Scheme for constructing graphs associated with stabilizer quantum codes

Scheme for constructing graphs associated with stabilizer quantum codes

Carlo Cafaro, Damian Markham, and Peter van Loock Max-Planck Institute for the Science of Light, Gunther-Scharowsky-Str.1/Bau 26, 91058 Erlangen, Germany Institute of Physics, University of Mainz, Staudingerweg 7, 55128 Mainz, Germany Department of Mathematics, Clarkson University, 8 Clarkson Ave, Potsdam, NY 13699-5815, USA CNRS LTCI, Département Informatique et Réseaux, Telecom ParisTech, 23 avenue d’Italie, CS 51327, 75214 Paris CEDEX 13, France
Abstract

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.

pacs:
03.67.-a (quantum information)

I Introduction

Classical graphs die (); west (); wilson () are closely related to quantum error correcting codes (QECCs) gotty (). The first construction of QECCs based upon the use of graphs and finite Abelian groups appears in werner () and is provided by Schlingemann and Werner (SW-work). However, while in werner () it is proved that all codes constructed from graphs are stabilizer codes, it remains unclear how to embed the usual stabilizer code constructions into the proposed graphical scheme. Therefore, although necessary and sufficient conditions are uncovered for the graph such that the resulting code corrects a certain number of errors, the power of the graphical approach to quantum coding for stabilizer codes cannot be fully exploited unless this embedding issue is resolved. In dirk (), Schlingemann (S-work) clarifies this issue by establishing that each quantum stabilizer code (both binary and nonbinary) could be realized as a graph code and vice-versa. Almost at the same time, inspired by the work presented in werner (), the equivalence of graphical quantum codes and stabilizer codes is also established by Grassl et al. in markus (). Despite being very important, the works in dirk () and markus () still suffer from the fact that no systematic scheme for constructing a graph of a stabilizer code or the stabilizer of a graphical quantum code is available. The solution of this point is especially important in view of the fact that although  any stabilizer code over a finite field has an equivalent representation as a graphical quantum code, unfortunately, this representation is not unique. Furthermore, the chosen representation does not reflect all the properties of the quantum code. A crucial step forward for the description and understanding of the interplay between properties of graphs and stabilizer codes is achieved thanks to the introduction of the notion of graph states (and cluster states, hans ()) into the graphical construction of QECCs as presented by Hein et al. in hein (). In this last work, it is shown how graph states are in correspondence to graphs and special focus is devoted to the question of how the entanglement in a graph state is related to the topology of its underlying graph. In hein (), it is also pointed out that codewords of various QECCs could be regarded as special instances of graph states and criteria for the equivalence of graph states under local unitary transformations entirely on the level of the underlying graphs are presented. Similar findings are uncovered by Van den Nest et al. in bart () (VdN-work) where a constructive scheme showing that each stabilizer state is equivalent to a graph state under local Clifford operations is discussed. Thus, the main finding of Schlingemann in dirk () is re-obtained in bart () for the special case of binary quantum states. Most importantly, in bart (), an algorithmic procedure for transforming any binary quantum stabilizer code into a graph code appears. However, to the best of our knowledge, nobody has fully and jointly exploited the results provided by either Schlingemann in dirk () or Van den Nest et al. in bart () to provide a more systematic procedure for constructing graphs associated with arbitrary binary stabilizer codes with special emphasis on the verification of their error-correcting capabilities. We emphasize that this last point constitutes one of the original motivations for introducing the concept of a graph into quantum error correction (QEC) werner ().

