Globally controlled universal quantum computation with arbitrary subsystem dimension

We introduce a scheme to perform universal quantum computation in quantum cellular automata (QCA) fashion in arbitrary subsystem dimension (not necessarily finite). The scheme is developed over a one spatial dimension $N$-element array, requiring only mirror symmetric logical encoding and global pulses. A mechanism using ancillary degrees of freedom for subsystem specific measurement is also presented.Comment: 7 pages, 1 figur

Minimum orbit dimension for local unitary action on n-qubit pure states

The group of local unitary transformations partitions the space of n-qubit quantum states into orbits, each of which is a differentiable manifold of some dimension. We prove that all orbits of the n-qubit quantum state space have dimension greater than or equal to 3n/2 for n even and greater than or equal to (3n + 1)/2 for n odd. This lower bound on orbit dimension is sharp, since n-qubit states composed of products of singlets achieve these lowest orbit dimensions.Comment: 19 page

Yang-Mills theory for bundle gerbes

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang-Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.Comment: Latex2e, 7 pages, some typos corrected, to appear in J. Phys. A: Math. and Ge

Cavity QED and Quantum Computation in the Weak Coupling Regime

In this paper we consider a model of quantum computation based on n atoms of laser-cooled and trapped linearly in a cavity and realize it as the n atoms Tavis-Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schr{\" o}dinger equation of the model in the case of n=2 and construct the controlled NOT gate by making use of a resonance condition and rotating wave approximation associated to it. Our method is not heuristic but completely mathematical, and the significant feature is a consistent use of Rabi oscillations. We also present an idea of the construction of three controlled NOT gates in the case of n=3 which gives the controlled-controlled NOT gate.Comment: Latex file, 22 pages, revised version. To appear in Journal of Optics B : Quantum and Semiclassical Optic

Quantum circuits with uniformly controlled one-qubit gates

Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any n-qubit gate of this type using at most 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit gates and a single diagonal n-qubit gate. The circuit is based on the so-called quantum multiplexor, for which we provide a modified construction. We illustrate the versatility of these gates by applying them to the decomposition of a general n-qubit gate and a local state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.Comment: 8 pages, 10 figures. v2 has simpler notation and sharpens some result

BRST, anti-BRST and their geometry

We continue the comparison between the field theoretical and geometrical approaches to the gauge field theories of various types, by deriving their Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST trasformation properties and comparing them with the geometrical properties of the bundles and gerbes. In particular, we provide the geometrical interpretation of the so--called Curci-Ferrari conditions that are invoked for the absolute anticommutativity of the BRST and anti-BRST symmetry transformations in the context of non-Abelian 1-form gauge theories as well as Abelian gauge theory that incorporates a 2-form gauge field. We also carry out the explicit construction of the 3-form gauge fields and compare it with the geometry of 2--gerbes.Comment: A comment added. To appear in Jour. Phys. A: Mathemaical and Theoretica

Modular Invariance and Characteristic Numbers

We show that a general miraculous cancellation formula, the divisibility of certain characteristic numbers and some other topologiclal results are con- sequences of the modular invariance of elliptic operators on loop spaces. Previously we have shown that modular invariance also implies the rigidity of many elliptic operators on loop spaces.Comment: 14 page

Crystal Graphs and qq-Analogues of Weight Multiplicities for the Root System AnA_n

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal graph attached to the corresponding U_q(\sl_{n+1})-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to appear in Lett. Math. Phy

Cohomological aspects on complex and symplectic manifolds

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non K\"ahler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the $\partial\overline\partial$-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to [6] and [29].Comment: The present paper is a proceeding written on the occasion of the "INdAM Meeting Complex and Symplectic Geometry" held in Cortona. It is going to be published on the "Springer INdAM Series

The diagonalization method in quantum recursion theory

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.Comment: 15 pages, completely rewritte