Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of new codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error correction performance and many new codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.
pacs:03.67.Hk, 03.67.Mn, 03.67.Pp
Also at ]Graduate School of System and Information Engineering, University of Tsukuba.
This paper develops a general combinatorial method for constructing quantum low-density parity-check (LDPC) codes under the entanglement-assisted stabilizer formalism established by Brun, Devetak, and Hsieh Brun et al. (2006). Our results include many new explicit constructions for entanglement-assisted quantum error-correcting codes for a wide range of parameters. We also prove a variety of new results for classical error-correcting codes, which directly apply to the quantum setting. Most of the quantum codes designed in this paper achieve high error correction performance and high rates while requiring prescribed amounts of entanglement. These codes can be efficiently decoded by message-passing algorithms such as the sum-product algorithm (for details of iterative probabilistic decoding, see MacKay (2003)).
The existence of quantum error-correcting codes was one of the most important discoveries in quantum information science Shor (1995); Steane (1996). Unfortunately, most of the known quantum error-correcting codes lack practical decoding algorithms.
In this paper, we focus on the use of LDPC codes in a quantum setting. Classical LDPC codes Gallager (1963) can be efficiently decoded while achieving information rates close to the classical Shannon limit Luby et al. (2001); Richardson and Urbanke (2001); Richardson et al. (2001). This extends to the quantum setting: the pioneering works of Hagiwara and Imai Hagiwara and Imai (2007) and MacKay, Mitchison, and McFadden MacKay et al. (2004) presented quantum LDPC codes which surpassed, in simulations, all previously known quantum error-correcting codes. Their quantum codes have nearly as low decoding complexity as their classical counterparts.
However, most of the previous results concerning quantum LDPC codes and related efficiently decodable codes have relied on the stabilizer formalism, which severely restricts the classical codes which can be used. The difficulty in developing constructions for non-stabilizer codes was also a substantial obstacle.
Our results use the newly developed theory of entanglement-assisted quantum error-correcting codes (EAQECCs) Bowen (2002); Brun et al. (2006, 2006); Devetak et al. (2009). The entanglement-assisted stabilizer formalism allows the use of arbitrary classical binary or quaternary linear codes for quantum data transmission and error correction by using shared entanglement Hsieh et al. (2007); Wilde and Brun (2008). Previous work related to entanglement-assisted quantum LDPC codes is due to Hsieh, Brun, and Devetak Hsieh et al. (2009) and Hsieh, Yen, and Hsu Hsieh et al. (2009).
The major difficulty in using classical LDPC codes in the entanglement-assisted quantum setting is that very little is known about methods for designing EAQECCs requiring desirable amounts of entanglement. While entanglement-assisted quantum LDPC codes can achieve both notable error correction performance and low decoding complexity, the resulting quantum codes might require too much entanglement to be usable; in general entanglement is a valuable resource Wilde and Brun (2008). In some situations, one might wish to effectively take advantage of high performance codes requiring a larger amount of entanglement Brun et al. (2006, 2006). To the best of the authors’ knowledge, no general methods have been developed which allow the code designer flexibility in choice of parameters and required amounts of entanglement.
Our primary focus in this paper is to show that it is possible to create infinite classes of EAQECCs which consume prescribed amounts of entanglement and achieve good error correction performance while allowing efficient decoding. Our methods are flexible and address various situations, including the extreme case when an EAQECC requires only one preexisting entanglement bit.
The entanglement-assisted quantum LDPC codes which we construct include quantum analogues of the well-known finite geometry LDPC codes originally proposed by Kou, Lin, and Fossorier Kou et al. (2001) (see also Tang et al. (2004, 2005)), and LDPC codes from balanced incomplete block designs that achieve the upper bound on the rate for a classical regular LDPC code with girth six proposed independently by several authors (see Johnson (2010) and references therein). Some classes of our codes outperform previously proposed quantum LDPC codes having the best known error correction performance Hagiwara and Imai (2007); MacKay et al. (2004); Hsieh et al. (2009, 2009).
Our primary tools come from combinatorial design theory, which plays an important role in classical coding theory Tonchev (1998) and also gave several classes of stabilizer codes in quantum coding theory Aly (2008); Djordjevic (2008, 2010); Tonchev (2008, 2009). The use of combinatorial design theory allows us to exactly determine or give tighter bounds on the parameters of the finite geometry LDPC codes in both quantum and classical settings. Comprehensive lists of the parameters of these codes are given in Tables 14 and 15 in Appendix B.
