Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on `qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states--machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their `natural' computational reference model.Comment: 17 pages, 1 figure; Workshop `Natural processes and models of computation' Bologna (Italy) June 16-18 2005; to appear in Natural Computin

PyDEC: Software and Algorithms for Discretization of Exterior Calculus

This paper describes the algorithms, features and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest order finite element exterior calculus and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability, removed redundant text, emphasized the role of matrix based algorithms, shortened length of pape

Spin networks, quantum automata and link invariants

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah--Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states--quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200

Quantum computing classical physics

In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so--quantum lattice gas automata--and show how to implement them efficiently on standard quantum computers.Comment: 13 pages, plain TeX, 10 PostScript figures included with epsf.tex; for related work see http://math.ucsd.edu/~dmeyer/research.htm