Related papers: Kummer surfaces for primality testing
The 5-qubit quantum computer prototypes that IBM has given open access to on the cloud allow the implementation of real experiments on a quantum processor. We present the results obtained in five experimental tests performed on these…
The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2].
We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time…
Based on an idea in Hironaka's proof of resolution of singularities, we present an algorithmic smoothness test for algebraic varieties. The test is inherently parallel and does not involve the calculation of codimension-sized minors of the…
A number of many-body problems can be formulated using Hamiltonians that are quadratic in the creation and annihilation operators. Here, we show how such quadratic Hamiltonians can be efficiently estimated indirectly, employing very few…
We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present…
We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares…
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and…
We study effective bounds for Brauer groups of Kummer surfaces associated to the Jacobians of curves of genus $2$ defined over number fields.
We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole…
We find formulas for the birational maps from a Kummer surface K and its dual K^* to their common minimal desingularization S. We show how the nodes of K blow up. Then we give a description of the group of linear automorphisms of S.
Feynman's prescription for a quantum simulator was to find a hamitonian for a system that could serve as a computer. P\'olya and Hilbert conjecture was to demonstrate Riemann's hypothesis through the spectral decomposition of hermitian…
For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function $N_V(B) = #{P\in V(k) \mid H(P)\leq B}$. In this paper, we calculate the counting function for Kummer…
We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one…
The study of graph drawings on 2-surfaces is an active area of mathematical research. Our main results are criteria for integer and modulo 2 embeddability of graphs to surfaces.
We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm…
Fundamental questions in chemistry and physics may never be answered due to the exponential complexity of the underlying quantum phenomena. A desire to overcome this challenge has sparked a new industry of quantum technologies with the…
Reliable probabilistic primality tests are fundamental in public-key cryptography. In adversarial scenarios, a composite with a high probability of passing a specific primality test could be chosen. In such cases, we need worst-case error…
This paper is about models for a vector of probabilities whose elements must have a multiplicative structure and sum to 1 at the same time; in certain applications, as basket analysis, these models may be seen as a constrained version of…
The quantum permutation algorithm provides computational speed-up over classical algorithms in determining the parity of a given cyclic permutation. For its $n$-qubit implementations, the number of required quantum gates scales…