24,957 research outputs found
Quantum stabilizer codes and beyond
The importance of quantum error correction in paving the way to build a
practical quantum computer is no longer in doubt. This dissertation makes a
threefold contribution to the mathematical theory of quantum error-correcting
codes. Firstly, it extends the framework of an important class of quantum codes
-- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes
to classical codes over quadratic extension fields, provides many new
constructions of quantum codes, and develops further the theory of optimal
quantum codes and punctured quantum codes. Secondly, it contributes to the
theory of operator quantum error correcting codes also called as subsystem
codes. These codes are expected to have efficient error recovery schemes than
stabilizer codes. This dissertation develops a framework for study and analysis
of subsystem codes using character theoretic methods. In particular, this work
establishes a close link between subsystem codes and classical codes showing
that the subsystem codes can be constructed from arbitrary classical codes.
Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum
codes and considers more realistic channels than the commonly studied
depolarizing channel. It gives systematic constructions of asymmetric quantum
stabilizer codes that exploit the asymmetry of errors in certain quantum
channels.Comment: Ph.D. Dissertation, Texas A&M University, 200
Quantum error control codes
It is conjectured that quantum computers are able to solve certain problems more
quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm
is able to factor large integers in polynomial time on a quantum computer.
A quantum computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the
quantum mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in addition to
families of subsystem codes. This dissertation is organized into three parts:
Quantum Block Codes. After introducing the theory of quantum block codes, I
establish conditions when BCH codes are self-orthogonal (or dual-containing)
with respect to Euclidean and Hermitian inner products. In particular, I derive
two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum
codes, as well as families of quantum codes derived from projective geometries.
Subsystem Codes. Subsystem codes form a new class of quantum codes in which
the underlying classical codes do not need to be self-orthogonal. I give an
introduction to subsystem codes and present several methods for subsystem
code constructions. I derive families of subsystem codes from classical BCH and
RS codes and establish a family of optimal MDS subsystem codes. I establish
propagation rules of subsystem codes and construct tables of upper and lower
bounds on subsystem code parameters.
Quantum Convolutional Codes. Quantum convolutional codes are particularly
well-suited for communication applications. I develop the theory of quantum
convolutional codes and give families of quantum convolutional codes based
on RS codes. Furthermore, I establish a bound on the code parameters of
quantum convolutional codes - the generalized Singleton bound. I develop a
general framework for deriving convolutional codes from block codes and use it
to derive families of non-catastrophic quantum convolutional codes from BCH
codes.
The dissertation concludes with a discussion of some open problems
On Subsystem Codes Beating the Hamming or Singleton Bound
Subsystem codes are a generalization of noiseless subsystems, decoherence
free subspaces, and quantum error-correcting codes. We prove a Singleton bound
for GF(q)-linear subsystem codes. It follows that no subsystem code over a
prime field can beat the Singleton bound. On the other hand, we show the
remarkable fact that there exist impure subsystem codes beating the Hamming
bound. A number of open problems concern the comparison in performance of
stabilizer and subsystem codes. One of the open problems suggested by Poulin's
work asks whether a subsystem code can use fewer syndrome measurements than an
optimal MDS stabilizer code while encoding the same number of qudits and having
the same distance. We prove that linear subsystem codes cannot offer such an
improvement under complete decoding.Comment: 18 pages more densely packed than classically possibl
Automated searching for quantum subsystem codes
Quantum error correction allows for faulty quantum systems to behave in an
effectively error free manner. One important class of techniques for quantum
error correction is the class of quantum subsystem codes, which are relevant
both to active quantum error correcting schemes as well as to the design of
self-correcting quantum memories. Previous approaches for investigating these
codes have focused on applying theoretical analysis to look for interesting
codes and to investigate their properties. In this paper we present an
alternative approach that uses computational analysis to accomplish the same
goals. Specifically, we present an algorithm that computes the optimal quantum
subsystem code that can be implemented given an arbitrary set of measurement
operators that are tensor products of Pauli operators. We then demonstrate the
utility of this algorithm by performing a systematic investigation of the
quantum subsystem codes that exist in the setting where the interactions are
limited to 2-body interactions between neighbors on lattices derived from the
convex uniform tilings of the plane.Comment: 38 pages, 15 figure, 10 tables. The algorithm described in this paper
is available as both library and a command line program (including full
source code) that can be downloaded from
http://github.com/gcross/CodeQuest/downloads. The source code used to apply
the algorithm to scan the lattices is available upon request. Please feel
free to contact the authors with question
A statistical mechanical interpretation of instantaneous codes
In this paper we develop a statistical mechanical interpretation of the
noiseless source coding scheme based on an absolutely optimal instantaneous
code. The notions in statistical mechanics such as statistical mechanical
entropy, temperature, and thermal equilibrium are translated into the context
of noiseless source coding. Especially, it is discovered that the temperature 1
corresponds to the average codeword length of an instantaneous code in this
statistical mechanical interpretation of noiseless source coding scheme. This
correspondence is also verified by the investigation using box-counting
dimension. Using the notion of temperature and statistical mechanical
arguments, some information-theoretic relations can be derived in the manner
which appeals to intuition.Comment: 5 pages, Proceedings of the 2007 IEEE International Symposium on
Information Theory, pp.1906 - 1910, Nice, France, June 24 - 29, 200
Subsystem Pseudo-pure States
A critical step in experimental quantum information processing (QIP) is to
implement control of quantum systems protected against decoherence via
informational encodings, such as quantum error correcting codes, noiseless
subsystems and decoherence free subspaces. These encodings lead to the promise
of fault tolerant QIP, but they come at the expense of resource overheads.
Part of the challenge in studying control over multiple logical qubits, is
that QIP test-beds have not had sufficient resources to analyze encodings
beyond the simplest ones. The most relevant resources are the number of
available qubits and the cost to initialize and control them. Here we
demonstrate an encoding of logical information that permits the control over
multiple logical qubits without full initialization, an issue that is
particularly challenging in liquid state NMR. The method of subsystem
pseudo-pure state will allow the study of decoherence control schemes on up to
6 logical qubits using liquid state NMR implementations.Comment: 9 pages, 1 Figur
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