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In this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so-called Kirillov Hamiltonian system. Moreover, we show that if we reduce first by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the configuration space of the symplectic Hamiltonian system is a Lie group G, which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie-Poisson structure on the dual space of the Lie algebra of G. We also discuss a reconstruction process for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.

A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solutions of the Hamilton-Jacobi Equation (HJE) related to these systems. Then we show that, for a given contact system, the knowledge of what we have called a complete pseudo-isotropic solution ensures the integrability by quadratures of its equations of motion. This extends to contact manifolds a recent result obtained in the context of general symplectic and Poisson manifolds.

Understanding quantum speed-up over classical computing is fundamental for the development of efficient quantum algorithms. In this paper, we study such problem within the framework of the Quantum Query Model, which represents the probability of output $x \in \{0,1\}^n$ as a function $\pi(x)$. We present a classical simulation for output probabilities $\pi$, whose error depends on the Fourier $1$-norm of $\pi$. Such dependence implies upper-bounds for the quotient between the number of queries applied by an optimal classical algorithm and our quantum algorithm, respectively. These upper-bounds show a strong relation between Fourier $1$-norm and quantum parallelism. We show applications to query complexity., Comment: 16 pages, 1 figure, changed title, improved text, added applications

The Quantum Query Model is a framework that allows us to express most known quantum algorithms. Algorithms represented by this model consist on a set of unitary operators acting over a finite Hilbert space, and a final measurement step consisting on a set of projectors. In this work, we prove that the application of these unitary operators before the measurement step is equivalent to decomposing a unit vector into a sum of vectors and then inverting some of their relative phases. We also prove that the vectors of that sum must fulfill a list of properties and we call such vectors a Block Set. If we define the measurement step for the Block Set Formulation similarly to the Quantum Query Model, then we prove that both formulations give the same Gram matrix of output states, although the Block Set Formulation allows a much more explicit form. Therefore, the Block Set reformulation of the Quantum Query Model gives us an alternative interpretation on how quantum algorithms works. Finally, we apply our approach to the analysis and complexity of quantum exact algorithms., Comment: 30 pages, 2 figures. In this version, we changed title, corrected typos, and added a new lower-bound result (Sec. 6)

Given a family of monodromy matrices {T_u; u=0,1,...,K-1} corresponding to integrable anisotropic vertex models of A_{(n_u)-1}-type, we build up a related mixed vertex model by means of glueing the lattices on which they are defined, in such a way that integrability property is preserved. Algebraically, the glueing process is implemented through one dimensional representations of rectangular matrix algebras A(R_p,R_q), namely, the `glueing matrices' zeta_u. Here R_n indicates the Yang-Baxter operator associated to the standard Hopf algebra deformation of the simple Lie algebra A_{n-1}. We show there exists a pseudovacuum subspace with respect to which algebraic Bethe ansatz can be applied. For each pseudovacuum vector we have a set of nested Bethe ansatz equations identical to the ones corresponding to an A_{m-1} quasi-periodic model, with m equal to the minimal range of involved glueing matrices., Comment: REVTeX 28 pages. Here we complete the proof of integrability for mixed vertex models as defined in the first version

Adapting the idea of twisted tensor products to the category of finitely generated algebras, we define on its opposite, the category QLS of quantum linear spaces, a family of objects hom(B,A)^{op}, one for each pair A^{op},B^{op} there, with analogous properties to its internal Hom ones, but representing spaces of transformations whose coordinate rings hom(B,A) and the ones of their respective domains B^{op} do not commute among themselves. The mentioned non commutativity is controlled by a collection of twisting maps \tau_{A,B}. We show that the (bi)algebras end(A)=hom(A,A), under certain circumstances, are 2-cocycle twistings of the quantum semigroups end(A) in the untwisted case. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups GL_{q}(n) and their multiparametric versions GL_{q,\phi}(n)., Comment: 23 pages; Added references and minor changes (including more accurate terminology)

We study a suitable q-deformed version of the Moriya's superexchange interaction theory by means of its underlying quantum group structure. We show that the one-dimensional chain case is associated with the non-standard quantum group $GL_{pq}(2) $, evidencing the integrability structure of the system. This biparametric deformation of $GL(2,{\bf C}) $ arise as a twisting of $GL_{q}(2) $ and it match exactly the local rotation appearing in the Shekhtman's work \cite{Sh}. This allow us to express the frustration condition in terms of this twisting, also showing that effect of the Moriya's vector amounts to a twisting of the boundary condition., Comment: 12 pages