25,399 research outputs found
Coprime Factor Reduction of H-infinity Controllers
We consider the efficient solution of the coprime factorization based H infinity controller approximation problems by using frequency-weighted balancing related model reduction approaches. It is shown that for a class of frequency-weighted performance preserving coprime factor reduction as well as for a relative error coprime factor reduction method, the computation of the frequency-weighted controllability and observability grammians can be done by solving Lyapunov equations of the order of the controller. The new approach can be used in conjunction with accuracy enhancing square-root and balancing-free techniques developed for the balancing related coprime factors based model reduction
Hypergeometric L-functions in average polynomial time
We describe an algorithm for computing, for all primes , the
mod- reduction of the trace of Frobenius at of a fixed hypergeometric
motive in time quasilinear in . This combines the Beukers--Cohen--Mellit
trace formula with average polynomial time techniques of Harvey et al.Comment: 15 pages, 1 figure; v4 several exposition improvements as suggested
the referee
There is entanglement in the primes
Large series of prime numbers can be superposed on a single quantum register
and then analyzed in full parallelism. The construction of this Prime state is
efficient, as it hinges on the use of a quantum version of any efficient
primality test. We show that the Prime state turns out to be very entangled as
shown by the scaling properties of purity, Renyi entropy and von Neumann
entropy. An analytical approximation to these measures of entanglement can be
obtained from the detailed analysis of the entanglement spectrum of the Prime
state, which in turn produces new insights in the Hardy-Littlewood conjecture
for the pairwise distribution of primes. The extension of these ideas to a Twin
Prime state shows that this new state is even more entangled than the Prime
state, obeying majorization relations. We further discuss the construction of
quantum states that encompass relevant series of numbers and opens the
possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo
Remarks on Quantum Modular Exponentiation and Some Experimental Demonstrations of Shor's Algorithm
An efficient quantum modular exponentiation method is indispensible for
Shor's factoring algorithm. But we find that all descriptions presented by
Shor, Nielsen and Chuang, Markov and Saeedi, et al., are flawed. We also remark
that some experimental demonstrations of Shor's algorithm are misleading,
because they violate the necessary condition that the selected number ,
where is the number of qubits used in the first register, must satisfy , where is the large number to be factored.Comment: 12 pages,5 figures. The original version has 6 pages. It did not
point out the reason that some researchers took for granted that quantum
modlar exponentiation is in polynomial time. In the new version, we indicate
the reason and analyze some experimental demonstrations of Shor's algorithm.
Besides, the author Zhenfu Cao is added to the version for his contribution.
arXiv admin note: text overlap with arXiv:1409.735
Implementing Shor's algorithm on Josephson Charge Qubits
We investigate the physical implementation of Shor's factorization algorithm
on a Josephson charge qubit register. While we pursue a universal method to
factor a composite integer of any size, the scheme is demonstrated for the
number 21. We consider both the physical and algorithmic requirements for an
optimal implementation when only a small number of qubits is available. These
aspects of quantum computation are usually the topics of separate research
communities; we present a unifying discussion of both of these fundamental
features bridging Shor's algorithm to its physical realization using Josephson
junction qubits. In order to meet the stringent requirements set by a short
decoherence time, we accelerate the algorithm by decomposing the quantum
circuit into tailored two- and three-qubit gates and we find their physical
realizations through numerical optimization.Comment: 12 pages, submitted to Phys. Rev.
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