Construction of quantum codes based on self-dual orientable embeddings of complete multipartite graphs
This paper presents four new classes of binary quantum codes with minimum distance and , namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the complete graphs and and by current graphs and rotation schemes. The parameters of two classes of quantum codes are and respectively, where and . For these quantum codes, the code rate approaches as and tend to infinity. The Class-III with rate and with minimum distance is constructed by using self-dual embeddings of complete bipartite graphs. The parameters of this class are , where and are both divisible by . The proposed Class-IV is of minimum distance and code length . This class is constructed based on self-dual embeddings of complete tripartite graph and its parameters are , where and .
Keywords: quantum codes; embedding; orientable; self-dual; complete graphs; complete bipartite graphs; complete tripartite graphs.
1 . Introduction
Quantum error-correcting codes (QECs) play an essential role in
various quantum informational processes. In the theory of quantum
computation, the information is stored in entangled states of
quantum systems. Since the interaction with the system environment
is inevitable, these interactions create noise that disrupt the
encoded data and make mistakes. One of the most useful techniques to
reduce the effects of these noise is applying the QECs. The first
quantum code was discovered by Shor . Calderbank et
al.  introduced a systematic way for constructing the QEC from
classical error-correcting code. The problem of constructing toric
quantum codes has motivated considerable interest in the literature.
This problem was generalized within the context of surface codes 
and color codes . The most popular toric code was proposed for
the first time by Kitaev’s . This code defined on a square lattic
of size on the torus. The parameters of this class of
codes are . In similar way, the authors in
 have introduced a construction of topological quantum codes in
the projective plane . They showed that the original
Shor’s -qubit repetition code is one of these codes which can be
constructed in a planar domain.
Leslie proposed a new type of sparse CSS quantum error correcting codes based on the homology of hypermaps defined on an square lattice . The parameters of hypermap-homology codes are . These codes are more efficient than Kitaev’s toric codes. This seemed suggests good quantum codes that is constructed by using hypergraphs. But there are other surface codes with better parameters than the toric code. There exist surface codes with parameters , called homological quantum codes. These codes were introduced by Bombin and Martin-Delgado .
Authors in  presented a new class of toric quantum codes with parameters , where . Sarvepalli  studied relation between surface codes and hypermap-homology quantum codes. He showed that a canonical hypermap code is identical to a surface code while a noncanonical hypermap code can be transformed to a surface code by CNOT gates alone. Li et al.  introduced a large number of good binary quantum codes of minimum distances five and six by Steane’s Construction. In  good binary quantum stabilizer codes obtained via graphs of Abelian and non-Abelian groups schemes. In , Qian presented a new method for constructing quantum codes from cyclic codes over finite ring . In  two new classes of binary quantum codes with minimum distance of at least three presented by self-complementary self-dual orientable embeddings of voltage graphs and Paley graphs. In Korzhik  studied generating nonisomorphic quadrangular embeddings of a complete graph. In  M. Ellingham gave techniques to construct graph embeddings.
Our aim in this work is to present four new classes of binary quantum codes with parameters , ,
and respectively, based on results of Pengelley , Archdeacon et al.  and Archdeacon  in self-dual orientable embeddings of the complete graphs and , ( and ), complete bipartite graphs and complete multipartite graphs by current graphs and rotation schemes . Binary quantum codes are defined by pair of -matrices with . These codes have parameters , where logical qubits are encoded into physical qubits with minimum distance . A minimum distance code can correct all errors up to qubits. The code rate for two classes of quantum codes of length and is determined by and , and this rate approaches as and tend to infinity.
The paper is organized as follows. The simplices definition , chain complexes and homology group are recalled in Section 2. In Section 3 we shall briefly present the current graphs and rotation schemes. In Section 4, we give a brief outline of self-dual orientable embeddings of the complete graph. Section 5 is devoted to present new classes of binary quantum codes by using self-dual orientable embeddings of the complete graphs and , complete bipartite graphs, and complete tripartite graph . The paper is ended with a brief conclusion.
2 . Homological algebra
In this section, we review some fundamental notions of homology
spaces. For more detailed information about homology spaces, refer
to , .