In Section II, we outline our framework for designing entanglement-assisted quantum LDPC codes by using combinatorial design theory. Section III gives explicit constructions for entanglement-assisted quantum LDPC codes based on finite geometries and related combinatorial structures. New results concerning the well-known classical finite geometry LDPC codes are also given in this section. Section IV presents simulation results of our entanglement-assisted quantum LDPC codes and discusses their performance over the depolarizing channel. Section V contains concluding remarks and discusses some related problems that can be treated with the techniques developed in this paper.
Ii Combinatorial entanglement-assisted quantum LDPC codes
In this section we give a general construction method for entanglement-assisted quantum LDPC codes based on combinatorial designs. We do not describe the theory of classical LDPC codes in detail here, instead referring the reader to MacKay (2003); Johnson (2010) and references therein. Relations between quantum error-correcting codes and LDPC codes are concisely yet thoroughly explained in MacKay et al. (2004); Hsieh et al. (2009). Basic notions related to LDPC codes and their relations to combinatorial designs can be found in Ammar et al. (2004). For a detailed treatment of the entanglement-assisted stabilizer formalism, we refer the reader to Brun et al. (2006, 2006); Devetak et al. (2009); Hsieh et al. (2007).
In Subsection II.1 we introduce necessary notions from coding theory and combinatorial design theory. A general method for designing entanglement-assisted quantum LDPC codes is presented in Subsection II.2.
An entanglement-assisted quantum error-correcting code (EAQECC) encodes logical qubits into physical qubits with the help of copies of maximally entangled states. As in classical coding theory, is the length of the EAQECC, and the dimension. We say that the EAQECC requires ebits. An EAQECC with distance will be referred to as an code.
The rate of an EAQECC is defined to be . The ratio is called the net rate. The latter figure describes the rate of an EAQECC when used as a catalytic quantum error-correcting codes to create new bits of shared entanglement Brun et al. (2006, 2006).
Throughout this paper, matrix operations are performed over , the finite field of order two. The ranks of matrices are also calculated over .
We employ the Calderbank-Shor-Steane (CSS) construction Calderbank and Shor (1996); Steane (1996); Brun et al. (2006); Hsieh et al. (2007). Usually the CSS construction uses a minimal set of independent generators to construct an EAQECC. Hence, the construction is often described by using a classical binary linear code with a parity-check matrix of full rank. However, in actual decoding steps, sparse-graph codes may take advantage of redundant parity-check equations to improve error correction performance. Because the extended syndrome can be obtained in polynomial time without additional quantum interactions, we use the following formulation of the CSS construction for EAQECCs.
Theorem 1 (Hsieh, Brun, and Devetak Hsieh et al. (2009))
If there exists a classical binary code with parity-check matrix , then there exists an EAQECC, where .
Note that may contain redundant rows which are related only to classical operations to infer the noise by a message-passing algorithm.
We apply Theorem 1 to classical sparse-graph codes. An LDPC code is typically defined as a binary linear code with parity-check matrix in which every row and column is sparse. In this paper we consider LDPC codes with parity-check matrices whose rows and columns contain only small numbers of ones so that simple message-passing algorithms can efficiently give good performance in decoding.
An LDPC code with parity-check matrix with columns and minimum distance defines a classical binary code, which yields an EAQECC.
The Tanner graph of an parity-check matrix is the bipartite graph consisting of bit vertices and parity-check vertices, where an edge joins a bit vertex to a parity-check vertex if that bit is included in the corresponding parity-check equation. A cycle in a graph is a sequence of connected vertices which starts and ends at the same vertex in the graph and contains no other vertices more than once. The girth of a parity-check matrix is the length of a shortest cycle in the corresponding Tanner graph. Short cycles can severely reduce the performance of an otherwise well-designed LDPC code. In fact, one of the greatest obstacles to the development of a general theory of LDPC codes in the quantum setting is the difficulty of avoiding cycles of length four (See, for example, MacKay et al. (2004); Poulin and Chung (2008); Camara et al. (2007); Hagiwara and Imai (2007)). In order to improve error correction performance, we generally only treat LDPC codes with girth at least six.
The weight of a row or column of a binary matrix is its Hamming weight, that is, the number of ones in it. An LDPC code is regular if its parity-check matrix has constant row and column weights, and irregular otherwise. Regular LDPC codes are known to be able to achieve high error correction performance. Irregular LDPC codes allow the code designer to optimize characteristics of performance by a careful choice of row weights and column weights Luby et al. (2001); Richardson and Urbanke (2001); Richardson et al. (2001).
