The quantum query complexity of learning multilinear polynomials

@article{Montanaro2012TheQQ,
  title={The quantum query complexity of learning multilinear polynomials},
  author={Ashley Montanaro},
  journal={Inf. Process. Lett.},
  year={2012},
  volume={112},
  pages={438-442}
}
  • A. Montanaro
  • Published 17 May 2011
  • Computer Science
  • Inf. Process. Lett.
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11 Citations
Highly Influential Citations
1
Background Citations
5
Methods Citations
1

11 Citations

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The biased quantum Fourier transform introduced in Kanade et al. (2018) is employed to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. and proves lower bounds on the classical and quantum sample complexities of the learning problem.
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TLDR
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Identifying Generalized Reed-Muller Codewords by Quantum Queries
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Preliminaries and notations ( a ) Notation and definitions
  • 2018
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TLDR
A quantum Viterbi algorithm (QVA) with better than classical performance under certain conditions is applied to decoding classical convolutional codes, for instance, large constraint length $$Q$$Q and short decode frames $$N$$N.
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References

SHOWING 1-10 OF 16 REFERENCES
Sharp Quantum versus Classical Query Complexity Separations
TLDR
A black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations).
Bound on the number of functions that can be distinguished with k quantum queries
TLDR
A bound for how many functions can be distinguished with k quantum queries is derived.
Quantum versus classical learnability
  • R. ServedioS. Gortler
  • Computer Science
    Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
TLDR
This work considers quantum versions of two well-studied models of learning Boolean functions: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Correct (PAC) model of learning from random examples to establish a polynomial relationship between the number of quantum versus classical queries required for learning.
Quantum Algorithms to Solve the Hidden Shift Problem for Quadratics and for Functions of Large Gowers Norm
Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used…
Quantum algorithms for some hidden shift problems
TLDR
The hidden coset problem is defined, which generalizes the hidden shift problem and the hidden subgroup problem and provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
Quantum computation beyond the circuit model
The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of…
Finite fields
Bounds for the quantity of information transmitted by a quantum communication channel
Elements of Information Theory
TLDR
The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Lower Bounds on Quantum Query Complexity
TLDR
This paper discusses here what quantum computers cannot do, and specifically how to prove limits on their computational power.
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