• Corpus ID: 15799593

Quantum Computing and Zeroes of Zeta Functions

@article{Dam2004QuantumCA,
  title={Quantum Computing and Zeroes of Zeta Functions},
  author={Wim van Dam},
  journal={arXiv: Quantum Physics},
  year={2004}
}
  • W. V. Dam
  • Published 17 May 2004
  • Mathematics
  • arXiv: Quantum Physics
A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the 'spectral approach' to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. The notion of universal, efficient quantum computation is used to model the desired quantum systems. Using eigenvalue estimation, such quantum circuits would be… 
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References

SHOWING 1-10 OF 28 REFERENCES
Zeroes of zeta functions and symmetry
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence
Quantum Algorithms for Estimating Gauss Sums and Calculating Discrete Logarithms
TLDR
An efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings is presented and a reduction from the discrete logarithm problem to Gauss sum estimation gives evidence that the latter is classically a hard problem.
Quantum algorithms for solvable groups
  • J. Watrous
  • Mathematics, Computer Science
    STOC '01
  • 2001
TLDR
An important byproduct of this polynomial-time quantum algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups.
Algorithms for quantum computation: discrete logarithms and factoring
  • P. Shor
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
TLDR
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
  • P. Shor
  • Computer Science
    SIAM Rev.
  • 1999
TLDR
Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.
Complexity limitations on quantum computation
  • L. Fortnow, J. Rogers
  • Computer Science
    Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247)
  • 1998
We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum
Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p
TLDR
A deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation is presented.
Quantum Computing Discrete Logarithms with the Help of a Preprocessed State
An alternative quantum algorithm for the discrete logarithm problem is presented. The algorithm uses two quantum registers and two Fourier transforms whereas Shor's algorithm requires three registers
Quantum Computing Discrete Logarithms with the Help of a Preprocessed State
An alternative quantum algorithm for the discrete logarithm problem is presented. The algorithm uses two quantum registers and two Fourier transforms whereas Shor's algorithm requires three registers
Random matrix theory, the exceptional Lie groups and L-functions
There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is
...
...