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

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

Degenerate quantum codes and the quantum Hamming bound

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 codes. In this paper we show that Calderbank-Shor-Steane (CSS) codes with alphabet $q\geq 5$ cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes which allows us to strengthen the Rains' bound that an $[[n,k,d]]_2$ code cannot correct more than \floor{(n+1)/6} errors to \floor{(n-k+1)/6}. Additionally, we also show that the general quantum codes $[[n,k,d]]_q$ with $k+d\leq {(1-2eq^{-2})n}$ cannot beat the quantum Hamming bound.Comment: Reformulated one of the results, corrected an erroneous remark and added a few more result