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An integral equation method for a boundary value problem arising in unsteady water wave problems
In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result
is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem.
Keywords. Boundary integral equation method, Water waves, Laplace’
Encoding One Logical Qubit Into Six Physical Qubits
We discuss two methods to encode one qubit into six physical qubits. Each of
our two examples corrects an arbitrary single-qubit error. Our first example is
a degenerate six-qubit quantum error-correcting code. We explicitly provide the
stabilizer generators, encoding circuit, codewords, logical Pauli operators,
and logical CNOT operator for this code. We also show how to convert this code
into a non-trivial subsystem code that saturates the subsystem Singleton bound.
We then prove that a six-qubit code without entanglement assistance cannot
simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an
arbitrary single-qubit error. A corollary of this result is that the Steane
seven-qubit code is the smallest single-error correcting CSS code. Our second
example is the construction of a non-degenerate six-qubit CSS
entanglement-assisted code. This code uses one bit of entanglement (an ebit)
shared between the sender and the receiver and corrects an arbitrary
single-qubit error. The code we obtain is globally equivalent to the Steane
seven-qubit code and thus corrects an arbitrary error on the receiver's half of
the ebit as well. We prove that this code is the smallest code with a CSS
structure that uses only one ebit and corrects an arbitrary single-qubit error
on the sender's side. We discuss the advantages and disadvantages for each of
the two codes.Comment: 13 pages, 3 figures, 4 table
Minimal-memory realization of pearl-necklace encoders of general quantum convolutional codes
Quantum convolutional codes, like their classical counterparts, promise to
offer higher error correction performance than block codes of equivalent
encoding complexity, and are expected to find important applications in
reliable quantum communication where a continuous stream of qubits is
transmitted. Grassl and Roetteler devised an algorithm to encode a quantum
convolutional code with a "pearl-necklace encoder." Despite their theoretical
significance as a neat way of representing quantum convolutional codes, they
are not well-suited to practical realization. In fact, there is no
straightforward way to implement any given pearl-necklace structure. This paper
closes the gap between theoretical representation and practical implementation.
In our previous work, we presented an efficient algorithm for finding a
minimal-memory realization of a pearl-necklace encoder for
Calderbank-Shor-Steane (CSS) convolutional codes. This work extends our
previous work and presents an algorithm for turning a pearl-necklace encoder
for a general (non-CSS) quantum convolutional code into a realizable quantum
convolutional encoder. We show that a minimal-memory realization depends on the
commutativity relations between the gate strings in the pearl-necklace encoder.
We find a realization by means of a weighted graph which details the
non-commutative paths through the pearl-necklace. The weight of the longest
path in this graph is equal to the minimal amount of memory needed to implement
the encoder. The algorithm has a polynomial-time complexity in the number of
gate strings in the pearl-necklace encoder.Comment: 16 pages, 5 figures; extends paper arXiv:1004.5179v
Virtual Data in CMS Analysis
The use of virtual data for enhancing the collaboration between large groups
of scientists is explored in several ways:
- by defining ``virtual'' parameter spaces which can be searched and shared
in an organized way by a collaboration of scientists in the course of their
analysis;
- by providing a mechanism to log the provenance of results and the ability
to trace them back to the various stages in the analysis of real or simulated
data;
- by creating ``check points'' in the course of an analysis to permit
collaborators to explore their own analysis branches by refining selections,
improving the signal to background ratio, varying the estimation of parameters,
etc.;
- by facilitating the audit of an analysis and the reproduction of its
results by a different group, or in a peer review context.
We describe a prototype for the analysis of data from the CMS experiment
based on the virtual data system Chimera and the object-oriented data analysis
framework ROOT. The Chimera system is used to chain together several steps in
the analysis process including the Monte Carlo generation of data, the
simulation of detector response, the reconstruction of physics objects and
their subsequent analysis, histogramming and visualization using the ROOT
framework.Comment: Talk from the 2003 Computing in High Energy and Nuclear Physics
(CHEP03), La Jolla, Ca, USA, March 2003, 9 pages, LaTeX, 7 eps figures. PSN
TUAT010. V2 - references adde
Protecting Quantum Information with Entanglement and Noisy Optical Modes
We incorporate active and passive quantum error-correcting techniques to
protect a set of optical information modes of a continuous-variable quantum
information system. Our method uses ancilla modes, entangled modes, and gauge
modes (modes in a mixed state) to help correct errors on a set of information
modes. A linear-optical encoding circuit consisting of offline squeezers,
passive optical devices, feedforward control, conditional modulation, and
homodyne measurements performs the encoding. The result is that we extend the
entanglement-assisted operator stabilizer formalism for discrete variables to
continuous-variable quantum information processing.Comment: 7 pages, 1 figur
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