1. Introduction

With the discovery of Shor’s algorithm, Quantum computing has become an active interdisciplinary field of research. Quantum computers are able to solve hard computational problems more efficiently than present classical computers. But reliability of the quantum computers is questionable since the quantum states are subjected to decoherence. Quantum error correcting codes are the means of protecting quantum information against external sources such as noise and decoherence. Many explicit constructions of quantum error-correcting codes have been proposed so far. Most of the codes known so far are additive or stabilizer codes which are constructed from classical binary code that are self-orthogonal with respect to a certain symplectic inner product. An code is an additive quantum code of minimum-distance d of length n encoding k quantum bits and an code refers to a general code encoding K states in n qubits with minimum distance d. A code is called nonadditive if it is not equivalent to any additive code.

The construction of additive quantum codes using additive classical codes C over GF(4) is given in ^[1]. An important class of quantum codes called Stabilizer codes is defined in ^[1] and ^[4] which are analogous to the quantum additive codes. Among the additive codes the minimum distance two codes are those which correct any single qubit erasure. These distance two codes have been extensively studied and several constructions of both additive and nonadditive distance 2 codes are available in ^{[1, 2, 5, 7, 8, 9, 11]}. In our earlier work ^[14], we have also studied these distance 2 codes and now are in a position to find their undetected error probability.

In classical coding theory decoding is done by observing the received vector. If the received vector is not contained in the code space then an error is detected. An error remains undetected if the sent vector and the error vector sum up to a code word in the code space itself. The probability of undetected error for a code is given by

where is the number of code words of weight i in code. It was shown in ^[13] that the undetected error probability, when used solely for error detection on binary symmetric channel with crossover probability is upper bounded by . In quantum case, the error will not be detected if the measured transmission results in the code itself and is not orthogonal or collinear to transmitted state vector. The probability of undetected error in this case, as shown by ^[3] can be computed via the weight enumerators of quantum codes. For a stabilizer code this probability is given by

where and are the weight distributions of the quantum codes as defined in ^[10].

The optimal distance 2 codes along with their stabilizer structures and their explicit basis were found in our earlier work ^[14]. In this paper, the probability of undetected error for both and code have been found. This probability function is further proved to be monotonic increasing having an upper bound which is same as classical codes as given in ^[13].

2. Probability of Undetected Error

In ^[14] we have shown that the code is constructed from a classical additive self dual code whose generator matrix is

The direct sum of with is used to construct the code whose generator matrix is

In this paper, the probability of undetected error for both code have been found. This probability function is further proved to be monotonic increasing having an upper bound which is same as classical codes.

2.1. Undetected Error Probability for Even Length Quantum Code

The weight enumerators of the quantum code are

and

Also by MacWilliams Identity ^[6]

We shall prove by induction that for

Now for

Let us assume for

Now

Thus

Thus by induction

and hence s an increasing function in this interval.

Now for

Thus is upper bounded by .

2.2. Undetected Error Probability of Odd Length Quantum Code

The odd length quantum codes are obtained by taking the direct sum of the classical even length code C over with .

Now from ^[12], the weight enumerator of the resulting classical code will be

Hence, the weight enumerator of the quantum code will be

and

We shall prove that

Now

if

When .

Let us assume

That is

We shall prove that for

Now

Thus

and hence is an increasing function in this interval.

Now for

Thus the is upper bounded by

Thus we have proved that quantum and codes obeys the bound, where This result is similar to the general rule for the classical linear block codes as given in ^[13].

We have also verified that this bound is satisfied by the quantum Hamming codes

The probability of undetected error of these codes is

This is a monotonic increasing function of p giving an upper bound . Many more quantum codes satisfy this bound.

In classical codes there are certain codes ^[13], in which this bound is not satisfied so such violation in quantum additive codes would be the topic of further investigation

3. Conclusion

The probability of undetected error for optimal distance 2 codes was found to be increasing functions and satisfies the upper bound which is same as the classical bound.

Acknowledgement

This research work is supported by National Board for Higher Mathematics (NBHM), Mumbai with Ref. No. 2/48(1)/2012/NBHM/R&D11/10924.

One of the author, Divya Taneja also acknowledges Punjab Technical University, Jalandhar, Punjab for providing the research facilities.

References

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