The CWS quantum code formalism presents a unifying approach for constructing both additive and nonadditive QECCs, for both binary cross () (CWS-work) and nonbinary states chen (). Furthermore, every CWS code in its canonical form can be fully characterized by a graph and a classical code. In particular, any CWS code is locally Clifford equivalent to a CWS code with a graph state stabilizer and word operators consistent only of s cross (). Since the notion of stabilizer codes, graph codes and graph states can be recast into the CWS formalism, it seems natural to investigate the graphical depiction of stabilizer codes as originally thought by Schlingemann and Werner within this generalized framework where stabilizer codes are realized as CWS codes. Proceeding along this line of investigation, we shall observe that the notion of graph state in QEC as presented in hein () emerges naturally. Furthermore, the algorithmic procedure for transforming any (binary) quantum stabilizer code into a graph code advocated in bart () can be exploited and jointly used with the results in dirk () where the notions of both coincidence and adjacency matrices of a classical graphs are introduced. For the sake of completeness, we point out that the CWS formalism has been already employed into the literature for the graphical construction of both binary yu1 () and nonbinary yu2 () (both additive/stabilizer and nonadditive) QECCs. For instance in yu1 (), regarding stabilizer codes as CWS codes and employing a graphical approach to quantum coding, a classification of all the extremal stabilizer codes up to eight qubits and the construction of the optimal code together with a family of -error detecting nonadditive codes with the highest encoding rate so far is presented. With a leap of imagination, in yu1 () it is also envisioned a graphical quantum computation based directly on graphical objects. Indeed, this vision became recently more realistic in the work of Beigi et al. beigi (). Here, being essentially within the CWS framework, a systematic method for constructing both binary and nonbinary concatenated quantum codes based on graph concatenation is developed. Graphs representing the inner and the outer codes are concatenated via a simple graph operation (the so-called generalized local complementation, beigi ()). Despite their very illuminating findings, in beigi () it is emphasized that the elusive role played by graphs in QEC is still not well-understood. In neither yu1 () nor beigi (), the Authors are concerned with the joint exploitation of the results provided by either Van den Nest et al. in bart () (algorithmic procedure for transforming any binary quantum stabilizer code into a graph code) or Schlingemann in dirk () (use of both the coincidence and adjacency matrices of a classical graphs in QEC) in order to provide a more systematic procedure for constructing graphs associated with arbitrary binary stabilizer codes with special emphasis on the verification of their error correcting capabilities which, as pointed out earlier, constituted a major driving motivation for the introduction of graphs in QEC werner (). Instead, we aim here at investigating such unexplored topics and hope to further advance our understanding of the role played by classical graphs in quantum coding.

In this article, we propose a systematic scheme for the construction of graphs with both input and output vertices associated with arbitrary binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a CWS quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs with only output vertices. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous variables quantum codes is briefly addressed.

The layout of the article is as follows. In Section II, we introduce some preliminary material. First, the notions of graphs, graph states and graph codes are presented. Second, local Clifford transformations on graph states and local complementations on graphs are briefly described. Third, the CWS quantum codes formalism is briefly explained. In Section III, we re-examine some basic ingredients of the Schlingemann-Werner work (SW-work, werner ()), the Schlingemann work (S-work, dirk ()) and, finally, the Van den Nest et al. work (VdN-work, bart ()). We focus on those aspects of these works that are especially important for our systematic scheme. In this Section IV, we formally describe our scheme and, for the sake of clarity, apply it to the graphical construction of the Leung et al. four-qubit quantum code for the error correction of single amplitude damping errors debbie (). Finally, concluding remarks and a brief discussion on possible extensions of our schematic graphical construction to both (stabilizer and nonadditive) nonbinary and continuous variables quantum codes appear in Section V.

Several explicit constructions of graphs for various stabilizer codes characterized by either single or multi-qubit encoding operators are worked out in the Appendices. Specifically, we discuss the graphical construction of the following quantum codes: the three-qubit repetition code, the perfect -erasure correcting four-qubit code, the perfect -error correcting five-qubit code, -error correcting six-qubit quantum degenerate codes, the CSS seven-qubit stabilizer code, the Shor nine-qubit stabilizer code, the Gottesman -error correcting eleven-qubit code,  stabilizer codes, and, finally, the Gottesman  stabilizer code.