We now define several combinatorial structures, which we will need in Subsection II.2 and the subsequent sections. For additional facts and design theoretical results, the interested reader is referred to Beth et al. (1999).
An incidence structure is an ordered pair such that is a finite set of points, and is a family of subsets of , called blocks. A point-by-block incidence matrix of an incidence structure is a binary matrix in which rows are indexed by points, columns are indexed by blocks, and if the th point is contained in the th block, and otherwise. A block-by-point incidence matrix of is the transposed point-by-block incidence matrix .
Any LDPC code can be associated with an incidence structure by interpreting its parity-check matrix as an incidence matrix. The converse also holds as long as the considered incidence matrix is sparse.
The current paper will focus on incidence structures which have been extensively studied in combinatorics. This allows us to effectively exploit combinatorial design theory to develop a framework for designing entanglement-assisted quantum LDPC codes.
A - design is an incidence structure , where is a set of cardinality and is a family of -subsets of such that each pair of points is contained in exactly blocks. We will refer to the parameters , , and as the order, block size, and index of a -design. Note that the block size of a -design is usually written as in the combinatorial literature. To avoid any confusion with the dimension of a code, we use instead.
The number of blocks in a - design is determined by the design parameters:
A -design is called symmetric if .
Every point of a - design occurs in exactly blocks, where
The number is called the replication number of the design. A point-by-block incidence matrix of a 2- design satisfies the equation
where is the identity matrix and is the all-one matrix. Since and are integers, it follows that the following two conditions
are necessary conditions for the existence of a - design.
If the block size and index are relatively small, an incidence matrix of a - design is sparse. Hence, a point-by-block incidence matrix of a - design can be viewed as a parity-check matrix of a regular LDPC code with constant row weight and constant column weight . Similarly, a block-by-point incidence matrix defines a code with constant row weight and constant column weight . In this paper, incidence matrices will generally be point-by-block unless it is specifically noted otherwise. In the cases when block-by-point matrices are desirable, the notation will be used.
A substantial part of this paper deals with one of the most fundamental incidence structures in combinatorial design theory. A Steiner -design, denoted by , is a - design. A Steiner triple system of order , denoted by STS, is a Steiner 2-design with block size three. The s are trivial Steiner -designs if . We generally do not consider trivial designs to be Steiner -designs unless they play an important role.
It is easy to see that both point-by-block and block-by-point incidence matrices of an give regular LDPC codes with girth six (see, for example, Johnson and Weller (2001)).
ii.2 General combinatorial constructions
In this subsection we present a general framework for designing entanglement-assisted quantum LDPC codes based on combinatorial design theory. Specialized construction methods for desirable EAQECCs in this framework will be illustrated in Section III.
The following propositions are derived from Theorem 1 by using incidence matrices as parity-check matrices of binary LDPC codes.
Let be a point-by-block incidence matrix of an incidence structure . Then there exists a EAQECC.
Let be a block-by-point incidence matrix of an incidence structure . Then there exists a EAQECC.
We employ the following two theorems.
Theorem 5 (Hillebrandt Hillebrandt (1992))
The rank of an incidence matrix of an satisfies the following inequalities:
Theorem 6 (Hamada Hamada (1973))
If is an incidence matrix of an with even replication number then
Theorem 7 (High-Rate 1-Ebit Code)
Let be a point-by-block incidence matrix of an . Suppose is odd. Then has row weight , column weight , girth 6, and the corresponding EAQECC satisfies the following conditions:
Theorem 8 (High-Rate High-Consumption Code)
Let be a point-by-block incidence matrix of an . Suppose is even. Then has row weight , column weight , girth 6, and the corresponding EAQECC satisfies the following conditions:
that is, a matrix containing zeros on the diagonal and ones in the other entries. Because is even, is odd. Hence, we have as desired.
Theorem 9 (Low-Rate High-Redundancy Code)
Let be a block-by-point incidence matrix of an . Then has row weight , column weight , girth 6, and the corresponding EAQECC satisfies the following conditions:
Proof. Let be a block-by-point incidence matrix of an . Since any non-trivial contains a pair of blocks that share exactly one point, we have . Applying Proposition 4 to Theorem 5 completes the proof.
