A construction of Quantum LDPC codes from Cayley graphs
We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of together with a set of generators regarded as the columns of the parity–check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in where is the minimum distance of the classical code. When the classical code is the repetition code, we are able to compute the exact parameters of the associated Quantum code which are .
MSC: 94C15, 05C99, 94B99
Key words: Quantum codes, LDPC codes, Cayley Graphs, Graph covers.
Notes. The material in this paper was presented in part at ISIT 2011 . This article is published in IEEE Transactions on Information Theory . We point out that the second step of the proof of Proposition VI.2 in the published version (Proposition 25 in the present version and Proposition 18 in the ISIT extended abstract ) is not strictly correct. This issue is addressed in the present version.
Classical LDPC codes, it hardly needs to be recalled, come together with very efficient and fast decoding algorithms and overall display extremely good performance for a variety of channels. Quantum error-correcting codes on the other hand, under the guise of the CSS [7, 22] scheme, are in some ways strikingly similar to classical codes, and in particular can be decoded with purely classical means. It is therefore natural to try to import the classical LDPC know-how to the Quantum setting. There is however a structural obstacle. A Quantum CSS code is defined by two binary parity-check matrices whose row-spaces must be orthogonal to each other. To have a Quantum LDPC code decodable by message-passing these two matrices should be sparse, as in the classical case. Therefore, randomly choosing these matrices, the generic method which works very well in the classical case, is simply not an option in the Quantum case, because the probability of finding two sparse row-orthogonal matrices is extremely small. A number of constructions have been suggested by classical coding theorists nevertheless [16, 1, 2, 8, 14, 21] but they do not produce families of Quantum LDPC codes with a minimum distance growing with the blocklength. While this may be tolerable for practical constructions of fixed size, this is clearly an undesirable feature of any asymptotic construction and it raises the intriguing theoretical question of how large can the minimum distance of sparse (or LDPC) CSS codes be. Families of sparse CSS codes with a growing minimum distance do exist, the most well-known of these being Kitaev’s toric code , which has been generalised to codes based on tesselations of surfaces (see e.g. [5, 12, 3, 4, 24]) and higher-dimensional objects. These constructions exhibit minimum distances that scale at most as a square root of the blocklength (to be precise, is achieved in ) though this often comes at the cost of a very low dimension (recall that the dimension of the toric code is ). It is an open question as to whether families of sparse CSS codes exist with a minimum distance that grows at least as for , even for Quantum codes with dimension . The recent construction  manages to reconcile a minimum distance of the order of with a dimension linear in the blocklength. All these constructions borrow ideas from topology and can be seen as some generalisation of Kitaev’s toric code.
In a follow-up to the paper  MacKay, Mitchison and Shokrollahi  proposed a construction that seemingly owes very little to the topological approach. They noticed that the adjacency matrix of any Cayley graph over with an even set of generators is self-dual and can therefore be used to define a sparse CSS code. Experiments with some Cayley graphs were encouraging. In the present work we take up the theoretical study of the parameters of these CSS codes which was left open by MacKay et al. The Quantum code in the construction is defined by a classical linear binary code where must be even. Its length is , and the row-weight of the parity-check matrix is . The dimension and the minimum distance of the Quantum code does not depend solely on the classical code’s parameters, but depend more subtly on its structure. We solve the problem in the first non-trivial case, which was an explicit question of MacKay et al., namely the case when the classical code is the repetition code. Computing the parameters of the associated Quantum code turns out to be not easy, even in this apparently simple case. Our main result, Theorem 18, gives the exact parameters for this Quantum code, namely:
The construction therefore hits the barrier for the minimum distance, but it is quite noteworthy that it does so using a construction that breaks significantly with the topological connection. For Quantum codes based on more complicated classical structures, similarly precise results seem quite difficult to obtain, but we managed to prove a lower bound on the Quantum minimum distance of the form for some constant (Theorem 16).
Notice that the constructed quantum LDPC codes have not a constant row-weight. Indeed, this weight is logarithmic in the blocklength. This has its drawbacks since decoding will be slightly more complex, we remark however that the best families of classical LDPC codes (i.e. capacity-achieving LDPC codes) all have row weights that grow logarithmically in the block length. We note also that it was recently proved in  that quantum LDPC stabilizer codes cannot achieve the capacity of the quantum erasure channel if their stabilizer matrices have constant row weight.
