Related papers: A Note on Quantum Hamming Bound
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this…
We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound…
We show that within any quantum stabilizer code there lurks a classical binary linear code with similar error-correcting capabilities, thereby demonstrating new connections between quantum codes and classical codes. Using this result --…
A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate…
In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a…
I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode $k=n-j-2$ qubits in $n=2^j$ qubits and correct $t=1$…
This paper examines linear binary codes capable of correcting one or more errors. For the single-error-correcting case, it is shown that the Hamming bound is achieved by a constructive method, and an exact expression for the minimal…
Quantum error correction requires the use of error syndromes derived from measurements that may be unreliable. Recently, quantum data-syndrome (QDS) codes have been proposed as a possible approach to protect against both data and syndrome…
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed…
The parameters of a nondegenerate quantum code must obey the Hamming bound. An important open problem in quantum coding theory is whether or not the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum…
In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of…
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.…
Good quantum codes, such as quantum MDS codes, are typically nondegenerate, meaning that errors of small weight require active error-correction, which is--paradoxically--itself prone to errors. Decoherence free subspaces, on the other hand,…
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…
The high overhead of fault-tolerant measurement sequences (FTMSs) poses a major challenge for implementing quantum stabilizer codes. Here, we address this problem by constructing efficient FTMSs for the class of quantum Hamming codes…
One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers.…
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been…
Linear programming approaches have been applied to derive upper bounds on the size of classical codes and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance, including nonadditive…
Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely…
Quantum convolutional code was introduced recently as an alternative way to protect vital quantum information. To complete the analysis of quantum convolutional code, I report a way to decode certain quantum convolutional codes based on the…