It is worth mentioning that a weaker version of Theorem 7 was used in the context of integrated optics and photonic crystal technology Djordjevic (2010). Also notable is that Theorems 7 and 8 can be easily extended to the case when preexisting entanglement is not available. For example, quantum LDPC codes that do not require entanglement can be obtained by applying the extra column method used in Construction U in MacKay et al. (2004) and the CSS construction to s in the same manner as in Proposition 3. Aly’s construction for quantum LDPC codes Aly (2008) is a special case of this extended method. Djordjevic’s construction for quantum LDPC codes Djordjevic (2008) can be obtained by applying the CSS construction to -designs of even index in the same way as in Proposition 3.
The existence of -designs is discussed in Appendix A, which provides Steiner -designs necessary to obtain several infinite families of new entanglement-assisted quantum LDPC codes from Theorems 7, 8, and 9. Before applying our theorems to specific s, we explore general characteristics of our EAQECCs and further develop methods for designing desirable codes.
Theorem 7 yields entanglement-assisted quantum LDPC codes with very high net rates and various lengths while requiring only one ebit. Theorem 8 gives codes which have very high net rates and naturally take advantage of larger numbers of ebits when there is an adequate supply of entanglement. Since holds for any parity-check matrix , the required amounts of entanglement of high rate codes in Theorem 8 are expected to be relatively low when compared with randomly chosen codes of the same lengths. Theorem 9 generates entanglement-assisted quantum LDPC codes which can correct many quantum errors by taking advantage of the higher redundancy. The high error correction performance of these codes will be demonstrated in simulations in Section IV.
Theorem 10 (MacKay and Davey MacKay and Davey (1999))
Let be a parity-check matrix of a classical regular LDPC code of length , column weight , and girth . Let also . Then it holds that , where equality holds if and only if is an incidence matrix of an .
It follows that EAQECCs based on Steiner 2-designs achieve the highest possible net rates for quantum LDPC codes with girth at least six constructed from full rank parity-check matrices with constant column weights through the CSS construction.
The rank of an incidence matrix of an may not be full depending on the structure of the design. If one wishes a parity-check matrix to be regular and full rank at the same time, it is important to choose an with a full rank incidence matrix. This can always be done for the case when except for Doyen et al. (1978). For a more detailed treatment of the ranks of s, we refer the reader to Hamada (1973, 1968); Assmus and Key (1992).
In general, the code minimum distance plays less of a role in the performance of sum-product decoding than maximum likelihood decoding MacKay et al. (2004). Therefore, we explore in detail the distance of EAQECCs based on LDPC codes only when it is of great theoretical interest. Because codes derived from finite geometries are of great importance in coding theory, the distances of EAQECCs obtained from finite geometries will be investigated in detail in Section III.
Here we briefly review the minimum distances of LDPC codes based on Steiner -designs. A pair of s which are not mutually isomorphic may give different minimum distances. The tightest known upper and lower bounds on the minimum distance of an LDPC code based on an STS can be found in the very large scale integration (VLSI) literature as bounds on even-freeness.
Theorem 11 (Fujiwara and Colbourn Fujiwara and Colbourn (2010))
The minimum distance of a classical binary linear code whose parity-check matrix forms an incidence matrix of a non-trivial STS satisfies .
A carefully chosen triple system can have a good topological structure which gives good decoding performance. If conditions require larger minimum distances, the code designer may use either block-by-point incidence matrices, or s of larger block sizes. For known results on minimum distances, girths, and related characteristics of LDPC codes based on combinatorial designs, the reader is referred to Colbourn and Fujiwara (2009); Fujiwara and Colbourn (2010); Johnson (2004) and references therein.
In what follows, we describe general guidelines for designing entanglement-assisted quantum LDPC codes with desired parameters and properties by exploiting codes we have presented in this section.
We first consider an EAQECC requiring only a small amount of entanglement. The extreme case is when . The following theorem gives infinitely many such EAQECCs having extremely high rates and low decoding complexity.
Let and be positive integers satisfying and . Suppose also that is odd. Then for all sufficiently large and some satisfying the condition of Theorem 7, there exists an EAQECC.
In general, the error floor of a well-designed LDPC code is not dominated by low-weight codewords. Nonetheless, it is desirable to carefully choose an when applying our simple constructions so that the resulting code has a promising topological structure. While incidence matrices of s have long been investigated in various fields, it appears to be difficult to achieve the known upper bounds on the minimum distance of an LDPC code based on an incidence matrix of an . In fact, it is conjectured that the known upper bounds are generally not achievable even for the case Colbourn and Fujiwara (2009).