Ii From graph theory to the CWS formalism

In this section, we present some preliminary material. First, the notions of graphs, graph states and graph codes are introduced. Second, local Clifford transformations on graph states and local complementations on graphs are briefly presented. Third, the CWS quantum codes formalism is briefly discussed.

ii.1 Graphs, graph states, and graph codes

A graph is characterized by a set of vertices and a set of edges specified by the adjacency matrix die (); west (); wilson (). This matrix is a symmetric matrix with vanishing diagonal elements and if vertices , are connected and otherwise. The neighborhood of a vertex is the set of all vertices that are connected to and is defined by . When the vertices , are the end points of an edge, they are referred to as being adjacent. An path is an ordered list of vertices , ,…, , , such that for all , and are adjacent. A connected graph is a graph that has an path for any two , . Otherwise it is referred to as disconnected. A vertex represents a physical system, e.g., a qubit (two-dimensional Hilbert space), qudit (-dimensional Hilbert space), or continuous variables (CV) (continuous Hilbert space). An edge between two vertices represents the physical interaction between the corresponding systems. In what follows, we shall take into consideration simple graphs only. These are graphs that contain neither loops (edges connecting vertices with itself) nor multiple edges. Furthermore, for the time being, we do not make a distinction between different types of vertices. However, later on we will assign some vertices as inputs, and some as outputs.

Graph states hans () are multipartite entangled states that play a key-role in graphical constructions of QECCs codes and, in addition, are very important in quantum secret sharing damian2008 () which is, to a certain extent, equivalent to error correction anne2013 (). For a very recent experimental demonstration of a graph state quantum error correcting code, we refer to damian2014 ().

Consider a system of qubits that are labeled by those vertices in and denote by , , , (or, equivalently, , , ) the identity matrix and the three Pauli operators acting on the qubit . The -qubit graph state associated with the graph is defined by hein (),

(1)

where is the joint eigenstate of with , is the controlled phase gate between qubits and given by,

(2)

and is the joint eigenstate of with and as eigenvalues. The graph-state basis of the -qubit Hilbert space is given by where is an element of the set of all the subsets of denoted by . A collection of subsets specifies a -dimensional subspace of that is spanned by the graph-state basis with ,…, . The graph state is the unique joint eigenstate of the -vertex stabilizers with defined as hein (),

(3)

A graph code, first introduced into the realm of QEC in werner () and later reformulated into the graph state formalism in hein (), is defined to be one in which a graph is given and the codespace (or, coding space) is spanned by a subset of the graph state basis. These states are regarded as codewords, although we recall that what is significant from the point of view of the QEC properties is the subspace they span, not the codewords themselves robert ().

ii.2 Local Clifford transformations and local complementations

ii.2.1 Transformations on quantum states 

The Clifford group is the normalizer of the Pauli group in , i.e., it is the group of unitary operators satisfying . The local Clifford group is the subgroup of and consists of all -fold tensor products of elements in . The Clifford group is generated by a simple set of quantum gates: the Hadamard gate , the phase gate and the CNOT gate gaitan (). Using the well-known representations of the Pauli matrices in the computational basis, it is straightforward to show that the action of on such matrices reads

(4)

The action of the phase gate on , and is given by,

(5)

Finally, the CNOT gate leads to the following transformations rules,

(6)

Observe that the CNOT gate propagates bit flip errors from the control to the target, and phase errors from the target to the control. As a side remark, we stress that another useful two-qubit gate is the controlled-phase gate . The controlled-phase gate has the following action on the generators of ,

(7)

We observe that a controlled-phase gate does not propagate phase errors, though a bit-flip error on one qubit spreads to a phase error on the other qubit.

We also point out that a unitary operator that fixes the stabilizer group  (we refer to daniel-phd () for a detailed characterization of the quantum stabilizer formalism in QEC) of a quantum stabilizer code under conjugation is an encoded operation. In other words, is an encoded operation that maps codewords to codewords whenever . In particular, if (every element of can be written as for some ) and is a codeword stabilized by every element in , then is stabilized by every stabilizer element in .

ii.2.2 Transformations on graphs

If there exists a local unitary (LU) transformation such that , the states and will have the same entanglement properties. If and are graph states, we say that their corresponding graphs and will then represent equivalent quantum codes, with the same distance, weight distribution, and other properties. Determining whether two graphs are LU-equivalent is a difficult task, but a sufficient condition for equivalence was given in hein (). Let the graphs and on vertices correspond to the -qubit graph states and . We define the two unitary matrices,

