• Corpus ID: 1409556

Efficient Quantum Algorithms for Shifted Quadratic Character Problems

@article{Dam2000EfficientQA,
  title={Efficient Quantum Algorithms for Shifted Quadratic Character Problems},
  author={Wim van Dam and Sean Hallgren},
  journal={ArXiv},
  year={2000},
  volume={quant-ph/0011067}
}
We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown despite the various results on the irregularity of Legendre Sequences. 
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46 Citations
Background Citations
15
Methods Citations
9

46 Citations

Citation Type

Citation Type

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Quantum Algorithms for Weighing Matrices and Quadratic Residues
  • W. V. Dam
  • Mathematics, Computer Science
    Algorithmica
  • 2002
TLDR
It is shown how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one, and designed a query problem that uses the Legendre symbol χ.
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The quantum Fourier transform (QFT) over Galois rings is studied and it is proved it can be efficiently preformed on a quantum computer and lead to the quantum algorithm for hidden linear structures overGalois rings.
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TLDR
The results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.
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This paper describes mathematically Deutsch Jozsa algorithm in detail for 2 and 3 qubits, and calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.
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References

SHOWING 1-10 OF 22 REFERENCES
Quantum measurements and the Abelian Stabilizer Problem
  • A. Kitaev
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 1996
TLDR
A polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm is presented, based on a procedure for measuring an eigenvalue of a unitary operator.
Normal subgroup reconstruction and quantum computation using group representations
TLDR
It is shown that an immediate generalization of the Abelian case solution to the non-Abelian case does not efficiently solve Graph Isomorphism.
On Quantum Algorithms for Noncommutative Hidden Subgroups
TLDR
A quantum algorithm is presented for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function.
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.
Quantum algorithms revisited
Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum…
The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer
TLDR
It is pointed out how certain instances of Abelian hidden subgroup problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits.
Quantum Algorithms for Weighing Matrices and Quadratic Residues
  • W. V. Dam
  • Mathematics, Computer Science
    Algorithmica
  • 2002
TLDR
It is shown how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one, and designed a query problem that uses the Legendre symbol χ.
Quantum Cryptanalysis of Hidden Linear Functions (Extended Abstract)
TLDR
It is shown that any cryptosystem based on what is referred to as a ‘hidden linear form’ can be broken in quantum polynomial time and the notion of ‘junk bits’ is introduced which are helpful when performing classical computations that are not injective.
Quantum Fourier transforms for extracting hidden linear structures in finite fields
TLDR
A problem that requires Ω(2) queries in the classical case, but can be solved exactly with a single query in the quantum case using a polynomial number of auxiliary operations is defined.
An improved quantum Fourier transform algorithm and applications
  • Lisa Hales, Sean Hallgren
  • Mathematics, Computer Science
    Proceedings 41st Annual Symposium on Foundations of Computer Science
  • 2000
We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary…
...
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