Symmetries of Codeword Stabilized Quantum Codes1footnote 11footnote 1This work was partially supported by NSERC, CIFAR, and IARPA.

Symmetries of Codeword Stabilized Quantum Codes111This work was partially supported by NSERC, CIFAR, and IARPA.

Salman Beigi School of Mathematics, Institute for Research in Fundamental Sciences (IPM)
Niavaran Square, Tehran, Iran
salman.beigi@gmail.com
Jianxin Chen Markus Grassl Centre for Quantum Technologies, National University of Singapore
3 Science Drive 2, Singapore 117543
Markus.Grassl@nus.edu.sg
Zhengfeng Ji Institute for Quantum Computing
200 University Avenue West, Waterloo, Ontario, Canada
jizhengfeng@gmail.com
Qiang Wang School of Mathematics and Statistics, Carleton University
1125 Colonel By Drive, Ottawa, Ontario, Canada
wang@math.carleton.ca
Bei Zeng
Abstract

Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code can be represented by a self-dual additive code and a classical code , i. e., , however this representation is in general not unique. We show that for any CWS code with certain permutation symmetry, one can always find a self-dual additive code with the same permutation symmetry as such that . As many good CWS codes have been found by starting from a chosen , this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form has been considered, where is a graph state. We analyze the symmetry of the corresponding graph of , which in general cannot possess the same permutation symmetry as . We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.

CWS Codes, Union Stabilizer Codes, Permutation Symmetry, Toric Code
\Copyright

[by]S. Beigi, J. Chen, M. Grassl, Z. Ji, Q. Wang & B. Zeng\subjclassE.4 Coding and Information Theory\serieslogo\EventShortNameTQC 2013 \DOI10.4230/LIPIcs.xxx.yyy.p

1 Introduction

Coding theory is an important component of information theory having a long history dating back to Shannon’s seminal 1948 paper that laid the ground for information theory [21]. Coding theory is at the heart of reliable communication, where codes with symmetry, especially cyclic codes, such as the Reed-Solomon codes, are among the most widely used codes in practice [19].

In recent years, it has become evident that quantum communication and computation offer the possibility of secure and high rate information transmission, fast computational solution of certain important problems, and efficient physical simulation of quantum phenomena. However, quantum information processing depends on the identification of suitable quantum error-correcting codes (QECC) to make such processes and machines robust against faults due to decoherence, ubiquitous in quantum systems. Quantum coding theory has hence been extensively developed during the past 15 years [3, 9, 20].

Codeword stabilized (CWS) quantum codes are by far the most general construction of QECC [6]. A CWS code can be represented by a stabilizer state (i. e. a self-dual additive code) and a classical code , i. e. . When is a linear code, the corresponding CWS code is actually a stabilizer code. Also, any CWS code is local Clifford equivalent to a standard form , where is a graph state [6].

The CWS construction encompasses stabilizer (additive) codes and all the known non-additive codes with good parameters. It also leads to many new codes with good parameters, or good algebraic/combinatorial properties, through both analytical and numerical methods. Alternative perspectives of CWS codes have also been analyzed, including the union stabilizer codes (USt) method [11, 12], and the codes based on graphs [18, 23]. Concatenated codes and their generalizations using CWS codes have been developed [1], and decoding methods for CWS codes have been studied as well [17].

Given all the evidence that the CWS framework is a powerful method to construct and analyze QECC, it remains unclear to what extent the stabilizer state and the classical code can represent the symmetry of the CWS code in general. Given the vital importance that the code symmetry plays in coding theory, this understanding becomes crucial since if such a correspondence exists, it can provide practical methods for constructing CWS codes with desired symmetry from and/or with corresponding symmetry.

Unfortunately, there is no immediate clue what answer one can hope for. First of all, the representation is not unique. So for a given CWS code , there might be some stabilizer states and/or classical codes which are more symmetric than others. Perhaps the best known example is the CWS representation for the five-qubit code , where in the ideal case can be chosen as a graph state corresponding to the pentagon graph, and the is chosen as the repetition code . In this case, both and nicely represent the cyclic symmetry of the five-qubit code.

However, there are known ‘bad cases’, too. One example is the seven-qubit Steane code , where although the code itself is cyclic, one cannot find any corresponding to a cyclic graph, even if local Clifford operations are allowed [10]. Nonetheless, we know that the stabilizer group for this code is invariant under cyclic shifts, and the logical operator can be chosen as , therefore the logical can be chosen as a cyclic stabilizer code. This is to say, there exists a representation for such that is cyclic. In general it remains unclear under which conditions a representation for cyclic CWS code with a cyclic stabilizer state exists.

