1. Introduction
Quantum error-correcting codes provide a key approach to a scalable quantum computer that is resilient against decoherence and operational noise. Quantum errors can be expressed in terms of unitary error bases [
1,
2,
3,
4,
5,
6,
7,
8,
9]. A particularly useful class of unitary error bases, called nice error bases, was introduced by Knill in [
10], which is the foundation of the theory of quantum error-correcting codes [
11]. Almost all of the quantum codes constructed so far are stabilizer or additive codes [
3]. These are not the most general quantum codes, and there exist nonadditive quantum codes that are strictly better than any additive code [
12]. Up to now, all good codes known have fallen into the class of what have been called pure quantum codes [
11]. However, there are still some interesting and important problems about impure quantum codes [
13]. Nearly all known quantum codes are constructed over finite fields [
11]. However, it has been recognised that quantum codes can be constructed over finite rings as well [
14].
When a new physical problem occurs, it is always desirable to find an appropriate framework for it, such as quantum mechanics for quantum physics. Since the occurrence of quantum codes, almost all researches are carried out on the specific types of quantum codes, for example, mainly on stabilizer codes, pure codes and codes over finite fields. In this paper, we are mainly interested in a universal framework for quantum codes, i.e., a framework that applies to all codes, no matter whether they are pure or not, stabilizer codes or not, over finite field or not. Firstly, we recall the properties of nice error bases. Then, we give the definitions of the group algebra and characters associated with nice error basis. Finally, based on the group algebra, we establish a universal framework for quantum codes. Through the discussion we show that this framework can characterize the properties of quantum codes as well as generate some new results about quantum codes. It is a powerful tool for use in future works on quantum codes.
2. Preliminaries
Quantum information can be protected by encoding it into a quantum error-correcting code. An quantum code is a -dimensional subspace of the state space of quantum systems with levels that can detect all errors affecting less than quantum systems, but cannot detect some errors affecting quantum systems.
Let be an -level quantum system and let be an additive group of order with identity element 0. A nice error basis of is a set of unitary operators on such that
- (i)
is the identity operator,
- (ii)
for all nonzero ,
- (iii)
for all ,
where complex numbers have modulus 1. is called the index group of . Moreover, is a nice error basis of quantum systems .
Lemma 1. If the index group is Abelian, we havefor any nonzero . Proof. By (iii) in the definition of nice error bases, it follows that
and
which means
Now, let
. Then, from (1)
is a subgroup of a cyclic group since all
generate a cyclic group [
15]. Thus,
itself is a nontrivial cyclic group for any nonzero
. □
In the next section, we provide the concept of the group algebra based on the nice error basis with the Abelian index group. For simplicity, we assume throughout the paper that the index group
is Abelian. This assumption is reasonable since such nice error basis exists for any finite dimensional quantum system [
10].
3. Group Algebra
We are going to describe the elements of
by formal polynomials in
. In general,
is represented by
, which we abbreviate
. We create the convention that
. This forms the set of all
into a multiplicative group denoted by
. Thus,
and
are isomorphic groups, with addition in
corresponding to multiplication in
Definition 1. The group algebra of over the complex numbers consists of all formal sums Addition and multiplication of elements of
are defined in the natural way by
and
To each
we associate the mapping
from
to the complex numbers given by
is called a character of
.
is extended to act on
by linearity
Note that
Let
be an arbitrary element of the group algebra
, with the property that
Definition 2. The transform of is the element of given bywhere was defined above. Suppose
Then,
And
. Now, we describe several weight enumerators of the group algebra
. Let the elements of
be denoted by
, in some fixed order.
The first weight enumerator to be considered specifies the group algebra completely by introducing enough variables. In general, the variables
means that the
place in the vector
is the
element
of
. The vector
is described by the polynomial
Thus,
is uniquely determined by
. This requires the use of
variables
,
.
What we shall call the exact enumerator of
is then defined as
Then, the exact enumerator of
is
Proof. From (2) and (3), the LHS is equal to
which is equal to the RHS. □
The next weight enumerator to be considered classifies vectors in according to the number of times each group element appears in .
Definition 3. The composition of , denoted by , is where is the number of components equal to . Clearly We call the set
the complete weight distribution of
where
is the sum of
with
. We also define the complete weight enumerator of
to be
Then, the complete weight distribution of
is
, where
is the sum of
with
, and the complete weight enumerator of
is
Proof. Set for in Theorem 1. □
By setting certain variables equal to each other in the complete weight enumerator, we obtain the Lee and Hamming weight enumerators, which give progressively less and less information about the group algebra, but become easier to handle.
Definition 4. Suppose now that is odd, and let the elements of be labeled , where for . The Lee composition of a vector , denoted by , is where , for .
