64 research outputs found
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
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 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
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
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation
The Turaev-Viro invariants are scalar topological invariants of compact,
orientable 3-manifolds. We give a quantum algorithm for additively
approximating Turaev-Viro invariants of a manifold presented by a Heegaard
splitting. The algorithm is motivated by the relationship between topological
quantum computers and (2+1)-D topological quantum field theories. Its accuracy
is shown to be nontrivial, as the same algorithm, after efficient classical
preprocessing, can solve any problem efficiently decidable by a quantum
computer. Thus approximating certain Turaev-Viro invariants of manifolds
presented by Heegaard splittings is a universal problem for quantum
computation. This establishes a novel relation between the task of
distinguishing non-homeomorphic 3-manifolds and the power of a general quantum
computer.Comment: 4 pages, 3 figure
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
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
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