Simplices. Let , . Let moreover the set of points of be geometrically independent. An -simplex is a subset of given by
Chain complexes. Let be a simplicial
complex and a dimension. A -chain is a formal sum of
-simplices in . The standard notation for this is
, where and
is a -simplex in . Let be the set of
all -chains in . The boundary homomorphism
is defined as
The chain complex is the sequence of chain groups connected
by boundary homomorphisms,
Cycles and boundaries. We are interested in
two subgroups of , cycle and boundary
groups. The -th cycle group is the kernel of
, and denoted as
. The -th boundary group is the image of
, and denoted as
Definition 2.1 (Homology group, Betti number). The -th homology group is the -th cycle group modulo the -th boundary group, . The -th Betti number is the rank (i.e. the number of generators) of this group, =rank . So the first homology group is given as
From the algebraic topology, we can see that the group only
depends, up to isomorphisms, on the topology of the surface . In
where is the genus of the surface, i.e. the number of “holes”
or “handles”. We then have
3 . Current graphs
Current graphs were invented before voltage graphs. These graphs are used in proof of Map Color Theorem, determination of minimum genus of complete graphs. Current graphs are dual of voltage graphs which apply to embedded voltage graphs. Faces of current graph correspond to vertices in voltage graph, and vice versa. We refer the reader to  for more details.
3.1 . Rotation schemes
Rotation schemes are important for applying the face-tracing. Let
be a connected graph. Denote the vertex set of by
. For each , let . Let be an
oriented cyclic order (or a cyclic permutation) on , of
length ; ia called a rotation
scheme, or rotation system. By combining the two
concepts of rotation schemes and -cell embeddings we have the
Theorem 3.1. Every rotation scheme for a graph induces a unique embedding of into an orientable surface. Conversely, every embedding of a graph into orientable surface induces a unique rotation scheme for .
Proof. The Proof of this theorem is found in .
In proof of this theorem , and as a permutation on the set of directed edges of is defined as
The orbits under determine the (-cell)faces of the
Often it is customary to represent a graph with rotation in the plane in such a way that a clockwise (or counterclockwise) reading of the edges incident with a vertex gives the rotation at that vertex. By convention, a solid vertex has its incident edges ordered clockwise; a hollow vertex, counterclockwise. For further information about rotation schemes, the reader is referred to Refs. [16, 24].
4 . Self-dual orientable embeddings of complete graph
Let be a compact, connected, oriented surface (i.e.
-manifold) with genus . As shown by Pengelley , Euler’s
Formula excludes self-dual orientable embeddings of unless
or (mod ). Let be a self-dual embedding of
and , , denote
respectively the number of vertices, edges and faces . If the
embedding is in , then
and Euler’s Formula follows that
Hence such an embedding can exist only if or (mod
). Since the embedding consists of faces, each face must be
adjacent to every other along exactly one edge. Hence each face is
an -gon. Pengelley presented how to use current graphs and
rotation schemes to describe an orientable embedding of
having -gons as faces, and then show such an embedding is
self-dual . Since we wish our faces to be -gons, by the
current graph construction principles is requiring that each vertex
in the current graph be of valence .
In the case (mod ), we select a group for which is a Cayley color graph; in this case, we can only pick . Label the vertices with the elements of . Denote the faces by the numbers , ,,. Then choose a certain orientation for each face. Write down the cyclic order of the faces adjacent to face . This gives a certain permutation of , ,,. Do the same for the other faces. This leads to a scheme. The scheme for is generated by the following .
Explicit calculation gives the following cyclic sequence for the
vertices of the face containing the directed edge from to :
In general, by explicit calculation from scheme, we obtain, for each , the cyclic sequence
for the vertices of the face containing the directed edge ,
where is an edge between vertices and .
In the case (mod ), Pengelly  has used the following group
where , is odd, and . There are precisely
elements of order two. Pengelly in Ref.  released
elements with distinct inverses, and his choose
, , ,, from this
collection such that they and their inverses deplete the collection.
We label the elements of order two , ,
,,. From the elementary group theory, we
can see that . The scheme for
is generated by the following .
This generates by the additive rule the entire scheme for
. Explicit calculation yield the following cyclic sequence
for the vertices of the face containing the directed edge from
By calculation from scheme, we obtain, for each , the
cyclic sequence for the vertices of the face containing the
directed edge from to :
5 . Quantum codes from graphs on surfaces
The idea of constructing CSS (Calderbank-Shor-Steane) codes from
graphs embedded on surfaces has been discussed in a number of
papers. See for detailed descriptions e.g. . Let be a
compact, connected, oriented surface (i.e. 2-manifold) with genus
. A tiling of is defined to be a cellular embedding of an
undirected (simple) graph in a surface. This embedding
defines a set of faces . Each face is described by the set of
edges on its boundary. This tiling of surface is denoted
. The dual graph is the graph such
i) One vertex of inside each face of ,
ii) For each edge of there is an edge of between the two vertices of corresponding to the two faces of adjacent to .
It can be easily seen that, there is a bijection between the edges of and the edges of .