(8)

where , and and are Pauli matrices. Given a graph , corresponding to the graph state , we define a local unitary transformation ,

(9)

where is any vertex, is the neighborhood of , and means that the transform should be applied to the qubit corresponding to vertex . Given a graph , if there exists a finite sequence of vertices such that , then and are LU-equivalent hein (). It was discovered by Hein et al. and by Van den Nest et al. that the sequence of transformations taking to can equivalently be expressed as a sequence of simple graph operations taking to . In particular, it was shown in bart () that a graph determines uniquely a graph state and two graph states ( and ) determined by two graphs ( and ) are equivalent up to some local Clifford transformations iff these two graphs are related to each other by local complementations (LCs). The concept of LC was originally introduced by Bouchet in france (). A LC of a graph on a vertex refers to the operation that in the neighborhood of we connect all the disconnected vertices and disconnect all the connected vertices. All the graphs on up to vertices have been classified under LCs and graph isomorphisms parker (). In summary, the relation between graphs and quantum codes can be rather complicated since one graph may provide inequivalent codes and different graphs may provide equivalent codes. However, it has been established that the family of codes given by a graph is equivalent to the family of codes given by a local complementation of that graph.

As pointed out earlier, unitary operations in the local Clifford group act on graph states . However, there exists also graph theoretical rules, transformations acting on graphs, which correspond to local Clifford operations. These operations generate the orbit of any graph state under local Clifford operations. The LC orbit of a graph is the set of all non-isomorphic graphs, including itself, that can be transformed into by any sequence of local complementations and vertex permutations. The transformation laws for a graph state and a graph stabilizer under local unitary transformations read,

(10)

respectively. Neglecting overall phases, it turns out that local Clifford operations are just the symplectic transformations of which preserve the symplectic inner product moor (). Therefore, the -matrices satisfy the relation where T denotes the transpose operation and is the -matrix that defines a symplectic inner product in ,

(11)

Furthermore, since local Clifford operations act on each qubit separately, they have the additional block structure

(12)

where the -blocks , , , are diagonal. It was shown in bart () that each binary stabilizer code is equivalent to a graph code. In particular, each graph code characterized by the adjacency matrix corresponds to a stabilizer matrix and transpose stabilizer (generator matrix) . The generator matrix for a graph state with adjacency matrix reads,

(13)

where,

(14)

Observe that in order to have properly defined generators matrices in Eq. (13), must be nonsingular and must have vanishing diagonal elements. The graphical analog of the transformation law in Eq. (14) was provided in bart (). Before stating this result, some additional terminology awaits to be introduced.

Two vertices and of a graph are called adjacent vertices, or neighbors, if . The neighborhood of a vertex is the set of all neighbors of . A graph which satisfies and is a subgraph of and one writes . For a subset of vertices, the induced subgraph is the graph with vertex set and edge set . If has an adjacency matrix , its complement is the graph with adjacency matrix , where is the -matrix which has all ones, except for the diagonal entries which are zero. For each vertex ,…, , a local complementation sends the -vertex graph to the graph which is obtained by replacing the induced subgraph by its complement. In other words,

(15)

where has a on the th diagonal entry and zeros elsewhere and is a diagonal matrix such that yields zeros on the diagonal of . Finally, the graphical analog of Eq. (14) becomes,

(16)

with,

(17)

and diagdiag. Observe that substituting (17) in (14) and using (15), Eq. (16) gives

(18)

that is,

(19)

The translation of the action of local Clifford operations on graph states into the action of local complementations on graphs as presented in Eq. (16) is a major achievement of bart ().

ii.3 The CWS-work

CWS codes include all stabilizer codes as well as several nonadditive codes. However, for the sake of completeness, we point out that there are indeed quantum codes that cannot be recast within the CWS framework as pointed out in cross () and shown in ruskai (). CWS codes in standard form can be specified by a graph and a (nonadditive, in general) classical binary code . The vertices of the graph correspond to the qubits of the code and its adjacency matrix is . Given the graph state and the binary code , a unique base state and a set of word operators are specified. The base state is a single stabilizer state stabilized by the word stabilizer , a maximal Abelian subgroup of the Pauli group .

