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arXiv:1312.6344 · Physical Review A

Relation between surface codes and hypermap-homology quantum codes

Pradeep Kiran Sarvepalli

Published May 13, 2014• 4 citations
arXivHTMLPDF

Abstract

Recently, a new class of quantum codes based on hypermaps has been proposed. These codes are obtained from embeddings of hypergraphs as opposed to surface codes which are obtained from the embeddings of graphs. It is natural to compare these two classes of codes and their relation to each other. In this context two related questions are addressed in this paper: Can the parameters of hypermap-homology codes be superior to those of surface codes, and what is precisely the relation between these two classes of quantum codes? We show that a canonical hypermap code is identical to a surface code, whereas a noncanonical hypermap code can be transformed to a surface code by cnot gates alone. Our approach is constructive; we construct the related surface code and the transformation involving cnot gates.

DOI
10.1103/physreva.89.052316

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Recently, a new class of quantum codes based on hypermaps has been proposed. These codes are obtained from embeddings of hypergraphs as opposed to surface codes which are obtained from the embeddings of graphs. It is natural to compare these two classes of codes and their relation to each other. In this context two related questions are addressed in this paper: Can the parameters of hypermap-homology codes be superior to those of surface codes, and what is precisely the relation between these two classes of quantum codes? We show that a canonical hypermap code is identical to a surface code, whereas a noncanonical hypermap code can be transformed to a surface code by cnot gates alone. Our approach is constructive; we construct the related surface code and the transformation involving cnot gates.
Topological hypermaps can be used to construct quantum stabilizer codes, although these codes are later shown to be just surface codes. In this article, we explore the quantum codes stem from the dual hypermaps and their relations with those from the original hypermaps. We find that the cell strucure of the surface codes from the dual hypermaps are just the dual cell structure of original hypermap itself with special darts erased. Then we propose a new kind of dual hypermaps called △\triangle△-duals, and also the concept of their contrary maps. We also propose an alternative route to construct quantum codes from hypermaps. With all these new constructions, we find that the homological code of the original hypermap and that of the dual hypermap can be transformed to the new quantum codes of the △\triangle△-dual and its contrary map, which makes the duality between these two codes more apparent both geometrically and algebraically.
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×mm \times mm×m toric code to hypermap-homology codes gives us a [(3/2)m2,2,m][(3/2)m^2,2,m][(3/2)m2,2,m] code as compared to the toric code which is a [2m2,2,m][2m^2,2,m][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 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 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.
Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph G embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.