function that is $\varpsilon$-far from being monotones.
Classical Lower Bounds from Quantum Upper Bounds
It is shown that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt n) times the approximatedegree of f, which is optimal, and a new proof of Razborov's celebrated Omega (sqrtn) lower bound on the quantum communication complexity of the disjointness problem is given.
Lower bounds on quantum query complexity for symmetric functions
The value of lower bounding techniques using polynomial method and adversary method for quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions is explored.


Quantum Algorithms for Learning and Testing Juntas
This article develops quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables, and establishes the following lower bound: any FS-based k-junta testing algorithm requires $$Omega(\sqrt{k})$$ queries.
An optimal quantum algorithm for the oracle identification problem
The algorithm considerably simplifies and improves the previous best algorithm due to Ambainis et al. and removes all log factors from the best known quantum algorithm for Boolean matrix multiplication.
Easy and hard functions for the Boolean hidden shift problem
It is demonstrated that the easiest instances of the Boolean hidden shift problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent.
Quantum Identification of Boolean Oracles
The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem, but the interest is in the quantum query complexity, for which several upper bounds are presented.
Quantum algorithms for search with wildcards and combinatorial group testing
A nearly optimal O(√n log n) quantum query algorithm is given for search with wildcards, beating the classical lower bound of Ω(n) queries.
Improved Bounds on Quantum Learning Algorithms
AbstractIn this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al.[Quantum Information
A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem
A quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$.
Negative weights make adversaries stronger
A stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function, and which is a lower bound on bounded-error quantum query complexity.
Span Programs and Quantum Query Complexity: The General Adversary Bound Is Nearly Tight for Every Boolean Function
  • B. Reichardt
  • Computer Science
    2009 50th Annual IEEE Symposium on Foundations of Computer Science
  • 2009
It is generally that properties of eigenvalue-zero eigenvectors in fact imply an "effective" spectral gap around zero, and a strong universality result for span programs follows.
Equivalences and Separations Between Quantum and Classical Learnability
These results contrast known results that show that testing black-box functions for various properties, as opposed to learning, can require exponentially more classical queries than quantum queries.