Outline of the article
Some prerequisites on Quantum and Quantum CSS codes together with some basic notions on Cayley graphs are recalled in Section 2. In Sections 3, we describe some basic properties of Cayley graphs associated to the group . In Section 4, we focus on the properties of the Hamming hypercube, that is the Cayley graph , where denotes the canonical basis. In particular, we observe some nice property: for almost all families of generators of , the Cayley graph looks locally like the Hamming hypercube of dimension . In Section 5, we study the minimum distance of a Quantum code associated to a Cayley graph of and show that this distance is at least quadratic in . Finally in Section 6, we focus on the example studied by Mitchison et al. in  and give the exact parameters of this family of Quantum codes.
In this article all codes, classical and quantum, are binary.
2.1 Self-Orthogonal Codes and Quantum codes
A classical code is said to be self-orthogonal if . It is said to be self-dual if . For convenience’s sake, we also say that a binary matrix is self-orthogonal (resp. self-dual) if (resp. and ).
Classical self-orthogonal codes provide a way of constructing quantum codes through a particular case of the CSS construction [7, 22]. Let us just recall that if is self-orthogonal with classical parameters , then it yields a quantum code with parameters , where , and where is the minimum weight of a codeword in .
Notice that this last characterization of implies that where denotes the dual distance of . One way of obtaining quantum codes with good parameters is therefore simply to use classical self-orthogonal codes with a large dual distance: this approach has been used repeatedly to obtain record parameters. However, our purpose is to construct CSS codes with a low-density stabilizer (parity-check) matrix, meaning that we need a sparse self-orthogonal matrix . Since we have for the self-orthogonal code generated by the rows of , the bound is of little use because it cannot bound from below by anything more than the (low) weight of the rows of . Obtaining a better lower bound on the quantum code’s minimum distance can be quite challenging.
In the present work we shall develop a method to obtain improved lower bounds on for some quantum codes based on sparse self-orthogonal matrices. We focus on MacKay et al.’s construction based on the adjacency matrices of some Cayley graphs. Let us first recall some basic notions on Cayley Graphs.
2.2 Cayley graphs and CSS codes
The general construction
Let be a group and be a subset of . The Cayley graph or , when there is no possible confusion, is the graph whose vertex-set equals and such that two vertices are connected by an edge if there exists such that .
The graph is oriented unless . In addition, if , then, the adjacency matrix of the graph is symmetric.
The graph is connected if and only of generates .
Our point is to get pairs such that the adjacency matrix of is self–orthogonal, i.e. such that . Notice that happens if and only if both conditions are satisfied.
Each row of is self-orthogonal, i.e. has even weight;
Any pair of distinct rows of are orthogonal, i.e. any two distinct rows of have an even number of ’s in common.
The following proposition translates the above conditions in terms of the pair .
Let be a finite group and be a system of generators of . Assume that
for all , there is an even number of distinct expressions of of the form , with .
Then, the adjacency matrix of the Cayley graph is self-orthogonal.
If , then the graph in undirected, its adjacency matrix is symmetric and (ii) can be replaced by
for all , there is an even number of distinct expressions of of the form , with .
It is worth noting that if and commute, then and correspond to distinct expressions.
The group algebra point of view
We still consider a pair , where is a group and is a generating set of . Recall that the group algebra of over denoted by is the –vector space with a basis in one–to–one correspondence with elements of together with a multiplication law induced by the group law, i.e. .
Given a pair , where is a group and a generating set. We denote respectively by and the elements of ,
Clearly, the two elements are equal when .
The adjacency matrix of represents the right multiplication by i.e. the application
In addition, the matrix represents the right multiplication by .
Caution. In Lemma 4 above, we suppose that matrices act on row-vectors, i.e. an binary matrix corresponds to an endomorphism of by , where is represented by a row-vector.
The adjacency matrix of is self–orthogonal if and only if . In particular, if , then is self-orthogonal if and only if .
In particular, the problem of finding sparse self-orthogonal matrices is equivalent with that of finding a -nilpotent element of having a low weight compared to .
2.3 Some examples
Let be the group and be the set . Then, the adjacency matrix of is self-orthogonal. The corresponding group algebra is isomorphic to and the element equals .
Motivated by MacKay et al.’s draft , the group we will focus on in the rest of the paper is . Since we are dealing with an abelian group we denote group operations additively rather than multiplicatively.
and is any system of generators with an even number of elements. The corresponding group algebra is isomorphic to , in which one sees easily that any element of even weight satisfies .
