Physics – Quantum Physics
Scientific paper
2006-05-10
J.Math.Phys.48:052105,2007
Physics
Quantum Physics
Revtex4 file
Scientific paper
10.1063/1.2731356
We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.
Bombin H.
Martin-Delgado Miguel Angel
No associations
LandOfFree
Homological Error Correction: Classical and Quantum Codes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Homological Error Correction: Classical and Quantum Codes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Homological Error Correction: Classical and Quantum Codes will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-705976