We call the set
the Lee weight distribution of
where
is the sum of
with
. We also define the Lee weight enumerator of
to be
Then the Lee weight distribution of
is
, where
is the sum of
with
, and the Lee weight enumerator of
is
Theorem 3. The Lee enumerator for the transform is obtained from the Lee enumerator of by replacing each byand dividing the result by . Proof. Set for in Theorem 2. □
The Hamming weight, or simply the weight, of a vector is the number of nonzero components , and is denoted by .
We call the set
the Hamming weight distribution of
where
is the sum of
with
. We also define the Hamming weight enumerator of
to be
Then, the Hamming weight distribution of
is
, where
is the sum of
with
, and the Hamming weight enumerator of
is
Proof. In Theorem 2, put , , and use Lemma 1. □
4. Universal Framework for Quantum Codes
In this section, we establish the universal framework for quantum codes based on the group algebra defined in the last section.
Given an arbitrary quantum code
, let
be the orthogonal projection onto
where
is an orthonormal basis of
, and let
be the index group of any nice error basis
of the quantum system with
levels. Then, we can formulize the quantum code
as an element
from the group algebra
where
We call
the element associated with the quantum code
in the group algebra
.
From (3), the transform of
is given by
where
where
. Since
, from (5), we obtain
. Thus,
From (4) and (6), and using the Cauchy–Schwartz inequality, we deduce that
for all
. Furthermore, from the definition of the minimum distance
we obtain that if
then
for all
satisfying
and there exist some
with
such that
; if
then
for all
and the minimum nonzero weight of
such that
is
.
So far we have established the universal framework for quantum codes:
For arbitrary quantum code we can characterize it as the element of the group algebra , called the element associated with the quantum code , and the transform of so that
- (1)
the dimension of equals where ,
- (2)
the minimum distance of equals the minimum weight of such that if ; the minimum nonzero weight of such that if .
5. Conclusions
The nicest thing about the framework is that we can characterize the properties of quantum codes by the properties of the group algebra. So the problems about unfamiliar quantum codes can be transformed into those about familiar classical group algebra.
For example, we can define the weight distributions of the quantum code
as the weight distributions of the element
associated with
in the group algebra
and define the dual weight distributions of
as the weight distributions of the transform
of
. Then, for any quantum code, its weight distributions and dual weight distributions must satisfy the identities in Theorems 1, 2, 3 and 4. Note that the results about exact enumerators, complete enumerators and Lee enumerators of quantum codes are completely new. For Hamming weight enumerators, the binary version was first proved for quantum stabilizer codes by Calderbank et al. in [
16], and later generalized by Rains in [
17]. The nonbinary version for stabilizer codes was proved by Ketkar et al. in [
18]. The result given here is a generalization to the most general quantum codes.
Again, the purity of quantum codes can also be characterized by the group algebra. For arbitrary quantum code , let be the element associated with in the group algebra . Then is pure if, and only if, for .
Finally if is a quantum stabilizer code, from the definition of stabilizer codes, the element associated with in the group algebra can be written as where the summation is over all such that the operator belongs to the stabilizer of . In addition, the transform of can be written as where the summation is over all such that the operator belongs to the normalizer of . Both forms imply the relationship between quantum stabilizer codes and classical codes.
Next, we will give some examples to illustrate the above conclusions. For qubits, it is well known that Pauli operators
constitute a nice error basis where
In our terminology, the underlying Abelian index group of this nice error basis is
satisfying
and
, and the nice error basis can be denoted as
where
.
Example 1. A nine-qubit code with a basis For this code, from (4) and (6), one can verify that for all with weight , for example, , and for some with weight , for example, but . Thus, the minimum distance . Moreover, since there exists nonzero with weight and , the code is impure. In fact, this code is the well-known stabilizer code [1]. Example 2. A seven-qubit code with a basis where the subscript “cyc” indicates that all cyclic shifts occur. For this code, from (4) and (6), one can verify that , for all nonzero with weight , and for some with weight , for example, but . Thus, the minimum distance . Moreover, since for all nonzero with weight , the code is pure. In fact, this code is the well-known CSS code [19]. Example 3. A five-qubit code with a basis together with all five cyclic shifts of For this code, from (4) and (6), one can verify that , for all nonzero with weight , and for some with weight , for example, but . Thus, the minimum distance . Moreover, since for all nonzero with weight , the code is pure. In addition, since , the code must be nonadditive by the above conclusion. In fact, this code is the first nonadditive code [12]. To sum up, we have presented a universal framework for quantum codes and shown how it characterizes the properties of quantum codes as well as generates new results about quantum codes. We can assert that this framework is a very useful and potential tool in studying the problems about quantum error-correcting codes. In fact, the idea of this framework has been used to solve the problem about the classification of perfect quantum codes [
20]. Recently, we realized that this tool might have potential application in finding out various bounds on the parameters of the most general quantum error-correcting codes. We plan to develop this application further, among others, in our future works.