Homological Error Correction: Classical and Quantum Codes

. 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.

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Homological error correction: Classical and quantum codes

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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 nonorientable 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.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 nonorientable 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.

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Topological quantum error correction with optimal encoding rate

Héctor Bombín, Miguel Angel Martin-Delgado

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.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.

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Homological Quantum Error Correction with Torsion

S. Novák

Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.

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Hypermap-Homology Quantum Codes (Ph.D. thesis)

Martin P. Leslie

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arXiv:1310.5376Oct 20, 20130 citations

Hypermap-homology quantum codes

Martin P. Leslie

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

International Journal of Quantum InformationJan 27, 201420 citations

Topological quantum codes on compact surfaces with genus g≥2

Clarice Dias de Albuquerque, Reginaldo Palazzo, E.B. Da Silva

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.

Journal of Mathematical PhysicsFeb 1, 200942 citations

D>2

, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension

D

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.

m \times m

toric code to hypermap-homology codes gives us a

[(3/2)m^2,2,m]

code as compared to the toric code which is a

[2m^2,2,m]

code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.