Universal Framework for Quantum Error-Correcting Codes

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.


I. INTRODUCTION
The quest to build a scalable quantum computer that is resilient against decoherence errors and operational noise has sparked a lot of interest in quantum error-correcting codes [1][2][3][4][5][6][7][8][9].Although arbitrary error operators might affect a quantum state, it is always possible to keep track of the error amplitudes by expressing them in terms of an error operator basis.A particularly useful class of unitary error bases, called nice error bases, has been introduced by Knill in [10].The nice error bases are the pillar of quantum error-correcting codes [11].
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 the researches are carried out on the specific types of quantum codes, for example, mainly on stabilizer codes, pure codes and codes over finite field.In this paper we are mainly interested in universal framework for quantum codes, i.e., the one which applies for all codes, no matter 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 this framework can characterizes the properties of quantum codes as well as generates some new results about quantum codes.It is a powerful tool in future works on quantum codes.

II. 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 a quantum system with levels and let G be an additive group of order with identity element 0. A nice error basis of is a set where complex numbers gh ω have modulus 1.We call the index group of the error basis .Moreover is a nice error basis of quantum systems Proof.From property iii) of the nice error basis, it follows that

( ) ah ha bh hb a b h h a b)
ω ω ω ω ω ω In the next section we shall give 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 of the error basis is Abelian.This assumption is reasonable because such nice error basis exists for a quantum system with arbitrary levels [10].

Now let { |
We are going to describe the elements of by formal polynomials in .
In general ⊗ ⊗ E is represented by 1 Definition 2. The group algebra Z C of Z over the complex numbers consists of all formal sums Addition and multiplication of elements of Z C are defined in the natural way by ( ) To each we associate the mapping given by † † ( ) tr ∑ be an arbitrary element of the group algebra Z C , with the property that where χ was defined above.Let the elements of be denoted by

Suppose
, 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 Thus is uniquely determined by g ( ) f g .This requires the use of variables , .
What we shall call the exact enumerator of is then defined as Then the exact enumerator of C′ is ( ) , , , ) , , , , Proof.From ( 2) and ( 3), the LHS is equal to which is equal to the RHS. Q.E.D.
The next weight enumerator to be considered classifies vectors in according to the number of times each group element  ( , , ) , and the complete weight enumerator of is C′ 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 7. Suppose now that 2 2 m δ = + is odd, and let the elements of be labeled We call the set the Lee weight distribution of where is the sum of . We also define the Lee weight enumerator of to be Then The Hamming weight, or simply the weight, of a vector is the number of nonzero components , and is denoted by .

IV. 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 (( , , )) ∑ be the orthogonal projection onto where { } is a set of 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 We call the element associated with the quantum code in the group algebra C C Z C .

From (3), the transform of is given by C
where From ( 4), (6), and using the Cauchy-Schwartz inequality we deduce that g g c c′ ≤ for all .Furthermore, from the definition of the minimum distance we get that if then version was first proved for quantum stabilizer codes by Calderbank et al. in [13], and later generalized by Rains in [14].The nonbinary version for stabilizer codes was proved by Ketkar et al. in [15].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.

C
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.
we describe several weight enumerators of the group algebra 0 1 c′ = Z C .
weight distribution of C′ is { } i A′ , where i A′ is the sum of with , and the Hamming weight enumerator of h c′ wt( ) = , and use Lemma 1. Q.E.D.

ForZC
So far we have established the universal framework for quantum codes: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 C C C 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 Theorem 4, 6, 8, and 9.Note that the results about exact enumerators, complete enumerators and Lee enumerators of quantum codes are completely new.For Hamming weight enumerators, the binary C C′ C

For
Finally ifis a quantum stabilizer code, from the definition of stabilizer codes, the element associated with in the group algebraC C ZC can be written as g g C= ∑ z where the summation is over all such that the operator of .Both forms imply the relationship between quantum stabilizer codes and classical codes.
the Lee weight distribution of C′ is { ( )}