In this work, we address the symmetry properties of CWS codes. We are interested in the permutation symmetry of CWS codes, which includes the important category of cyclic codes. Our main question is, to which extent can the representation and the standard form reflect the symmetry of the corresponding CWS code . We show that for any CWS code with permutation symmetry, one can always find a stabilizer state with the same permutation symmetry as such that . As many good CWS codes are found by starting from a chosen , this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of with the same symmetry will suffice. A key step to reach this main result is to obtain a canonical representation for CWS codes, which is in terms of a unique decomposition as union stabilizer codes.

We know that for the standard form of CWS codes using graph states, it is not always possible to find a graph with the same permutation symmetry. This is partially due to the fact that the local Clifford operation transforming the CWS code into the standard form may break the permutation symmetry of the original code. Also, the graphs usually can only represent the symmetry of the stabilizer generators of the stabilizer state, but not the symmetry of the stabilizer state in general. We show that this is indeed the case for the toric code on a two-dimensional square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.

However, we show that the converse always holds, i. e., any graph and classical code with certain permutation symmetry yields a CWS code with the same symmetry.

2 Preliminaries

The single-qudit (generalized) Pauli group is generated by the operators and acting on the qudit Hilbert space , satisfying , where . For simplicity, throughout the paper, we assume that is a prime, although our results naturally extend to prime powers. Denote the computational basis of by . Then, without loss of generality, we can fix the operators and such that and , respectively. Let be the identity operator. The set of operators forms a so-called nice unitary error basis which is a particular basis for the vector space of matrices [15, 16].

The -qudit Pauli group consists of all local operators of the form , where for some integer is an overall phase factor, and for some , is an element of the single-qudit Pauli group of qudit . We can write as or when it is clear what the qudit labels are. The weight of an operator is the number of tensor factors that differ from identity.

The -qudit Clifford group is the group of unitary matrices that map to itself under conjugation. The -qudit local Clifford group is a subgroup in containing elements of the form , where each is a single qudit Clifford operation, i. e., .

A stabilizer group in the Pauli group is defined as an abelian subgroup of which does not contain . A stabilizer consists of Pauli operators for some . As the operators in a stabilizer commute with each other, they can be simultaneously diagonalized. The common eigenspace of eigenvalue is a stabilizer quantum code with length , dimension , and minimum distance . The projection onto the code can be expressed as

(1)

The centralizer of the stabilizer is given by the elements in which commute with all elements in . For , the minimum distance of the code is the minimum weight of all elements in .

If , then there exists a unique -qudit state such that for every . Such a state is called a stabilizer state, and the group is called the stabilizer of . A stabilizer state can also be viewed as a self-dual code over the finite field under the trace inner product [7]. For a stabilizer state, the minimum distance is defined as the minimum weight of the non-trivial elements in  [7].

A union stabilizer (USt) code of length is characterized by a stabilizer code with stabilizer , where are independent generators, and a classical code over of length . Note that for a given , the choice of the generators is not unique. Now for a classical code of length with codewords, for each codeword , the corresponding quantum code is given by the subspace stabilized by , , …, . Note that for , the subspaces and are mutually orthogonal. The corresponding USt code is then given by the subspace .

Therefore, the combination of (more precisely, the generators of ) and gives an USt quantum code . Hence we denote a USt code by . The projection onto can be expressed as

(2)

where we identify the elements of the finite field with integers modulo .

A CWS code of length is a USt code with . That is, it is characterized by a stabilizer state with stabilizer and a classical code of length . For a CWS code given by , the stabilizer always corresponds to a unique stabilizer state. We will then refer to as the stabilizer state when no confusion arises.

For a CWS code, the projection onto the code space is given by

(3)

where we again identify the elements of the finite field with integers modulo .

A CWS code has a permutation symmetry if

(4)

where is the projection onto the space obtained by permuting the qudits of the code according to .

3 Canonical form of CWS codes

For a given a CWS code , there might exist another stabilizer state and another classical code such that . In other words, the representation of a CWS code by the stabilizer state and the classical code is non-unique.

In order to discuss the relationship between the symmetry of the CWS code and that of the stabilizer state , we first need to explore the relationship between the different representations of (i. e., the relationship between and , as well as the relationship between and ).

