586,925 research outputs found
Topological Qubit Design and Leakage
We examine how best to design qubits for use in topological quantum
computation. These qubits are topological Hilbert spaces associated with small
groups of anyons. Op- erations are performed on these by exchanging the anyons.
One might argue that, in order to have as many simple single qubit operations
as possible, the number of anyons per group should be maximized. However, we
show that there is a maximal number of particles per qubit, namely 4, and more
generally a maximal number of particles for qudits of dimension d. We also look
at the possibility of having topological qubits for which one can perform
two-qubit gates without leakage into non-computational states. It turns out
that the requirement that all two-qubit gates are leakage free is very
restrictive and this property can only be realized for two-qubit systems
related to Ising-like anyon models, which do not allow for universal quantum
computation by braiding. Our results follow directly from the representation
theory of braid groups which means they are valid for all anyon models. We also
make some remarks on generalizations to other exchange groups.Comment: 13 pages, 3 figure
Lefschetz Contact Manifolds and Odd Dimensional Symplectic Geometry
In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology groups of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Cappelletti-Montano, De Nicola, and Yudin, holds for the De Rham cohomology groups of a compact Sasakian manifold. In the current paper, using the formalism of odd dimensional symplectic geometry, we prove a Hard Lefschetz theorem for compact K-contact manifolds, which implies immediately that the two existing versions of Hard Lefschetz theorems are mathematically equivalent to each other.
Our method sheds new light on the Hard Lefschetz property of a Sasakian manifold. It enables us to give a simple construction of simply-connected K-contact manifolds without any Sasakian structures in any dimension ≥ 9, and answer an open question asked by Boyer and late Galicki concerning the existence of such examples
Profinite isomorphisms and fixed-point properties
We describe a flexible construction that produces triples of finitely
generated, residually finite groups , where the maps induce isomorphisms of profinite completions
, but and have
Serre's property FA while does not. In this construction, is finitely
presented and is of type . More generally, given any
positive integer , one can demand that and have a fixed point
whenever they act by semisimple isometries on a complete CAT space of
dimension at most , while acts without a fixed point on a tree.Comment: 10 pages, no figure
The 4-string Braid group has property RD and exponential mesoscopic rank
We prove that the braid group on 4 strings, as well as its central
quotient , have the property RD of Haagerup-Jolissaint. It follows
that the automorphism group \Aut(F_2) of the free group on 2 generators
has property RD. We also prove that the braid group is a group of
intermediate rank (of dimension 3). Namely, we show that both and its
central quotient have exponential mesoscopic rank, i.e., that they contain
exponentially many large flat balls which are not included in flats.Comment: reference added, minor correction
Infinite groups with fixed point properties
We construct finitely generated groups with strong fixed point properties.
Let be the class of Hausdorff spaces of finite covering
dimension which are mod- acyclic for at least one prime . We produce the
first examples of infinite finitely generated groups with the property that
for any action of on any , there is a global fixed
point. Moreover, may be chosen to be simple and to have Kazhdan's property
(T). We construct a finitely presented infinite group that admits no
non-trivial action by diffeomorphisms on any smooth manifold in
. In building , we exhibit new families of hyperbolic
groups: for each and each prime , we construct a non-elementary
hyperbolic group which has a generating set of size , any proper
subset of which generates a finite -group.Comment: Version 2: 29 pages. This is the final published version of the
articl
- …