An STS is -even-free (or anti-Pasch) if its incidence matrix gives a classical LDPC code with minimum distance five or greater. A -even-free STS exists for all satisfying the necessary conditions (4) Grannell et al. (2000). It is conjectured that an incidence matrix of a -even-free STS gives the largest possible minimum distance Colbourn and Fujiwara (2009).
There exists a EAQECC with and for every except for .
Proof. If , then the replication number of an STS is odd. Applying Theorem 7 to a -even-free STS completes the proof.
A block-by-point incidence matrix of a symmetric can also be viewed as a point-by-block incidence matrix of a Steiner -design of the same parameters Colbourn and Dinitz (2007). Hence, Theorems 7 and 9 can overlap when symmetric designs are employed. This special case gives the EAQECCs with and good error correction performance originally presented in Hsieh et al. (2009). For completeness, we give a simple proof by using the following two theorems.
For every integer there exists a symmetric whose incidence matrix satisfies .
Proof. Take as the Desarguesian projective plane of order , whose incidence matrix has rank Graham and MacWilliams (1966).
Theorem 15 (Calkin, Key, and de Resmini Calkin et al. (1999))
Let be a block-by-point incidence matrix of a symmetric being the Desarguesian projective plane . Then defines a classical binary linear code.
For every integer there exists a EAQECC.
EAQECCs of this kind can be seen as quantum analogues of special Type I PG-LDPC codes, which have notable error correction performance in the classical setting Kou et al. (2001); Tang et al. (2004, 2005). Because of the direct correspondence between entanglement-assisted quantum codes and classical codes, these EAQECCs inherit excellent error correction performance while consuming only one initial ebit. We will further investigate entanglement-assisted quantum LDPC codes based on s with large minimum distances in Section III.
Next we present general combinatorial methods for designing EAQECCs with relatively small and better error correction performance. The main idea is that we discard some columns from an incidence matrix of an and then apply Proposition 3 as we did in Theorem 7. Our methods encompass the rate control technique for classical LDPC codes proposed in Johnson and Weller (2003) as a special case.
Let be an . Take two subsets and . The pair is called a proper subdesign of block size if it is an . Since we do not consider other kinds of subdesigns, we simply call a proper subdesign of block size a subdesign. A pair of subdesigns and of an are point-wise disjoint if .
Let be an with odd . Assume that contains point-wise mutually disjoint subdesigns , , such that and each has odd replication number. Then there exists an EAQECC satisfying the following conditions:
Proof. Take an arbitrary incidence matrix of an with odd . Delete point-wise mutually disjoint subdesigns each of which has odd replication number. It is always possible to reorder the rows and columns of the resulting incidence matrix such that has the form:
where is a zero matrix and each is an all-one matrix of appropriate size. It is easy to see that . Applying Proposition 3 to completes the proof.
Deleting subdesigns always shortens the length of the corresponding code. Discarding columns will not decrease the minimum distance or the girth. The rank of the parity-check matrix is unlikely to change. In this sense, we expect EAQECCs obtained through subdesign deletion to have better error correction performance than the original code. We will demonstrate this effect in simulations in Section IV.
In general, deleting a subdesign makes a parity-check matrix slightly irregular. If this irregularity is not desirable because of particular circumstances or conditions, it can be alleviated by discarding more point-wise disjoint subdesigns. In fact, if we delete subdesigns of the same order such that each point belongs to one deleted subdesign, we obtain a regular parity-check matrix again. The following construction demonstrates this.
Let be an and a set of Steiner -designs , where partition , that is, and for all . Then is called a Steiner spread in if each forms a subdesign of .
Let be an with odd replication number . Assume that contains a Steiner spread , where each subdesign has odd replication number. Then there exists an EAQECC satisfying the following conditions:
Moreover, if for all and , then the parity-check matrix of the corresponding LDPC code is regular and has row weight and column weight .
Proof. Let be an incidence matrix of an with odd which contains a Steiner spread . Delete all members of the Steiner spread from . By following the same argument as in the proof of Theorem 17, it is straightforward to see that when is odd, and otherwise. If for all and , each subdesign has the same replication number . Hence, the resulting code is regular.
When there is an adequate supply of entanglement, it may be acceptable to exploit a relatively large amount of entanglement to improve error correction performance while keeping similar characteristics of high rate codes. Deleting an with even replication number increases the required amount of entanglement to a slightly larger extent.