3 Basic properties of CSS codes from Cayley Graphs of
As we have just seen in the last example, any even number of generators of defines a Cayley graph whose adjacency matrix is a self-orthogonal matrix, from which we have a quantum code. The row weight of the matrix is equal to the cardinality of the set of generators: when this cardinality is chosen proportional to , we have a row weight that is logarithmic in the row length, hence the LDPC character of the quantum code. As put forward in , note also that the matrix is , i.e. has a highly redundant number of rows, which is beneficial for decoding. It also makes the computation of its rank, and hence the dimension of the quantum code, non-trivial. The present paper strives to compute or estimate parameters, dimension and minimum distance, of the resulting quantum LDPC code.
3.1 Context and notation
One of the main difficulties of the following work is that we juggle with different kinds of classical codes. Roughly speaking, we deal with small codes of length and big codes of length .
This is the reason why we first need to describe carefully the landscape and the notation we choose.
The “small” and “big” objects
For a positive integer , the canonical basis of is denoted by . In what follows, words of are denoted by letters in lower case such as or . Such words are referred as small words and subspaces of are referred as small codes.
Given a set of generators of we denote by , or when no confusion is possible, an adjacency matrix of the Cayley graph . From Proposition 3, if is even, then is self–orthogonal. We denote by or the code with generator matrix . Words of this code or more generally of its ambient space, namely will be denoted by letters in Gothic font such as or . In what follows and to help the reader, we frequently refer to big words and big codes when dealing with such words or codes. Gothic fonts are dedicated to big objects, such as the matrices , the Cayley graphs , the corresponding big codes and so on…
In a graph , we say that two connected vertices have distance if the the shortest path between them consists of edges. This defines a natural metric on .
For this distance, a ball centred at a vertex of radius is denoted by , it is the set of vertices at distance of . A sphere of centre and radius is denoted by .
We will say that a graph is a cover or a lift of if it comes together with a surjective map called a covering map such that for any vertex of , the map , restricted to the set of neighbours of , is a one-to-one mapping onto the set of neighbours of . The covering map is a local isomorphism. It can be shown that when is connected, the cardinality of the preimage of any vertex is constant: we will refer to this number as the degree of the cover.
Consider the particular case when for some set of generators of . A natural covering map of is
which can be thought of as removing linear dependencies between elements of (see §4.4). Any Cayley graph associated to is therefore locally isomorphic to some hypercube . This covering construction was used by Tillich and Friedman in . Starting with a code of generating matrix , they used the set of columns of to define a graph : relating the eigenvalues of to those of its cover (1) they derived upper bounds on the minimum distance of . Here we shall rather view the set of generators as the set of columns of a code ’s parity-check matrix (rather than a generating matrix). The minimum distance of is therefore the minimum weight of a linear relation between generators of , and for the balls in and in are isomorphic.
The dictionary relating big codes and graphs
We keep the notation of §3.1.1. It is worth noting that elements of the ambient space of are in one–to–one correspondence with subsets of the vertex–set of . In what follows, we frequently allow ourselves to regard big words as sets of vertices, while vertices are nothing but elements of . In particular we allow ourselves notation such as “”, where and . From this point of view, we frequently use the elementary lemma below. Recall that, given two subsets of a set , the symmetric difference of and is defined by . This operation is associative.
Regarding elements of the ambient space of as subsets of the vertex–set of ,
a row of is nothing but a sphere of centre and radius , where is the index of the row;
a word of is a symmetric difference of spheres of radius , or equivalently an –formal sum of such spheres;
a word is a set of vertices such that for every sphere of radius , the intersection has even cardinality.
3.2 Automorphisms of the big codes and the graphs
Given a positive integer , recall that the Hamming–isometries are of the form , where is a permutation of the coordinates and is the affine translation for some fixed .
Let be a family of generators of and be a Hamming–isometry of , then induces a permutation of which is an automorphism of and an element of the permutation group of (and hence in that of ).
For all small word , the sphere is the big word whose nonzero entries are the small words with . The code is generated by the ’s for and one sees easily that . ∎
Let , if there exists a nonzero big word in (resp. ), then, there exists a big word (resp. ) with the same weight and which contains the small word .
4 The Hamming hypercube
In this section, denotes an even integer and we study the properties the Cayley graph . Recall that denotes the canonical basis of .
First, we show that , which means that the corresponding big code is self–dual and hence that the corresponding CSS code is trivial. However, the properties of are of interest because of its role in the covering construction (1).
4.1 The corresponding Quantum code is trivial
Let be an even integer. The adjacency matrix of , satisfies
Therefore, , or equivalently , is self-dual.
The group algebra of is . Using Notation 1, the element is . Thus, the cokernel of the endomorphism is
This last algebra is isomorphic to and one sees easily that if is even, then . Thus, this cokernel is isomorphic to whose –dimension is exactly the half of that of . ∎
4.2 The graph is bipartite
Another very useful and nice property of this family of graphs is given by the following statement.
