Sometimes you write a paper and think it’s all ready for submission and then after you submit it to the archive you find that it is lacking for quite a few reasons. On Friday I posted the paper quant-ph/0506023 (and did the new paper dance!) But after communications from Michael Nielsen and David Poulin, I realized that I had made a mistake in one of my claims (the proof I had did not work) and that I had very much misrepresented what is new in this paper (in particular in relationship to quant-ph/0504189 and quant-ph/0412076.) Luckily the mistake in my proof was not a big deal for the paper and also luckily one can correct one’s foolishness and clarify what’s new and interesting in the paper. Here is the updated title and abstract:
Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
Authors: Dave Bacon
Comments: 17 pages, 3 figures, title change, rewrite of connection to operator quantum error correction, references added
The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error correcting codes, one requires the ability to apply a procedure which exactly reverses on the error correcting subspace any correctable error. In contrast, for operator error correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform correction only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error correction procedures.