On the universal p-adic sigma and Weierstrass zeta functions

For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.Comment: 23 page

Perfectoid covers of abelian varieties

For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger class of abeloid varieties

On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions

We generalize a paper of Mazur and Tate on p-adic sigma functions attached to elliptic curves of ordinary reduction over a p-adic field.We begin by generalizing the theory of division polynomials attached to an isogeny of elliptic curves, developed by Mazur and Tate, to isogenies of prinicipally polarized abelian varieties.As an application, we produce a notion of a p-adic sigma function attached to a prinicipally polarized abelian variety of good ordinary reduction over a complete non-archimedean field of residue characteristic p.Furthermore, we derive some the properties of the sigma function, many of which uniquely characterize the function.Independently, a notion of a pair of p-adic Weierstrass zeta functions is produced for a smooth projective curve C of genus two with invertible Hasse--Witt matrix over a p-adically complete field of characteristic zero.Using the explicit function theory afforded by Jacobians of genus two, general results about p-adic sigma functions are made more descriptive and the zeta functions on C are compared to the second logarithmic derivatives of the sigma function on the Jacobian of C

Quantum communication between trapped ions through a dissipative environment

We study two trapped ions coupled to the axial phonon modes of a one-dimensional Coulomb crystal. This system is formally equivalent to the "two spin-boson" model. We propose a scheme to dynamically generate a maximally entangled state of two ions within a decoherence-free subspace. Here the phononic environment of the trapped ions, whatever its temperature and number of modes, serves as the entangling bus. The efficient production of the pure singlet state can be exploited to perform short-ranged quantum communication which is essential in building up a large-scale quantum computer.Comment: 4 pages, 2 figure

Complete methods set for scalable ion trap quantum information processing

Large-scale quantum information processors must be able to transport and maintain quantum information, and repeatedly perform logical operations. Here we demonstrate a combination of all the fundamental elements required to perform scalable quantum computing using qubits stored in the internal states of trapped atomic ions. We quantify the repeatability of a multi-qubit operation, observing no loss of performance despite qubit transport over macroscopic distances. Key to these results is the use of different pairs of beryllium ion hyperfine states for robust qubit storage, readout and gates, and simultaneous trapping of magnesium re-cooling ions along with the qubit ions.Comment: 9 pages, 4 figures. Accepted to Science, and thus subject to a press embarg

Transport of charged particles by adjusting rf voltage amplitudes

We propose a planar architecture for scalable quantum information processing (QIP) that includes X-junctions through which particles can move without micromotion. This is achieved by adjusting radio frequency (rf) amplitudes to move an rf null along the legs of the junction. We provide a proof-of-principle by transporting dust particles in three dimensions via adjustable rf potentials in a 3D trap. For the proposed planar architecture, we use regularization techniques to obtain amplitude settings that guarantee smooth transport through the X-junction.Comment: 16 pages, 10 figure