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arXiv:quant-ph/0602063 · Physical Review A

Topological quantum error correction with optimal encoding rate

Héctor Bombín, Miguel Angel Martin-Delgado

Published Jun 6, 2006• 41 citations
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Abstract

We prove the existence of topological quantum error correcting codes with encoding rates $k∕n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.

DOI
10.1103/physreva.73.062303

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We prove the existence of topological quantum error correcting codes with encoding rates k∕nk∕nk∕n asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.
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Topological quantum error-correcting codes (QECC) encode a variety of topological invariants in their code space. A classic structure that has not been encoded directly is that of obstruction classes of a fiber bundle, such as the Chern or Euler class. Here, we construct and analyze extensions of toric codes. We then analyze the topological structure of their errors and finally construct a novel code using these errors to encode the obstruction class to a fiber bundle. In so doing, we construct an encoding of characteristic classes such as the Chern and Pontryagin class in topological QECC. An example of the Euler class of S2S^2S2 is constructed explicitly.
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>2D>2D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension DDD. 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.
We study the resources needed to construct topological two-dimensional stabilizer codes as a way to estimate in part their efficiency, and this leads us to perform a comparative study of surface codes and color codes. This study clarifies the similarities and differences between these two types of stabilizer code. We compute the topological error-correcting rate $C\ensuremath{\mathrel{:=}}n∕{d}^{2}$ for surface codes Cs{C}_{s}Cs​ and color codes Cc{C}_{c}Cc​ in several instances. On the torus, typical values are Cs=2{C}_{s}=2Cs​=2 and Cc=3∕2{C}_{c}=3∕2Cc​=3∕2, but we find that the optimal values are Cs=1{C}_{s}=1Cs​=1 and Cc=9∕8{C}_{c}=9∕8Cc​=9∕8. For planar codes, a typical value is Cs=2{C}_{s}=2Cs​=2, while we find that the optimal values are Cs=1{C}_{s}=1Cs​=1 and Cc=3∕4{C}_{c}=3∕4Cc​=3∕4. In general, a color code encodes twice as many logical qubits as does a surface code.