90 research outputs found
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
Conformal variations and quantum fluctuations in discrete gravity
After an overview of variational principles for discrete gravity, and on the
basis of the approach to conformal transformations in a simplicial PL setting
proposed by Luo and Glickenstein, we present at a heuristic level an improved
scheme for addressing the gravitational (Euclidean) path integral and
geometrodynamics.Comment: 11 pages, 3 figure
Projective Ponzano-Regge spin networks and their symmetries
We present a novel hierarchical construction of projective spin networks of
the Ponzano-Regge type from an assembling of five quadrangles up to the
combinatorial 4-simplex compatible with a geometrical realization in Euclidean
4-space. The key ingrendients are the projective Desargues configuration and
the incidence structure given by its space-dual, on the one hand, and the
Biedenharn--Elliott identity for the 6j symbol of SU(2), on the other. The
interplay between projective-combinatorial and algebraic features relies on the
recoupling theory of angular momenta, an approach to discrete quantum gravity
models carried out successfully over the last few decades. The role of Regge
symmetry --an intriguing discrete symmetry of the which goes beyond the
standard tetrahedral symmetry of this symbol-- will be also discussed in brief
to highlight its role in providing a natural regularization of projective spin
networks that somehow mimics the standard regularization through a
q-deformation of SU(2).Comment: 14 pages, 19 figure
Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials
The action of the quantum mechanical volume operator, introduced in
connection with a symmetric representation of the three-body problem and
recently recognized to play a fundamental role in discretized quantum gravity
models, can be given as a second order difference equation which, by a complex
phase change, we turn into a discrete Schr\"odinger-like equation. The
introduction of discrete potential-like functions reveals the surprising
crucial role here of hidden symmetries, first discovered by Regge for the
quantum mechanical 6j symbols; insight is provided into the underlying
geometric features. The spectrum and wavefunctions of the volume operator are
discussed from the viewpoint of the Hamiltonian evolution of an elementary
"quantum of space", and a transparent asymptotic picture emerges of the
semiclassical and classical regimes. The definition of coordinates adapted to
Regge symmetry is exploited for the construction of a novel set of discrete
orthogonal polynomials, characterizing the oscillatory components of
torsion-like modes.Comment: 13 pages, 5 figure
Quantum Tetrahedra
We discuss in details the role of Wigner 6j symbol as the basic building
block unifying such different fields as state sum models for quantum geometry,
topological quantum field theory, statistical lattice models and quantum
computing. The apparent twofold nature of the 6j symbol displayed in quantum
field theory and quantum computing -a quantum tetrahedron and a computational
gate- is shown to merge together in a unified quantum-computational SU(2)-state
sum framework
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
Invariants of spin networks with boundary in Quantum Gravity and TQFT's
The search for classical or quantum combinatorial invariants of compact
n-dimensional manifolds (n=3,4) plays a key role both in topological field
theories and in lattice quantum gravity. We present here a generalization of
the partition function proposed by Ponzano and Regge to the case of a compact
3-dimensional simplicial pair . The resulting state sum
contains both Racah-Wigner 6j symbols associated with
tetrahedra and Wigner 3jm symbols associated with triangular faces lying in
. The analysis of the algebraic identities associated with the
combinatorial transformations involved in the proof of the topological
invariance makes it manifest a common structure underlying the 3-dimensional
models with empty and non empty boundaries respectively. The techniques
developed in the 3-dimensional case can be further extended in order to deal
with combinatorial models in n=2,4 and possibly to establish a hierarchy among
such models. As an example we derive here a 2-dimensional closed state sum
model including suitable sums of products of double 3jm symbols, each one of
them being associated with a triangle in the surface.Comment: 9 page
- …