Let denote a quantum code on qubits that encodes dimensions with distance . Following cross (), it can be shown that a codeword stabilized code with word operators with and codeword stabilizer is locally Clifford equivalent to a codeword stabilized code with word operators ,

(20)

and codeword stabilizer ,

(21)

where s are codewords defining the classical binary code and is the th row vector of the adjacency matrix of the graph . For the sake of clarity, we stress that in Eq. (21) is the notational shorthand for

(22)

where is a binary -vector. Thus, any CWS code is locally Clifford equivalent to a CWS code with a graph-state stabilizer and word operators consisting only of s. Moreover, the word operators can always be chosen to include the identity. Eqs. (20) and (21) characterize the so-called standard form of a CWS quantum code. For a CWS code in standard form, the base state is a graph state. Furthermore, the codespace of a CWS code is spanned by a set of basis vectors which result from applying the word operators on the base state ,

(23)

Therefore, the dimension of the codespace equals the number of word operators. These operators are Pauli operators in that anticommute with one or more of the stabilizer generators for the base state. Thus, word operators map the base state onto an orthogonal state. The only exception is that in general the set of word operators also includes the identity operator so that the base state is a codeword of the quantum code as well. These basis states are also eigenstates of the stabilizer generators, but with some of the eigenvalues differing from . In addition, it turns out that a single qubit Pauli error , or acting on a codeword of a CWS code in standard form is equivalent up to a sign to another multi-qubit error consisting of s. Therefore, since all errors become s, the original quantum error model is transformed into a classical (induced by the CWS formalism) error model characterized, in general, by multi-qubit errors. The map that defines this transformation reads,

(24)

where denotes the set of Pauli errors , is the th row of the adjacency matrix for the graph and is the th bit of the vector . Finally, it was shown in cross () that any stabilizer code is a CWS code. Specifically, a quantum stabilizer code (where the parameters , , denote the length, the dimension and the distance of the quantum code, respectively) with stabilizer where with denote the stabilizer generators and logical operations ,…, and ,…, is equivalent to a CWS code defined by,

(25)

and word operators ,

(26)

The vector denotes a -bit string and with is the th bit of the vector . For further details on binary CWS quantum codes, we refer to cross (). Finally, for a very recent investigation on the symmetries of CWS codes, we refer to tqc ().

Iii From graphs to stabilizer codes and vice-versa

In this section, we revisit some basic ingredients of the Schlingemann-Werner work (SW-work, werner ()), the Schlingemann work (S-work, dirk ()) and, finally, the Van den Nest et al. work (VdN-work, bart ()). We focus on those aspects of these works that will be especially relevant for our proposed scheme.

iii.1 The Schlingemann-Werner work

The basic graphical construction of quantum codes within the SW-work werner () can be described as follows. Quantum codes are completely characterized by a unidirected graph characterized by a set of vertices and a set of edges specified by the coincidence matrix with both input and output vertices and a finite Abelian group with a nondegenerate symmetric bicharacter . We remark that there are various types of matrices that can be used to specify a given graph (for instance, incidence and adjacency matrices die ()). The coincidence matrix introduced in werner () is simply the adjacency matrix of a graph with both input and output vertices (and, it should not be confused with the so-called incidence matrix of a graph). The sets of input and output vertices will be denoted by and , respectively. Let be any finite additive Abelian group of cardinality with the addition operation denoted by and null element . A nondegenerate symmetric bicharacter is a map satisfying the following properties partha (): (i) , , ; (ii) , , , ; (iii) . If (the cyclic group of order ) with addition modulo as the group operation, the bicharacter can be chosen as

(27)

with , . The encoding operator of an error correcting code is an isometry (a bijective map between two metric spaces that preserve distances),

(28)

where is the -fold tensor product with (the Hilbert space is realized as the space of integrable functions over ) and in the qubit case. Similarly, is the -fold tensor product . The Hilbert space is defined as,

