12,924 research outputs found
The Gross-Saccoman Conjecture is True
Consider a graph with perfect nodes but independent edge failures with identical probability ρ. The reliability is the connectedness probability of the random graph. A graph with n nodes and e edges is uniformly optimally reliable (UOR) if it has the greatest reliability among all graphs with the same number of nodes and edges, for all values of ρ. In 1997, Gross and Saccoman proved that the simple UOR graphs for e = n, e = n + 1 and e = n + 2 are also optimal when the classes are extended to include multigraphs [6]. The authors conjectured that the UOR simple graphs for e = n + 3 are optimal in multigraphs as well. A proof of the Gross-Saccoman conjecture is introduced.Agencia Nacional de Investigación e Innovació
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
Fault-Tolerant, but Paradoxical Path-Finding in Physical and Conceptual Systems
We report our initial investigations into reliability and path-finding based
models and propose future areas of interest. Inspired by broken sidewalks
during on-campus construction projects, we develop two models for navigating
this "unreliable network." These are based on a concept of "accumulating risk"
backward from the destination, and both operate on directed acyclic graphs with
a probability of failure associated with each edge. The first serves to
introduce and has faults addressed by the second, more conservative model.
Next, we show a paradox when these models are used to construct polynomials on
conceptual networks, such as design processes and software development life
cycles. When the risk of a network increases uniformly, the most reliable path
changes from wider and longer to shorter and narrower. If we let professional
inexperience--such as with entry level cooks and software developers--represent
probability of edge failure, does this change in path imply that the novice
should follow instructions with fewer "back-up" plans, yet those with
alternative routes should be followed by the expert?Comment: 8 page
Adaptive multiscale detection of filamentary structures in a background of uniform random points
We are given a set of points that might be uniformly distributed in the
unit square . We wish to test whether the set, although mostly
consisting of uniformly scattered points, also contains a small fraction of
points sampled from some (a priori unknown) curve with -norm
bounded by . An asymptotic detection threshold exists in this problem;
for a constant , if the number of points sampled from the
curve is smaller than , reliable detection
is not possible for large . We describe a multiscale significant-runs
algorithm that can reliably detect concentration of data near a smooth curve,
without knowing the smoothness information or in advance,
provided that the number of points on the curve exceeds
. This algorithm therefore has an optimal
detection threshold, up to a factor . At the heart of our approach is
an analysis of the data by counting membership in multiscale multianisotropic
strips. The strips will have area and exhibit a variety of lengths,
orientations and anisotropies. The strips are partitioned into anisotropy
classes; each class is organized as a directed graph whose vertices all are
strips of the same anisotropy and whose edges link such strips to their ``good
continuations.'' The point-cloud data are reduced to counts that measure
membership in strips. Each anisotropy graph is reduced to a subgraph that
consist of strips with significant counts. The algorithm rejects
whenever some such subgraph contains a path that connects many consecutive
significant counts.Comment: Published at http://dx.doi.org/10.1214/009053605000000787 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Decentralized Erasure Codes for Distributed Networked Storage
We consider the problem of constructing an erasure code for storage over a
network when the data sources are distributed. Specifically, we assume that
there are n storage nodes with limited memory and k<n sources generating the
data. We want a data collector, who can appear anywhere in the network, to
query any k storage nodes and be able to retrieve the data. We introduce
Decentralized Erasure Codes, which are linear codes with a specific randomized
structure inspired by network coding on random bipartite graphs. We show that
decentralized erasure codes are optimally sparse, and lead to reduced
communication, storage and computation cost over random linear coding.Comment: to appear in IEEE Transactions on Information Theory, Special Issue:
Networking and Information Theor
- …