Identifying Generalized Reed-Muller Codewords by Quantum Queries

2017 ◽  
Vol 28 (02) ◽  
pp. 185-194
Author(s):  
Stefan Arnold
Keyword(s):  
Finite Field ◽  
Probability Of Error ◽  
Exact Quantum ◽  
Multilinear Polynomials ◽  
Query Algorithm ◽  
Quantum Query

We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qn over the finite field of size q. When d is constant, the algorithm needs 𝒪(nd-1) quantum queries, which is optimal. Classically, Ω(nd) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals On Exact Quantum Query Complexity

Algorithmica ◽  
2013 ◽  
Vol 71 (4) ◽  
pp. 775-796 ◽  
Author(s):  
Ashley Montanaro ◽  
Richard Jozsa ◽  
Graeme Mitchison
Keyword(s):  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Quantum Query

scholarly journals Quantum Speedup Based on Classical Decision Trees

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 241
Author(s):  
Salman Beigi ◽  
Leila Taghavi
Keyword(s):  
Decision Tree ◽  
Main Tool ◽  
Upper Bounds ◽  
Query Complexity ◽  
Bipartite Matching ◽  
Quantum Query Complexity ◽  
Maximum Bipartite Matching ◽  
Query Algorithm ◽  
Binary Functions ◽  
Quantum Query

Lin and Lin \cite{LL16} have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function f:{0,1}n→[m] whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for f which makes at most O(GT) quantum queries where T is the depth of the decision tree and G is the maximum number of mistakes of the guessing algorithm. In this paper we give a simple proof of and generalize this result for functions f:[ℓ]n→[m] with non-binary input as well as output alphabets. Our main tool for this generalization is non-binary span program which has recently been developed for non-binary functions, and the dual adversary bound. As applications of our main result we present several quantum query upper bounds, some of which are new. In particular, we show that topological sorting of vertices of a directed graph G can be done with O(n3/2) quantum queries in the adjacency matrix model. Also, we show that the quantum query complexity of the maximum bipartite matching is upper bounded by O(n3/4m+n) in the adjacency list model.

scholarly journals Quantum query complexity of symmetric oracle problems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 403
Author(s):  
Daniel Copeland ◽  
Jamie Pommersheim
Keyword(s):  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Random Permutation ◽  
Query Complexity ◽  
Learning Problems ◽  
Unitary Matrices ◽  
Query Algorithm ◽  
Quantum Query

We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn. More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G, we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.

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