(29)

with scalar product between two elements and in given by,

(30)

The action of on is defined as werner (),

(31)

where , the integral kernel of the isometry , is given by werner (),

(32)

The product in Eq. (32) must be taken over each two elementary subsets in . Substituting Eq. (32) into Eq. (31), the action of on finally becomes,

(33)

We recall that the sequential steps of a QEC cycle can be described as follows,

(34)

that is,

(35)

Furthermore, the traditional Knill-Laflamme error-correction conditions read,

(36)

where the multiplicative factor does not depend on the states and . The graphical analog of Eq. (36) is given by,

(37)

for all operators in , the set of all operators in which are localized in . Thus, operators in are given by the tensor product of an arbitrary operator on with the identity on . A graph code corrects errors if and only if it detects all error configurations with . Given this graphical construction of the encoding operator in Eq. (33) and the graphical quantum error-correction conditions in Eq. (37), the main finding provided by Schlingemann and Werner can be restated as follows: given a finite Abelian group and a weighted graph , an error configuration is detected by the quantum code if and only if given that

(38)

then,

(39)

with . In general, the condition  is a set of equations, one for each integration vertex : for each vertex , we have to sum the  for all vertices  connected to , and equate it to zero. Furthermore, we underline that the fact that  is an isometry is equivalent to the detection of zero errors. In graph-theoretic terms, the detection of zero errors requires that  implies . A code that satisfies Eq. (39) given Eq. (38) can be either nondegenerate or degenerate. We shall assume that Eq. (39) with the additional constraints in Eq. (38) denotes the weak version (necessary and sufficient conditions) of the graph-theoretic error detection conditions. However, sufficient graph-theoretic error detection conditions can be introduced as well. Specifically, an error configuration is detectable by a quantum code if,

(40)

We shall denote conditions in Eq. (40) without any additional set of graph-theoretic constraints (like the ones provided in Eq. (38)) the strong version (sufficient conditions) of the graph-theoretic error detection conditions. We finally emphasize, as originally pointed out in werner (), that a code that satisfies Eq. (40) is nondegenerate.

iii.2 The Schlingemann-work

Schlingemann was able to show that stabilizer codes, either binary or nonbinary, are equivalent to graph codes (and vice-versa). However, as far as our proposed scheme concerns, the main finding uncovered in the S-work dirk () may be stated as follows. Consider a graph code with only one input and -output vertices. Its corresponding coincidence matrix can be written as,

(41)

where denotes the -symmetric adjacency matrix . Then, the graph code with symmetric coincidence matrix in Eq. (41) is equivalent to stabilizer codes being associated with the isotropic subspace defined as,

(42)

that is, omitting unimportant phase factors, with the binary stabilizer group ,

(43)

Observe that a stabilizer operator for an -vertex graph has a -dimensional binary vector space representation such that .

More generally, consider a binary quantum stabilizer code associated with a graph characterized by the symmetric coincidence matrix ,

(44)

To attach the input vertices, has to be constructed in such a manner that the following conditions are satisfied: i) first, (); ii) second, the matrix must define a -dimensional subspace in spanned by linearly independent binary vectors of length not included in the Span of the raw-vectors defining the symmetric adjacency matrix ,

(45)

where for and for ; iii) third, Span contains a vector such that for any . Condition i) is needed to avoid disconnected graphs. Condition ii) is required to have a properly defined isometry capable of detecting zero errors. Finally, condition iii) is needed to generate an isotropic subspace (or, in other words, an Abelian subgroup of the Pauli group, the so-called stabilizer group) with,

(46)

for any pair in where the symbol denotes the symplectic product gaitan ().

