29 research outputs found
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
Fidelity approach to the disordered quantum XY model
We study the random XY spin chain in a transverse field by analyzing the
susceptibility of the ground state fidelity, numerically evaluated through a
standard mapping of the model onto quasi-free fermions. It is found that the
fidelity susceptibility and its scaling properties provide useful information
about the phase diagram. In particular it is possible to determine the Ising
critical line and the Griffiths phase regions, in agreement with previous
analytical and numerical results.Comment: 4 pages, 3 figures; references adde
Bipartite quantum states and random complex networks
We introduce a mapping between graphs and pure quantum bipartite states and
show that the associated entanglement entropy conveys non-trivial information
about the structure of the graph. Our primary goal is to investigate the family
of random graphs known as complex networks. In the case of classical random
graphs we derive an analytic expression for the averaged entanglement entropy
while for general complex networks we rely on numerics. For large
number of nodes we find a scaling where both
the prefactor and the sub-leading O(1) term are a characteristic of
the different classes of complex networks. In particular, encodes
topological features of the graphs and is named network topological entropy.
Our results suggest that quantum entanglement may provide a powerful tool in
the analysis of large complex networks with non-trivial topological properties.Comment: 4 pages, 3 figure
Fidelity in topological quantum phases of matter
Quantum phase transitions that take place between two distinct topological
phases remain an unexplored area for the applicability of the fidelity
approach. Here, we apply this method to spin systems in two and three
dimensions and show that the fidelity susceptibility can be used to determine
the boundary between different topological phases particular to these models,
while at the same time offering information about the critical exponent of the
correlation length. The success of this approach relies on its independence on
local order parameters or breaking symmetry mechanisms, with which
non-topological phases are usually characterized. We also consider a
topological insulator/superconducting phase transition in three dimensions and
point out the relevant features of fidelity susceptibility at the boundary
between these phases.Comment: 7 pages, 7 figures; added references; to appear on PR