75 research outputs found
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
Quantum states of elementary three-geometry
We introduce a quantum volume operator in three--dimensional Quantum
Gravity by taking into account a symmetrical coupling scheme of three SU(2)
angular momenta. The spectrum of is discrete and defines a complete set of
eigenvectors which is alternative with respect to the complete sets employed
when the usual binary coupling schemes of angular momenta are considered. Each
of these states, that we call quantum bubbles, represents an interference of
extended configurations which provides a rigorous meaning to the heuristic
notion of quantum tetrahedron. We study the generalized recoupling coefficients
connecting the symmetrical and the binary basis vectors, and provide an
explicit recursive solution for such coefficients by analyzing also its
asymptotic limit.Comment: 15 pages, LaTe
Post Quantum Cryptography from Mutant Prime Knots
By resorting to basic features of topological knot theory we propose a
(classical) cryptographic protocol based on the `difficulty' of decomposing
complex knots generated as connected sums of prime knots and their mutants. The
scheme combines an asymmetric public key protocol with symmetric private ones
and is intrinsecally secure against quantum eavesdropper attacks.Comment: 14 pages, 5 figure
Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
Loop Quantum Gravity defines the quantum states of space geometry as spin
networks and describes their evolution in time. We reformulate spin networks in
terms of harmonic oscillators and show how the holographic degrees of freedom
of the theory are described as matrix models. This allow us to make a link with
non-commutative geometry and to look at the issue of the semi-classical limit
of LQG from a new perspective. This work is thought as part of a bigger project
of describing quantum geometry in quantum information terms.Comment: 16 pages, revtex, 3 figure
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
Global constants in (2+1)--dimensional gravity
The extended conformal algebra (so)(2,3) of global, quantum, constants of
motion in 2+1 dimensional gravity with topology R x T^2 and negative
cosmological constant is reviewed. It is shown that the 10 global constants
form a complete set by expressing them in terms of two commuting spinors and
the Dirac gamma matrices. The spinor components are the globally constant
holonomy parameters, and their respective spinor norms are their quantum
commutators.Comment: 14 pages, to appear in Classical and Quantum Gravity, Spacetime
Safari: Essays in Honor of Vincent Moncrief on the Classical Physics of
Strong Gravitational Field
The modular geometry of Random Regge Triangulations
We show that the introduction of triangulations with variable connectivity
and fluctuating egde-lengths (Random Regge Triangulations) allows for a
relatively simple and direct analyisis of the modular properties of 2
dimensional simplicial quantum gravity. In particular, we discuss in detail an
explicit bijection between the space of possible random Regge triangulations
(of given genus g and with N vertices) and a suitable decorated version of the
(compactified) moduli space of genus g Riemann surfaces with N punctures. Such
an analysis allows us to associate a Weil-Petersson metric with the set of
random Regge triangulations and prove that the corresponding volume provides
the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio
Entanglement and Density Matrix of a Block of Spins in AKLT Model
We study a 1-dimensional AKLT spin chain, consisting of spins in the bulk
and at both ends. The unique ground state of this AKLT model is described
by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a
contiguous block of bulk spins in this ground state. It is shown that the
density matrix is a projector onto a subspace of dimension . This
subspace is described by non-zero eigenvalues and corresponding eigenvectors of
the density matrix. We prove that for large block the von Neumann entropy
coincides with Renyi entropy and is equal to .Comment: Revised version, typos corrected, references added, 31 page
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