As a final remark, we point out that in a more general framework like the one presented in dirk (), we could consider three types of vertices: input, auxiliary and output vertices. The input vertices label the input systems and are used for encoding. The auxiliary vertices are inputs used as auxiliary degrees of freedom for implementing additional constraints for the protected code subspace. Finally, output vertices simply label the output quantum systems.

iii.3 The Van den Nest-work

The main achievement of the VdN-work in bart () is the construction of a very useful algorithmic procedure for transforming any binary quantum stabilizer code into a graph code. Before describing this procedure, we remark that it is straightforward to check that a graph code given by the adjacency matrix corresponds to a stabilizer matrix and transpose stabilizer . That said, consider a quantum stabilizer code with stabilizer matrix,

(47)

and transpose stabilizer given by,

(48)

Let us define in Eq. (47). Given a set of generators of the stabilizer, the stabilizer matrix  is constructed by assembling the binary representations of the generators as the rows of a full rank -matrix. The transpose of the binary stabilizer matrix (i.e., the transpose stabilizer)   is simply the full rank -matrix obtained from  after exchanging rows with columns. The goal of the algorithmic procedure is to convert the transpose stabilizer in Eq. (48) of a given stabilizer code into the transpose stabilizer of an equivalent graph code. Then, the matrix will represent the adjacency matrix of the corresponding graph. Two scenarios may occur: i) is a invertible matrix; ii) is not an invertible matrix. In the first scenario where is invertible, a right-multiplication of the transpose stabilizer by will perform a basis change, an operation that provides us with an equivalent stabilizer code,

(49)

Then, the matrix will denote the resulting adjacency matrix of the corresponding graph. Furthermore, if the matrix has nonzero diagonal elements, we can simply set these elements to zero in order to satisfy the standard requirements for a correct definition of an adjacency matrix of simple graphs. In the second scenario where is not invertible, we can always find a suitable local Clifford unitary transformation such that bart (),

(50)

and,

(51)

with . Therefore, right-multiplying with , we get

(52)

Thus, the adjacency matrix of the corresponding graph becomes .

The above-described algorithmic procedure for transforming any binary quantum stabilizer code into a graph code is very important for our proposed scheme as it will become clear in the next section.

Iv The scheme

In this section, we formally describe our scheme and apply it to the graphical construction of the Leung et al. four-qubit quantum code for the error correction of single amplitude damping errors.

iv.1 Description of the scheme

We emphasize that our ultimate goal is the construction of classical graphs with both input and output vertices defined by the coincidence matrix in order to verify the error-correcting capabilities of the corresponding quantum stabilizer codes via the graph-theoretic error correction conditions advocated in the SW-work. To achieve this goal, we propose a systematic scheme based on a very simple idea. The CWS-, VdN- and S-works must be combined in such a manner that, with respect to our ultimate goal, the weak-points of one method should be compensated by the strong-points of another method.

iv.1.1 Step one

The CWS formalism offers a very general framework where both binary/nonbinary and/or additive/nonadditive quantum codes can be described. For this reason, the starting point of our scheme is the realization of binary stabilizer codes as CWS quantum codes. Although this is a relatively straightforward step, the CWS code that one obtains is not, in general, in the standard canonical form. From the CWS-work in cross (), it is known that there does exist a local (unitary) Clifford operations that allows in principle to write down the CWS code that realizes the binary stabilizer code in standard form. However, the CWS-work does not suggest any algorithmic procedure to achieve this standard form. In the absence of a systematic procedure, uncovering a local Clifford unitary such that (every element can be written as for some ) may constitute a very tedious challenge. Fortunately, we can avoid this. Before explaining how, let us introduce the codeword stabilizer matrix corresponding to the codeword stabilizer .

iv.1.2 Step two

Two main achievements of the VdN-work in bart () are the following: first, each stabilizer state is equivalent to a graph state under local Clifford operations; second, an algorithmic procedure for transforming any binary quantum stabilizer code into a graph code is provided. Observe that a stabilizer state can be regarded as a quantum code with parameters . Our idea is to exploit the algorithmic procedure provided by the VdN-work by translating the starting point of the algorithmic procedure in the CWS language. To achieve this, we replace the generator matrix of the stabilizer state with the codeword stabilizer matrix corresponding to the codeword stabilizer of the CWS code that realizes the binary stabilizer code whose graphical depiction is being sought. This way, we can simply apply the VdN algorithmic procedure to uncover the standard form of the CWS code and, if necessary, the explicit expression for the local (unitary) Clifford operation that links the non-standard to the standard forms of the CWS code. After applying this VdN algorithmic procedure adapted to the CWS formalism, we can construct a graph characterized by a symmetric adjacency matrix with only output vertices. How do we attach possible input vertices to this graph associated with the binary stabilizer codes with ?

