26,934 research outputs found

    Algebraic Distributed Differential Space-Time Codes with Low Decoding Complexity

    Full text link
    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

    Get PDF
    For a given class F{\cal F} of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation <fk,fl>|< f_k,f_l >| among all frames {fk}kIF\{f_k\}_{k \in {\cal I}} \in {\cal F}. 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

    Full text link
    Andries Brouwer maintains a public database of existence results for strongly regular graphs on n1300n\leq 1300 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 nn.Comment: 18 pages, LaTe

    Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup

    Full text link
    At its core a tt-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 nn-qubits giving rise to ε\varepsilon-approximate unitary tt-designs efficiently in O(n3t12)O(n^3t^{12}) 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 tt-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 nn-qubit graph states with fixed assignments of measurements (graph gadgets) giving rise to unitary tt-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 nn-qubit cluster state give efficient tt-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

    Full text link
    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 π\pi, π/2\pi/2, 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

    Full text link
    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 (Ω,σ)(\Omega,\sigma) and appropriate spaces of functions inside L2(Ω,σ)L^2(\Omega,\sigma). 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
    corecore