There is an interesting relationship between the number of elements of a lattice embedded in a surface and its genus. The Euler characteristic of is defined as its number of vertices () minus its number of edges () plus its number of faces (), i.e.,
For closed orientable surfaces we have
The surface code associated with a tiling is the CSS code defined by the matrices and such that is the vertex-edge incidence matrix of the tiling and is the face-edge incidence matrix of the tiling. Therefore, from is constructed a CSS code with parameters . where is the number of edges of , (by (2.6)) and is the shortest non-boundary cycle in or . In this work, the minimum distance of quantum codes by a parity check matrix (or generator matrix) is obtained. For a detailed information to compute the minimum distance, we refer the reader to .
5.1 . New class of binary quantum codes from embeddings of
Our aim in this subsection is to construct
a new class of binary quantum codes by using self-dual orientable
embeddings of complete graphs. Let be an embedding in an
orientable surface . We know that .
Also, from (4.2) with a self-dual embedding of complete graph on an
orientable surface of genus , we know that if
(mod ), then . Therefore,
. By finding the vertex-edge incidence matrix
using the relation (4.3) and rotation schemes, and the
face-edge incidence matrix by using (4.5), one can easily
see that and . Thus the code
parameters are given by: the code minimum distance is ;
the code length is and .
Consequently, the class of codes with parameters
, is obtained.
Example 5.1.1. Let (mod ). Then and . For determining by using rotation schemes, by the following Figure we have:
By rotation schemes, we get the following vertex-edge and
face-edge matrices respectively:
One can easily see that and .
Therefore, the code with parameters is obtained.
In the above figure the torus is recovered from the rectangle by identifying its left and right sides and simultaneously identifying its top and bottom sides.
Example 5.1.2. Let (mod ). Then and . For determining and by using (4.3) and rotation schemes, we have:
Also, by using (3.1), (4.4) and (4.5), we obtain the following
cyclic sequence for the vertices of the face containing the directed
edge for each
By using (5.3) and (5.4) we obtain the following vertex-edge and
face-edge matrices respectively:
One can easily see that and . Therefore, the code with parameters is obtained.
5.2 . New class of binary quantum codes from embeddings of
The construction of this class will be
based on self-dual orientable embeddings of complete graph
Let be a self-dual graph on vertices. From (4.1), we know that . Also, from (4.2) we know that . Therefore, the code length is and . With finding the matrices and using the relations (4.6), (4.7) and (4.8), one can see that and . Consequently, the class of codes with parameters , () is constructed.
5.3 . New class of binary quantum codes from embeddings of complete bipartite graphs
Let the complete
bipartite graph be an embedding in (the
orientable -manifold of genus ). The graph has
vertices divided into two subsets, one of size and the
other of size . The number of edges in a complete bipartite
graph is . From Ref.  and the following
theorem, the genus of the complete bipartite graph is
where and are both divisible by
Theorem 5.3.1. has both an orientable and a nonorientable self-dual embedding for all even integers and exceeding , except that there is no orientable self-dual embedding of .
Proof. The Proof of this theorem is found in .
In the complete bipartite graph , where and are both divisible by , we know that . Since in this self-dual embedding on an orientable surface the code minimum distance is four. Thus, the code parameters are given by: the code minimum distance ; the code length is and . Consequently, the new class of codes with parameters is obtained.
Example 5.3.1. In Figure
we give a self-dual embedding of into the torus. In this
figure the top of the rectangle is identified with the bottom and
the left with the right to recover the torus.
In this figure the primal graph is shown by solid lines, while the dual graph is drawn with dotted lines. The vertex bipartitions are vs. . Since in this figure the shortest non-boundary cycle in the primal graph or the dual graph is . So, . Also, we have , the number of edges and (by (5.3)). Hence, a code is obtained.
5.4 . New class of binary quantum codes from embeddings of complete tripartite graph
be a complete tripartite embedded on a closed
surface. The graph has vertices. The number
of edges in a complete tripartite graph is
. From Ref. , the genus of the
complete tripartite graph is given as
Theorem 5.4.1. The complete multipartite
graph , , has both an orientable and a
nonorientable self-dual embeddings except in the following cases:
(1) When the surface is orientable: is odd and or (mod ), or when the graph is , or possibly when the graph is or .
(2) When the surface is nonorientable: the graph is , where , or when the graph is .
Proof. The Proof of this theorem is found in .
In the complete tripartite graph , we know that . Since in this self-dual embedding (except the Case of the above theorem) on an orientable surface the code minimum distance is three. Thus, the code parameters are given by: the code minimum distance ; the code length is and . Consequently, the new class of codes with parameters is constructed.
6 . Conclusion
We considered presentation of four new classes of binary quantum codes based on self-dual orientable embeddings of the complete graphs and , ( and ), complete bipartite graph , and complete tripartite graph by using current graphs and rotation schemes. These codes are superior to quantum codes presented in other references. We point out the classes and of quantum codes achieve the best ratio . For new classes and of codes, the code rate approaches to .
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