Related papers: Hopf algebra description of quantum circuits
An overview and synthesis of results and criteria for open-loop controllability of Hamiltonian quantum systems obtained using Lie group and Lie algebra techniques is presented. Negative results for open-loop controllability of dissipative…
In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld-Jimbo quantum enveloping algebra $U_q(\mathfrak{g})$ for an arbitrary semi-simple Lie algebra $\mathfrak{g}$ with coefficients in a modular…
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear…
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum…
We define an algebra $\mathcal{U}_0$ using a simplified set of generators for the quantum toroidal algebra $U_q(sl_{n+1}, tor)$ and show that there exists an epimorphism from $\mathcal{U}_0$ to $U_q(sl_{n+1}, tor)$. We derive a closed…
Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a…
We give a pedagogical introduction to integration techniques appropriate for non-commutative spaces while presenting some new results as well. A rather detailed discussion outlines the motivation for adopting the Hopf algebra language. We…
An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths…
Following a suggestion of A. Connes (see [Co] {\S} I.1), we build up a (first) simple natural structure of a no finitely generated braided non-commutative Hopf algebra, suggested by elementary quantum mechanics.
We exhibit some simple gadgets useful in designing shallow parallel circuits for quantum algorithms. We prove that any quantum circuit composed entirely of controlled-not gates or of diagonal gates can be parallelized to logarithmic depth,…
In the qubit semantics the \emph{meaning} of any sentence $\alpha$ is represented by a \emph{quregister}: a unit vector of the $n$--fold tensor product $\otimes^n \C^2$, where $n$ depends on the number of occurrences of atomic sentences in…
Linear-Optical Passive (LOP) devices and photon counters are sufficient to implement universal quantum computation with single photons, and particular schemes have already been proposed. In this paper we discuss the link between the…
Super-Hopf algebra structure on the function algebra on the extended quantum symplectic superspace ${\rm SP}_q^{2|1}$ has been defined. The dual Hopf algebra is explicitly constructed.
A universal quantum computing scheme, with a universal set of logical gates, is proposed based on networks of 1D quantum systems. The encoding of information is in terms of universal features of gapped phases, for which effective field…
The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.
The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible transformation in a connected matrix group;…
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.
We first quantize the Witt algebra in characteristic 0. Then, we consider the reduction modulo p of our formulas. This gives polynomial deformations of the restricted envelopping algebra of the Witt algebra. By this way, we get new families…
We introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a $\lambda$-Hopf algebra. Moreover, a method to…
Existing abstract models of quantum computation make reference to circuit elements, much in contrast to their classical counterparts. Circuits, as a model of computation, substantially limit algorithmic expression and obscure high-level…