Consider the partition of by of small words of having respectively even and odd Hamming weight. Then, is bipartite, i.e. any edge links an element of with one of .
For all and all , the small words and have weights of distinct parities. ∎
In matrix terms, this means, that, for a suitable ordering of the elements of , there exists a binary matrix such that
In addition, one shows easily by induction on that .
The former result has interesting consequences on the code for even.
Let be an even integer. The code splits in a direct sum of two isomorphic subcodes with disjoint supports
corresponding to big words whose supports are the small words of even and odd weight respectively. Both subcodes are self–dual.
The two codes come respectively from the upper and lower halves of the row-set of in (2). They are obviously isomorphic since they have the same generator matrix . The self–orthogonality is clear since entails . In addition, it is clear that , which yields self–duality. ∎
Since is self-dual, a big word is in if and only if it is orthogonal to any sphere of radius . If is a small word of even weight, then the elements of have odd weight and hence is obviously orthogonal to any big word supported in . Thus, a big word with support in (resp. in ) is in if and only if it is orthogonal to any sphere of radius centred at a small word of even (resp. odd) weight. ∎
Consequently, the graph can be regarded as a Tanner graph for where is the set of bit nodes and the set of check nodes. It can conversely be regarded as a Tanner graph for by switching bit and check nodes.
Actually, this property of being bipartite is satisfied by any Cayley graph as soon as for all and all , the weights of the small words and have distinct parities. It holds for instance for , where is odd.
4.3 A property of bounded codewords
The following statement is crucial in the study of the minimum distance of Quantum codes from graphs covered by .
Let be a codeword in the row-space of . Regarding as a subset of the vertex–set of , assume that is contained in the ball for some vertex and some integer . Then is a sum of rows of with support contained in . Equivalently, is the –formal sum of spheres of radius contained in .
From Corollary 8, one can assume that and hence . Let us prove the result by induction on .
We will consider the extremal points of , that is the vertices of whose distance to 0 is maximal. For every extremal vertex of , we will add a sphere included in the ball to to obtain a new codeword which does not contain the vertex . This procedure will lead to a decomposition of as a sum of spheres included in the ball .
If , then is either the zero codeword or the unique big word with support equal to the vertex . But the big word of with support equal to the vertex cannot be in . Indeed, this big word has weight and since is self–dual, if it had such a big word in its row–space, the word would lie in its kernel. Thus, would have a zero column which is impossible. Thus, is the big word zero which is the empty formal sum of spheres of radius .
Let and assume that the result holds for all radius .
Claim. Let be the least integer such that . If , then, clearly, is nonempty. Then, for all , there exists whose -th entry is nonzero.
Proof of the claim. Assume the claim is false. Without loss of generality, one can assume that the –th entry of any element of is zero. Thus, the elements of are of the form , where the ’s and the “” denotes the concatenation. From, Proposition 9, we have . Thus, regarding as an element of and using Lemma 6(3), we see that the intersection of with any sphere of radius has an even cardinality. However, the spheres contain one and only one element of , namely . This yields the contradiction.
Thanks to the claim, we know that there exists at least one element of with a nonzero –th entry. Let be these elements. Clearly, the small words have weight and hence the spheres are contained in . For all , the only element of whose –th entry is nonzero is . Thus, the big word
is contained in and the elements have all a zero –th entry. Indeed, the ’s have been cancelled and no other element of the form have been added while adding the spheres of radius . The claim entails that . By the induction hypothesis, is a sum of spheres of radius contained in . Since the spheres are also contained in , Equation (3) yields the result. ∎
4.4 The hypercube cover
In what follows, denotes an integer. Recall that denotes the canonical basis of . Let be a family of distinct nonzero elements of with cardinality and assume that is even. From Proposition 3, the code is self-orthogonal and hence provides a Quantum CSS code with parameters , where is the minimum weight of a codeword of .
Regarding the elements of as column vectors, we introduce the binary matrix whose columns correspond to the elements of , that is
where denotes the identity matrix and is the matrix whose columns are the elements of .
Let be the code with parity–check matrix . There is a natural graph cover
The degree of is . In addition, denoting by the minimum distance of , the restriction of to any ball of radius is an isomorphism of graphs.
Recall that we denote the elements of the canonical basis by . Denote by the elements of . Consider the linear map
that sends . The covering map is naturally constructed from the above map. One sees easily that the fibre (preimage) of a vertex of is nothing but the coset and hence has cardinality .
