# 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} }

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…

## 12 Citations

Quantum computation of zeta functions of curves

- Mathematics, Computer Sciencecomputational complexity
- 2006

An algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from enough of its cyclic resultants to effectivize a result of Fried in a restricted setting.

On zeros of exponential polynomials and quantum algorithms

- MathematicsQuantum Inf. Process.
- 2010

The zeros of an exponential polynomial of some variables are calculated by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski and the ratio of the exponent in the time complexity is considered.

Quantum Physics, Algorithmic Information Theory and the Riemanns Hypothesis

- Mathematics
- 2017

In the present work the Riemanns hypothesis (RH) is discussed from four different perspectives. In the first case, coherent states and the Stengers approximation to Riemann-zeta function are used to…

Miscellaneous Quantum Algorithms

- Computer Science, Mathematics
- 2015

This last and short chapter shall discuss some various other quantum algorithms and methods for more number-theoretic problems, and will not emphasize on the introduction of the details quantum algorithms for number- theoretics problems, rather it shall concentrated on new ideas and new developments in quantum algorithms in general.

A Fowler-Nordheim Integrator can Track the Density of Prime Numbers

- PhysicsArXiv
- 2017

It is reported for the first time that any hypothetical prime number generator, to the authors' knowledge, has to be a special case of a dynamical system that is governed by the physics of Fowler-Nordheim quantum-tunneling and how such a dynamicals system can be implemented using a counting process that naturally arises from sequential FN tunneling and integration of electrons on a floating-gate (FG) device.

From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

- Mathematics46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

The results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.

Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis

- Mathematics
- 2005

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate…

Will a physicist prove the Riemann hypothesis?

- MathematicsReports on progress in physics. Physical Society
- 2019

The Riemann Hypothesis is formulated and some physical problems related to this hypothesis are reviewed: the Polya--Hilbert conjecture, the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function and phase transitions, random walks, billiards etc.

From non-abelian anyons to quantum computation to coin-flipping by telephone

- Computer Science
- 2005

This thesis derives a tight lower bound on the bias for a large family of quantum weak coin-flipping protocols, proving such a protocol optimal within the family.

## References

SHOWING 1-10 OF 28 REFERENCES

Zeroes of zeta functions and symmetry

- Mathematics
- 1999

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

- Computer Science
- 2003

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

- Mathematics, Computer ScienceSTOC '01
- 2001

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

- Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

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

- Computer ScienceSIAM Rev.
- 1999

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

- Computer ScienceProceedings. 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

- Mathematics, Computer Science
- 1985

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

- Computer Science
- 2003

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

- Computer Science
- 2003

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

- Mathematics
- 2003

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…