Let be an with odd replication number . Assume that contains point-wise mutually disjoint subdesigns , , such that and each has even replication number. Then there exists an EAQECC satisfying the following conditions:
Moreover, if the subdesigns for form a Steiner spread with for all and , then the parity-check matrix of the corresponding LDPC code is regular and has row weight and column weight .
Proof. Take an arbitrary incidence matrix of an with odd . Delete point-wise mutually disjoint subdesigns each of which has even replication number. If , it is always possible to reorder the columns of the resulting incidence matrix such that is of the form:
where is the identity matrix and each is an all-one matrix of appropriate size. Because each has independent rows and each is odd, . Applying Proposition 3 to gives . If , we have identity matrices across the diagonal of . Hence, we have again. If each is of the same size, it is straightforward to see that the resulting code is regular.
When irregularity in a parity-check matrix is acceptable or favorable, the code designer can combine the techniques of Theorems 17, 18, and 19. The required amount of entanglement is readily computed by the same argument as above.
In general, subdesign deletion changes the parameters of a code in a gradual manner. Hence, these techniques are also useful when one would like an EAQECC of specific length or dimension. While we only employed Theorem 7 in the above arguments, Theorem 8 can also be used in a straightforward manner to fine-tune the parameters of EAQECCs.
In order to exploit the subdesign deletion techniques, one needs Steiner -designs having subdesigns or preferably Steiner spreads of appropriate sizes. We conclude this section with a brief review of known general results and useful theorems for finding with subdesigns and Steiner spreads. For a more thorough treatment, the reader is referred to Colbourn and Dinitz (2007); Beth et al. (1999) and references therein.
The well-known Doyen-Wilson theorem Doyen and Wilson (1979) states that one can always find an STS containing an STS as a subdesign as long as both and satisfy the necessary conditions for the existence of an STS and . The following is a general asymptotic theorem on Steiner -designs having subdesigns.
Theorem 20 (Fujiwara Fujiwara (2007))
Let be a positive integer and (mod ). Then there exist a constant number depending on , and a constant number depending on and such that if and satisfies the conditions (mod ) and (mod ), then there exists an having an as a subdesign.
Theorem 20 states that one can always find an having an as a subdesign as long as is a sufficiently large integer satisfying the necessary conditions (4) and is a sufficiently large integer satisfying (mod ).
Steiner spreads are closely related to a special kind of combinatorial design. A group divisible design (GDD) with index one is a triple , where
is a finite set of elements called points,
is a family of subsets of , called groups, which partition ,
is a collection of subsets of , called blocks, such that every pair of points from distinct groups occurs in exactly one block,
for all and .
If all groups are of the same size , all blocks are of the same size , and , one refers to the design as a -GDD of type .
The existence of an and a -GDD of type with index one implies the existence of an having a Steiner spread , where each member of is an .
Proof. Let be a -GDD of type with index one and an . For each , we construct an , , by mapping each point of to an element of by an arbitrary bijection . Define . It is straightforward to check that is an having a Steiner spread whose members are all s.
The above theorem is useful to obtain regular LDPC codes through Theorems 18 and 19 and similar subdesgin deletion techniques based on Theorem 8. One can also modify Theorem 21 for the case when a GDD has different group sizes by a similar argument. The existence of GDDs and their constructions have been extensively investigated in combinatorial design theory. For a comprehensive list of known existence results on GDDs, we refer the reader to Colbourn and Dinitz (2007).
Iii Finite geometry codes
In this section, we demonstrate applications of our general designing methods by using combinatorial designs arising from finite geometries.
The classical LDPC codes obtained from finite geometries are known to have remarkable error correction abilities. By using these codes, we generate infinitely many new high performance entanglement-assisted quantum LDPC codes having numerous Steiner spreads of various sizes. The various Steiner spreads in each code allow the code designer to flexibly fine-tune the parameters and error correction performance.
This section is divided into three subsections. Subsection III.1 studies entanglement-assisted quantum LDPC codes of girth six obtained from projective geometries. Codes based on affine geometries are investigated in Subsection III.2. In Subsection III.3 we investigate slightly modified affine geometry codes, called Euclidean geometry codes. Classical LDPC codes based on these three kinds of finite geometries are called finite geometry LDPC codes or simply FG-LDPC codes.