To conclude, consider a ball of of radius . Notice that two vertices of have the same image by if and only if , i.e. if and only if with . In particular, two such vertices have the same image only if their distance is . Since the distance between any two vertices in a ball of radius is then, they have distinct images by . Thus, the restriction of to the ball is an isomorphism. ∎
5 On the minimum distance of the Quantum code
We keep the notation of §4.4. Given a set of generators of as before, our point is to bound below the minimum distance of the corresponding CSS Quantum code, that is the minimum weight of the set .
We keep the notation of Theorem 14. A codeword in is not contained in a ball of radius .
First, let us quickly sketch this proof. Set . Assume that is contained in a ball of radius . Then, using Theorem 14, we construct a lift of satisfying
is contained in a ball of radius ;
, where is the graph covering map introduced in Theorem 14.
From Proposition 9, the code is self–orthogonal and hence . From Proposition 13, is a sum of spheres contained in the ball of radius . From Theorem 14, the covering map restricted to a ball of radius is an isomorphism. Thus, is a sum of spheres and hence is a codeword of which leads to a contradiction.
The non-obvious part of the proof is the construction of the lift . It is worth noting that, despite inducing an isomorphism between balls of radius , it is however not possible to lift all such big words in a ball of radius . A counter-example is given in Example 3.
Let us prove the existence of such a lift. Without loss of generality, one can assume that is contained in the ball . Clearly induces an isomorphism between this ball and the ball centred at zero of radius of . Let be the inverse image of by this isomorphism. The above conditions (2) and (3) are obviously satisfied. There remains to prove that has an even number of common elements with any sphere of radius . Since , any sphere which is not contained in has an empty intersection with . On the other hand, any sphere of radius contained in corresponds thanks to and Theorem 14 to a unique sphere of radius contained in the ball of radius centred at of . Thanks to this ball-isomorphism and by definition of , it is clear that has an even number of common elements with such a sphere. This yields (1). ∎
Suppose that and . Theorem 14 asserts the existence of a graph covering map . The classical code defined in Theorem 14 is nothing but the pure repetition code of length which has minimum distance . Therefore, induces isomorphisms between balls of radius . Let us show that some big words contained in in cannot lift as in the previous proof as a word . Let
Let us show that . Let , we have to prove that is orthogonal to , that is has an even number of common elements with . Notice that for the graph , we have
It is clear that if or , then contains no element of weight and hence is obviously orthogonal to . If , then for some and contains four elements of weight , namely all the with . If , that it for distinct to each other, then contains also four elements of weight , namely and , where .
Thus, . From Theorem 14, the map induces an isomorphism from the ball of and the ball of . Let us consider the lift of by this isomorphism
This big word is not an element of (which equals ). Indeed, let be a small word of weight . Then, has exactly three common elements with , namely and . Thus, is not orthogonal to this sphere of radius .
Let be a family of vectors of with . Let be as in (4) and be the small code of length and parity–check matrix . Recall that denotes the canonical basis of . Let be the minimum distance of . Then, the minimum distance of and hence of the corresponding Quantum code satisfies
This theorem is proved further thanks to the following technical lemma.
Let and be as in Theorem 16. Assume moreover that the minimum distance of the small code is at least . Let be a big word of minimum weight in and let then
From Corollary 8, one can assume that the small word of the statement is . From Theorem 14, the ball of is isomorphic to that of . Therefore, as soon as we stay inside , we can reason as if we were inside that of . Thus, set and let us reason in .
Step 1. First, it is important to notice that is supposed to have a minimum weight as a word of , therefore,
Step 2. Since by assumption and , this word must be orthogonal to any sphere of radius . Denote by the elements of the canonical basis of . Then must be orthogonal to the spheres . Thus,
Thus, contains at least small words in .
Now, consider the maximal subset of elements of with disjoint supports. After reordering the indexes, one can assume that these elements are , for some . We will get the result by considering separately the situations “ is large” and “ is small”.
Step 3. If , then for all odd and , consider the sphere . Since is orthogonal to any sphere of radius and contains , it should contain at least one other element of . This other element is either in or in .
If this other element of is in , then it is either or . This additional element is in at most one other sphere of the form with odd, and .
If this other element is in , then it is of the form for some . For obvious degree reasons this additional element is in at most spheres of degree centred at a small word of weight .
Finally, Since there are spheres of the form with odd, and , there are at least additional elements in lying in . Therefore, considering also and the elements we get
Since, by assumption we conclude that
Step 4. Now, assume that . From (6) and by maximality of the set , for all , there exists at least an integer