344 research outputs found
High-Rate Quantum Low-Density Parity-Check Codes Assisted by Reliable Qubits
Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find
Catalytic quantum error correction
We develop the theory of entanglement-assisted quantum error correcting
(EAQEC) codes, a generalization of the stabilizer formalism to the setting in
which the sender and receiver have access to pre-shared entanglement.
Conventional stabilizer codes are equivalent to dual-containing symplectic
codes. In contrast, EAQEC codes do not require the dual-containing condition,
which greatly simplifies their construction. We show how any quaternary
classical code can be made into a EAQEC code. In particular, efficient modern
codes, like LDPC codes, which attain the Shannon capacity, can be made into
EAQEC codes attaining the hashing bound. In a quantum computation setting,
EAQEC codes give rise to catalytic quantum codes which maintain a region of
inherited noiseless qubits.
We also give an alternative construction of EAQEC codes by making classical
entanglement assisted codes coherent.Comment: 30 pages, 10 figures. Notation change: [[n,k;c]] instead of
[[n,k-c;c]
Quantum Coding with Entanglement
Quantum error-correcting codes will be the ultimate enabler of a future
quantum computing or quantum communication device. This theory forms the
cornerstone of practical quantum information theory. We provide several
contributions to the theory of quantum error correction--mainly to the theory
of "entanglement-assisted" quantum error correction where the sender and
receiver share entanglement in the form of entangled bits (ebits) before
quantum communication begins. Our first contribution is an algorithm for
encoding and decoding an entanglement-assisted quantum block code. We then give
several formulas that determine the optimal number of ebits for an
entanglement-assisted code. The major contribution of this thesis is the
development of the theory of entanglement-assisted quantum convolutional
coding. A convolutional code is one that has memory and acts on an incoming
stream of qubits. We explicitly show how to encode and decode a stream of
information qubits with the help of ancilla qubits and ebits. Our
entanglement-assisted convolutional codes include those with a
Calderbank-Shor-Steane structure and those with a more general structure. We
then formulate convolutional protocols that correct errors in noisy
entanglement. Our final contribution is a unification of the theory of quantum
error correction--these unified convolutional codes exploit all of the known
resources for quantum redundancy.Comment: Ph.D. Thesis, University of Southern California, 2008, 193 pages, 2
tables, 12 figures, 9 limericks; Available at
http://digitallibrary.usc.edu/search/controller/view/usctheses-m1491.htm
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
Communicating over adversarial quantum channels using quantum list codes
We study quantum communication in the presence of adversarial noise. In this
setting, communicating with perfect fidelity requires using a quantum code of
bounded minimum distance, for which the best known rates are given by the
quantum Gilbert-Varshamov (QGV) bound. By asking only for arbitrarily high
fidelity and allowing the sender and reciever to use a secret key with length
logarithmic in the number of qubits sent, we achieve a dramatic improvement
over the QGV rates. In fact, we find protocols that achieve arbitrarily high
fidelity at noise levels for which perfect fidelity is impossible. To achieve
such communication rates, we introduce fully quantum list codes, which may be
of independent interest.Comment: 6 pages. Discussion expanded and more details provided in proofs. Far
less unclear than previous versio
Entanglement-assisted Coding Theory
In this dissertation, I present a general method for studying quantum error
correction codes (QECCs). This method not only provides us an intuitive way of
understanding QECCs, but also leads to several extensions of standard QECCs,
including the operator quantum error correction (OQECC), the
entanglement-assisted quantum error correction (EAQECC). Furthermore, we can
combine both OQECC and EAQECC into a unified formalism, the
entanglement-assisted operator formalism. This provides great flexibility of
designing QECCs for different applications. Finally, I show that the
performance of quantum low-density parity-check codes will be largely improved
using entanglement-assisted formalism.Comment: PhD dissertation, 102 page
Quantum Computing with Very Noisy Devices
In theory, quantum computers can efficiently simulate quantum physics, factor
large numbers and estimate integrals, thus solving otherwise intractable
computational problems. In practice, quantum computers must operate with noisy
devices called ``gates'' that tend to destroy the fragile quantum states needed
for computation. The goal of fault-tolerant quantum computing is to compute
accurately even when gates have a high probability of error each time they are
used. Here we give evidence that accurate quantum computing is possible with
error probabilities above 3% per gate, which is significantly higher than what
was previously thought possible. However, the resources required for computing
at such high error probabilities are excessive. Fortunately, they decrease
rapidly with decreasing error probabilities. If we had quantum resources
comparable to the considerable resources available in today's digital
computers, we could implement non-trivial quantum computations at error
probabilities as high as 1% per gate.Comment: 47 page
Entanglement-assisted quantum convolutional coding
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits. © 2010 The American Physical Society
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