26,934 research outputs found
Algebraic Distributed Differential Space-Time Codes with Low Decoding Complexity
The differential encoding/decoding setup introduced by Kiran et al,
Oggier-Hassibi and Jing-Jafarkhani for wireless relay networks that use
codebooks consisting of unitary matrices is extended to allow codebooks
consisting of scaled unitary matrices. For such codebooks to be usable in the
Jing-Hassibi protocol for cooperative diversity, the conditions involving the
relay matrices and the codebook that need to be satisfied are identified. Using
the algebraic framework of extended Clifford algebras, a new class of
Distributed Differential Space-Time Codes satisfying these conditions for power
of two number of relays and also achieving full cooperative diversity with a
low complexity sub-optimal receiver is proposed. Simulation results indicate
that the proposed codes outperform both the cyclic codes as well as the
circulant codes. Furthermore, these codes can also be applied as Differential
Space-Time codes for non-coherent communication in classical point to point
multiple antenna systems.Comment: To appear in IEEE Transactions on Wireless Communications. 10 pages,
5 figure
Grassmannian Frames with Applications to Coding and Communication
For a given class of uniform frames of fixed redundancy we define
a Grassmannian frame as one that minimizes the maximal correlation among all frames . We first analyze
finite-dimensional Grassmannian frames. Using links to packings in Grassmannian
spaces and antipodal spherical codes we derive bounds on the minimal achievable
correlation for Grassmannian frames. These bounds yield a simple condition
under which Grassmannian frames coincide with uniform tight frames. We exploit
connections to graph theory, equiangular line sets, and coding theory in order
to derive explicit constructions of Grassmannian frames. Our findings extend
recent results on uniform tight frames. We then introduce infinite-dimensional
Grassmannian frames and analyze their connection to uniform tight frames for
frames which are generated by group-like unitary systems. We derive an example
of a Grassmannian Gabor frame by using connections to sphere packing theory.
Finally we discuss the application of Grassmannian frames to wireless
communication and to multiple description coding.Comment: Submitted in June 2002 to Appl. Comp. Harm. Ana
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
At its core a -design is a method for sampling from a set of unitaries in
a way which mimics sampling randomly from the Haar measure on the unitary
group, with applications across quantum information processing and physics. We
construct new families of quantum circuits on -qubits giving rise to
-approximate unitary -designs efficiently in
depth. These quantum circuits are based on a relaxation of technical
requirements in previous constructions. In particular, the construction of
circuits which give efficient approximate -designs by Brandao, Harrow, and
Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys.
(2016).) required choosing gates from ensembles which contained inverses for
all elements, and that the entries of the unitaries are algebraic. We reduce
these requirements, to sets that contain elements without inverses in the set,
and non-algebraic entries, which we dub partially invertible universal sets. We
then adapt this circuit construction to the framework of measurement based
quantum computation(MBQC) and give new explicit examples of -qubit graph
states with fixed assignments of measurements (graph gadgets) giving rise to
unitary -designs based on partially invertible universal sets, in a natural
way. We further show that these graph gadgets demonstrate a quantum speedup, up
to standard complexity theoretic conjectures. We provide numerical and
analytical evidence that almost any assignment of fixed measurement angles on
an -qubit cluster state give efficient -designs and demonstrate a quantum
speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in
newest version. new References added.Proofs unchanged. Results unchange
Rolling quantum dice with a superconducting qubit
One of the key challenges in quantum information is coherently manipulating
the quantum state. However, it is an outstanding question whether control can
be realized with low error. Only gates from the Clifford group -- containing
, , and Hadamard gates -- have been characterized with high
accuracy. Here, we show how the Platonic solids enable implementing and
characterizing larger gate sets. We find that all gates can be implemented with
low error. The results fundamentally imply arbitrary manipulation of the
quantum state can be realized with high precision, providing new practical
possibilities for designing efficient quantum algorithms.Comment: 8 pages, 4 figures, including supplementary materia
Cubature formulas, geometrical designs, reproducing kernels, and Markov operators
Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces and
appropriate spaces of functions inside . The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups
- …