Let us start by recalling that a stabilizer code can be viewed as a CWS code where the classical code is a linear code [6]. A simple way to see this is that for a given stabilizer code with stabilizer generated by , which is a code of dimension , we can choose the larger stabilizer , where mutually commute. Now choose the classical code of length with codewords, where the first coordinates of each codeword are zero. Then we have , i. e., the stabilizer code can then be viewed as a CWS code with stabilizer state and classical code . However, note that the choice of (and hence ) is non-unique, as in particular the choice of is non-unique.

{example}

As an example, consider the five-qubit code with stabilizer

(5)

In the CWS picture, the stabilizer state can be chosen as

(6)

where is the logical operator. Alternatively, one can choose the stabilizer state

(7)

where is the logical operator. For both and , the classical code can be chosen as .

Similarly, a USt code can be viewed as a CWS code with the classical code of length possessing some coset structure, i. e., , where is a linear code. This linear code of length can be readily chosen as the classical code for the CWS representation of the stabilizer code . The code of length can be derived from of length by appending zero coordinates. However, again, the choices of and are non-unique.

In the general situation, we have some freedom in choosing the stabilizer state when representing a stabilizer code or a USt code in the CWS framework. Consequently, for a given CWS code , there are also many different ways to write it in terms of a USt code in general. We will show, however, that we can always obtain a unique stabilizer , when expressing a given CWS code as a USt code. The following theorem gives a canonical form for any CWS code.

{theorem}

Every CWS code has a unique representation as a union stabilizer code.

Proof.

To prove this theorem, we will need some lemmas.

{lemma}

[translational invariant codes] Let be a code over with and assume that for some non-zero we have , i. e., the code is invariant with respect to translation by . Then can be written as a disjoint union of cosets of the one-dimensional space generated by , i. e.,

where with .

Proof.

By assumption, for every , the vector is in the code as well. Hence we can arrange the elements of as follows:

Every column in this arrangements is a coset . ∎

{lemma}

[vanishing character sum] Let be an arbitrary code of length . Assume that the function

where , vanishes outside a proper subspace . Then there exists a non-zero vector such that . What is more, the code can be written as a union of cosets of the linear code , i. e.,

(8)
Proof.

Let denote the characteristic function of the code , i. e., , and if and only if . Define . Then .

The Fourier transform of over reads

where if , and otherwise.

This shows that for , the Fourier transform is proportional to the function , and hence vanishes outside of as well. Recall that that , as is a proper subspace by assumption. Let be a non-zero vector that is orthogonal to all vectors in . Furthermore, let denote the set-complement of in the full vector space.

We want to show that the code is invariant with respect to translations by , i. e., or equivalently, . This is in turn equivalent to showing that . In the following, denotes the inverse Fourier transform:

Here we have used the fact that vanishes outside of and that is orthogonal to all vectors in .

From Lemma 3, it follows that the code can be written as a union of cosets of the code generated by all vectors that are orthogonal to . ∎

Now we are ready to prove Theorem 3. Let denote the projection operator onto a CWS code , i. e.

(9)

where are the generators of the stabilizer, and is a classical code.

First note that the coefficients in (3) are uniquely determined since the operators are a subset of the error-basis of linear operators on the space . The coefficient is proportional to . On the other hand, , where is the function appearing in Lemma 3. So if the coefficients vanish outside of a proper subspace , the classical code can be decomposed as union of cosets of . Then (3) can be re-written as follows:

(10)

In the last step we have used the fact that the spaces and are orthogonal to each other, i. e., the inner product vanishes. Now assume that the space has dimension and that is a basis of . Then every vector can be expressed as . For every we define the vectors with , forming another classical code . Further, we define the operators . This allows us to express (3) as

(11)

Hence, whenever the classical code associated to a CWS code has some non-trivial shift invariance, the projection onto a CWS code can be expressed as a projection onto a USt code (cf. (2)), thereby increasing the dimension of the underlying stabilizer code and reducing the size of the classical code. In order to obtain a unique representation, we may assume that the stabilizer code is of maximal dimension, and hence the classical code is “without any linear structure.”

In order to show uniqueness, consider the coefficients of the expansion of the projection in terms of the operator basis formed by the -qudit Pauli matrices . Clearly, we have . If the group was a proper subgroup of , the coefficients would vanish for outside a proper subspace , contradicting the assumption the classical code has no linear structure.

Note that the stabilizer is only unique up to the choice of some phase factors of the error basis. For example, replacing by will introduce some phase factor which has to be compensated by changing the first coordinate of the codewords of the classical code . To finally fix the degree of freedom, we can enforce , with for and . ∎

4 Symmetries of the stabilizer state of a CWS code

We are now ready to discuss the relationship between the symmetries of the CWS code and that of the corresponding stabilizer state .