Many of the results presented in this section can also be seen as new results on classical finite geometry LDPC codes. In particular, properties of finite geometries have been independently studied in the combinatorial literature, and hence many of the “known” results are new results in the field of LDPC codes. For the convenience of the reader, we summarize our results on fundamental parameters of LDPC codes from finite geometries in Tables 14 and 15 in Appendix B. Lengths, dimensions, and minimum distances of the FG-LDPC codes with girth six from projective geometry , affine geometry , and Euclidean geometry are all determined. Specifically for EAQECCs based on FG-LDPC codes, we also determine the required amounts of entanglement for most cases. For a few cases, we give upper bounds on the required amount of entanglement.
iii.1 Projective geometry codes
We begin with EAQECCs obtained from finite projective geometries. The use of projective geometries for constructing EAQECCs first appeared in the work of Hsieh, Yen, and Hsu Hsieh et al. (2009). This subsection illustrates how our combinatorial framework generalizes their method and determines fundamental parameters of quantum and classical LDPC codes obtained from .
Points of the -dimensional projective geometry over are the 1-dimensional subspaces of . The -dimensional projective subspaces of are the -dimensional vector subspaces of . The points and lines of form an , denoted by , having blocks and replication number
One can obtain two types of EAQECCs from projective geometry designs: Type II (using a point-by-block incidence matrix) and Type I (using a block-by-point incidence matrix of the design). Applying Proposition 3 to an incidence matrix of , we obtain a Type II EAQECC. This type of EAQECC belongs to the high rate entanglement-assisted quantum LDPC codes given in Theorems 7 and 8. If we apply Proposition 4 to a block-by-point incidence matrix, we obtain a Type I EAQECC. This kind of EAQECC belongs to the high redundancy entanglement-assisted quantum LDPC codes given in Theorem 9.
The rank of an incidence matrix determines the dimension of the corresponding FG-LDPC code, hence it is one of the key values in the quantum setting as well. Exact values for many sporadic examples have been computed in the fields of quantum and classical LDPC codes. The following two theorems give the exact rank for all projective geometry designs.
Theorem 22 (Hamada Hamada (1968))
The rank of is given by
where , the sum is taken over all ordered sets with , such that and for each , and
We will use the notation for the rank of when is even, that is, . When is odd, the rank of is given by a formula of Frumkin and Yakir Frumkin and Yakir (1990).
Theorem 23 (Frumkin and Yakir Frumkin and Yakir (1990))
Let be odd and an incidence matrix of the design with points. Then .
Hence the exact dimensions of the corresponding FG-LDPC codes obtained from projective geometries can be calculated for all cases.
The rank of was conjectured by Hamada Hamada (1973) to be the lowest rank among all Steiner -designs of the same order and block size. This has been confirmed in a number of cases, although in general the conjecture is still open. Thus we expect that the designs should provide codes with the best possible dimensions among all non-isomorphic s.
We will now examine the codes obtained from in detail. This subsection is divided into two portions based on the orientation of the incidence matrix.
iii.1.1 Point-by-block (Type II) Projective geometry codes
In this portion, we consider the EAQECCs corresponding to a point-by-block incidence matrix of .
We first consider the case for some positive integer . The following theorem gives an infinite family of entanglement-assisted quantum LDPC codes which consume only one initial ebit and have extremely large net rate.
For every pair of integers and there exists an entanglement-assisted quantum LDPC codes with girth six whose parameters are
To prove Theorem 24, we first prove a new result on the distance of EAQECCs obtained from an incidence matrix of . We use a special set of lines. A dual hyperoval is a set of lines of , such that each point of lies on either zero or two lines of . Dual hyperovals exist if and only if is even. An example is the set of projective lines with equations
Let be an incidence matrix of . The minimum distance of the classical binary linear code with parity-check matrix is .
Proof. First, we note that coordinates of the codewords correspond to lines of the geometry, and a codeword corresponds to a set of lines in such that every point is contained in an even number of lines of . Assume that is a non-zero codeword, and let denote the support of , that is, the set of indices of the nonzero coordinates of . Since , the support of contains at least one line . Through each point of , there pass an even number of lines from . In particular, each of the points on lies on at least one other line of , and all these lines are different as they have different intersections with . Hence, there are at least lines in , that is, minimum distance is at least . Let be a plane in and the set of the lines of a dual hyperoval in . Then corresponds to a codeword of weight , hence .
