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Relation between homological quantum codes of a hypermap and its dual hypermap

Zihan Lei

Published May 4, 2021• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Homological quantum codes (also called topological codes) are low density parity check error correcting codes that come from surfaces and higher dimension manifolds. Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. We will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom, i.e., systolic rigidity.
If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce {thirteen} new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type $ [k^k]$, (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of \Echar{-1}, along with {their} covering maps. The encoding rate of the class of codes associated with the maps in (i) is such that $ \frac{k}{n}\rightarrow 1 $ as $ n\rightarrow\infty $, and for the remaining classes of codes, the encoding rate is $ \frac{k}{n}\rightarrow α$ as $ n\rightarrow \infty $ with $ α< 1 $.