{theorem}

For any CWS code with permutation symmetry , there exists a stabilizer state with the same permutation symmetry such that .

Proof.

To prove this theorem, we will need some lemmas. {lemma} If the projection operator given in Eq. (3) is invariant under a permutation of the qudits, then the stabilizer code related to expressing in terms of a USt code as in Eq. (11) is invariant with respect to the permutation as well.

Proof.

The statement follows directly from the uniqueness of the stabilizer group generated by the operators in Eq. (11). ∎

We now prove a lemma for a special case of Theorem 4, when the CWS code is a Calderbank–Shor–Steane (CSS) code [4, 22].

{lemma}

For a CSS code with permutation symmetry , there exists a stabilizer state such that has the same permutation symmetry as .

Proof.

For a CSS code , the stabilizer generators can always be chosen such that every generator is either a tensor product of powers of (denoted by ) or a tensor product of powers of (denoted by ). We can use the following matrix form:

As the permutation symmetry of does not change the type of an operator, both and have necessarily the same symmetry . Furthermore, the logical operators can also be chosen as either tensor products of powers of or tensor products of powers of , which correspond to the dual of the classical codes associated to either the stabilizers or the stabilizers, respectively. Without loss of generality let us choose a set of commuting logical operators which are all of type. Then group generated by the set of mutually commuting operators is again invariant under the permutation . As the stabilizer group is maximal, it stabilizes a unique state . Hence is the stabilizer state with the desired symmetry, and the CSS code can be expressed as CWS code in terms of and some classical code . ∎

We now prove a lemma for the stabilizer code case of Theorem 4, which improves the result of Lemma 4.

{lemma}

For a stabilizer code with permutation symmetry , there exists a stabilizer state such that has the same permutation symmetry as .

Proof.

To prove this lemma, we shall use a standard form for stabilizers (see [20, Section 10.5.7]):

where is an matrix, is an matrix, is an matrix, is an matrix, is an matrix, and is an matrix. Similar as in the CSS case, we can choose a set of commuting logical operators which are all of type. In matrix form, they are given by . Then the group generated by the mutually commuting operators in stabilizes a unique state which is invariant with respect to the permutation . Hence is the stabilizer state with the desired symmetry that can be used to express as CWS code with some classical code . ∎

To prove Theorem 4, given a CWS code , we first find its unique decomposition as a USt code , based on Theorem 3. Here is in general a stabilizer code with generators. If has a permutation symmetry , then according to Lemma 4, the stabilizer code must also have the symmetry . Now according to Lemma 4, there exists a quantum state in the stabilizer code which also has the symmetry . Hence is the stabilizer state with the desired symmetry. Note that the stabilizer of the state contains the original stabilizer . Therefore, common eigenspaces of are further decomposed into one-dimensional joint eigenspaces of , and we can rewrite the projection onto the USt code in the form corresponding to a CWS code. ∎

5 Symmetries of the Classical Code

Theorem 4 does not make any statement about the symmetry of the classical code. In general, if we insist to use the canonical form of the CWS code as given by Theorem 3, we cannot expect that the (non-linear) classical code associated with the CWS code has the same symmetry as . That is, in this case, even if the stabilizer has the same permutation symmetry as the quantum code , one might not be able to find a classical code with the same symmetry in general. Let us look at an example.

{example}

Consider the stabilizer state (hence a CWS code, denoted by ), which is invariant under all permutations. Using the canonical form of as given by Theorem 3, the group is generated by and all pairs of , which is permutation invariant. However, the classical code consists of the vector which is one in the first coordinate and zero elsewhere, i. e., is a code with a single codeword , which has a smaller symmetry group than that of .

On the other hand, if we choose the group generated by and all pairs of , the corresponding classical code consists just of the zero vector. So in the representation , both and have the same permutation symmetries as .

This example indicates that exploiting the phase factor freedom in the USt code decomposition of a CWS code, and thereby deviating slightly from the canonical form, there is some chance to find both a stabilizer and a classical code with the same permutation symmetry as the CWS code.

To study the properties of the classical code associated with a CWS code , consider the case where the stabilizer state has some permutation symmetry . Then for given generators of the stabilizer , the permuted operators generate the same stabilizer . The transformation can be characterized by a -valued, invertible matrix given by

(12)

Let us write the