Proof of Theorem 24. Let be an incidence matrix of . The rank of is given by Theorem 22. The index of is one. The replication number is odd. By Equation (3) and Theorem 7, we have . By Theorem 25, the minimum distance of the binary linear code with parity-check matrix is .
Next, we examine EAQECCs obtained from an incidence matrix of with odd. This case also gives very high rate entanglement-assisted quantum LDPC codes.
Let be an incidence matrix of , odd. Then the classical binary linear code defined by parity-check matrix consists of only the zero vector and the all-one vector.
Proof. This follows directly from Theorem 23.
A hyperbolic quadric is a substructure of with points and lines, such that each point of lies on exactly two lines of and every plane of contains zero or two lines of . Hyperbolic quadrics exist for every odd prime power .
Let be an incidence matrix of , , odd. Then the minimum distance of the classical binary linear code with a parity-check matrix is .
Proof. Let be a -dimensional subspace of and a hyperbolic quadric in . The set of lines determines a codeword of weight , since each point of is contained in zero or two lines of . Hence minimum distance is at least .
We show that there are no codewords of weight smaller than . Assume that there exists a codeword of weight smaller than , that is, is a set of less than lines of , such that each point lies on an even number of lines of . We will show that for any -dimensional subspace one has either or .
First, let . For each , each of the points on has to lie on at least one other line of , and at most of them can lie on a line of . Hence, at least of them are lines in and since they all have different intersections with , this yields lines in . Together with the lines of , we have
and solving this quadratic inequality for gives us that either or . Since is an integer, hence or .
Now, let be any line of . Each point of must lie on at least one other line, hence there certainly exist planes with , and we have . Let be such a plane. We will now show that all lines of are contained in . Assume the contrary, that there exists a line . Through each of the points on , we need at least one other line of which is not contained in . Since there are at least points on , one has
a contradiction. Hence, does not exist and is contained within a single plane . However, is a and by Lemma 26 we need lines in this case, a contradiction. Hence, there are no codewords of weight less than .
We now give another infinite family of Type II entanglement-assisted quantum LDPC codes.
Let be an odd prime power. Then for every integer there exists an entanglement-assisted quantum LDPC code with girth six whose parameters are
Therefore in the case where is odd, we have another infinite class of EAQECCs which consume only one ebit. If is even, we obtain infinitely many high rate codes which consume reasonable numbers of ebit. Tables 1 and 2 give a sample of the parameters of the Type II codes obtained from with even and odd respectively.
In the reminder of this portion, we examine Steiner spreads of projective geometry designs. These substructures can be used in Theorems 17, 18, and 19 and their analogous techniques based on Theorem 8 to fine-turn the rates and distances of the EAQECCs.
An -spread of is a set of -dimensional projective subspaces which partition the points of the geometry. In other words, an -spread consists of a set of -dimensional vector subspaces of which contain every nonzero vector exactly once. It is known that admits an -spread if and only if divides (see Segre (1964) and (Dembowski, 1968, p. 29)).
Take and suppose is chosen so that divides . Then an -spread of exists. Each -dimensional subspace in the spread contains an isomorphic copy of , and hence this forms a Steiner spread. Note that the blocks of have size and are also blocks of . Therefore we have the following result.
Let , be positive integers such that divides . Then contains disjoint copies of whose point sets partition the point of .
Thus, we can find a set of disjoint subdesigns which partition the points of whenever has a nontrivial factor. Naturally, we may further sub-divide each subdesign of dimension into smaller subdesigns, based on the nontrivial factors of . Hence, the s from are very flexible in that they have Steiner spreads of various sizes.
In general, the length, dimension, required ebits, and rate each change gradually as we delete subdesigns in a Steiner spread. The minimum distance and rank are either remain the same or improve slightly. Table 3 lists the example parameters of EAQECCs created by deleting subdesigns from . The first and last rows correspond to regular LDPC codes.
|Subs111This column denotes the number of subdesigns removed.||Rate|
iii.1.2 Block-by-point (Type I) Projective geometry codes
Next we consider EAQECCs obtained via Theorem 9 by using the block-by-point incidence matrix of . The codes obtained in this manner correspond to the classical Type I LDPC codes. As in the classical setting, Type I entanglement-assisted quantum regular LDPC codes can correct many quantum errors. Because an incidence matrix of for odd is almost full rank, the corresponding Type I code is not of much interest. Hence, in this portion we always assume that for some positive integer .
For every pair of integers and there exists an entanglement-assisted quantum LDPC code with girth six whose parameters are