iv.1.3 Step three

Unlike the VdN-work whose findings are limited to the binary quantum states, the S-work extends its applicability to both binary and nonbinary quantum codes. In particular, in dirk () it was shown that any stabilizer code is a graph code and vice-versa. However, in the S-work an analog of the algorithmic procedure for transforming any binary quantum stabilizer code into a graph code is missing. Despite this fact, the S-work does provide a very useful result for our proposed scheme. Namely, it is shown that a graph code with associated graph with both input and output vertices and corresponding symmetric coincidence matrix is equivalent to stabilizer codes being associated with a suitable isotropic subspace space . Recall that at the end of the above-mentioned step two, we are basically given both the isotropic subspace and the graph without input vertices, that is the symmetric adjacency matrix embedded in the more general coincidence matrix . Therefore, by exploiting the just mentioned very useful specific finding of the S-work in a reverse direction (we are allowed to do so since a graph code is equivalent to a stabilizer code and vice-versa), in some sense, we can construct the full coincidence matrix and finally attach the input vertices to the graph. What can we do with a graphical depiction of a binary stabilizer code?

iv.1.4 Step three+one

In the SW-work, outstanding graphical QEC conditions were introduced werner (). However, these conditions were only partially employed for quantum codes associated with graphs and the codes needed not be necessarily stabilizer codes. By logically combining the CWS-, VdN- and S-works, the power of the graphical QEC conditions in werner () can be fully exploited in a systematic manner in both directions: from graph codes to stabilizer codes and vice-versa.

In summary, given a binary quantum stabilizer code , the systematic procedure that we propose can be described in points as follows:

  • Realize the stabilizer code as a CWS quantum code ;

  • Apply the VdN-work adapted to the CWS formalism to identify the standard form of the CWS code that realizes the stabilizer code whose graphical depiction is being sought. In other words, find the graph with only output vertices characterized by the symmetric adjacency matrix associated with in the standard form;

  • Exploit the S-work as explained to identify the extended graph with both input and output vertices characterized by the symmetric coincidence matrix associated with the isometric encoding map that defines ;

  • Use the SW-work to apply the graph-theoretic error-correction conditions to the extended graph in order to explicitly verify the error-correcting capabilities of the corresponding realized as a quantum code.

iv.2 Application of the scheme

We think there is no better way to describe and understand the effectiveness of our proposed scheme than by simply working out in detail a simple illustrative example. In what follows, we wish to uncover the graph associated with the Leung et al. four-qubit stabilizer (nondegenerate) quantum code debbie (). Several explicit constructions of graphs for various stabilizer codes characterized by either single or multi-qubit encoding operators are added in the Appendices: the three-qubit repetition code, the perfect -erasure correcting four-qubit code, the perfect -error correcting five-qubit code, -error correcting six-qubit quantum degenerate codes, the CSS seven-qubit stabilizer code, the Shor nine-qubit stabilizer code, the Gottesman -error correcting eleven-qubit code,  stabilizer codes, and, finally, the Gottesman  stabilizer code.

iv.2.1 Step one

Recall that the stabilizer of the Leung et al. code is given by fletcher (),

(53)

with a suitable logical operation given by . Therefore, when regarded within the CWS framework cross (), the Leung et al. code is equivalent to a CWS code defined with codeword stabilizer,

(54)

iv.2.2 Step two

Taking into consideration Eq. (54), we observe that is local Clifford equivalent to given by,

(55)

with where denotes the Hadamard transformation. We notice that the codeword stabilizer matrix associated with the codeword stabilizer reads,

(56)

with . Therefore, we can find a suitable graph with output vertices only that is associated with the Leung et al. code by applying the VdN algorithmic procedure. The transpose of becomes,