Keywords: additive codes, stabilizer, pure and impure codes, weight enumerator, probability of undetected error
American Journal of Applied Mathematics and Statistics, 2015 3 (2),
pp 7679.
DOI: 10.12691/ajams326
Received April 02, 2015; Revised April 10, 2015; Accepted April 15, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
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 errorcorrecting 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 selforthogonal with respect to a certain symplectic inner product. An code is an additive quantum code of minimumdistance 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 CodeThe 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 CodeThe 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]}.
2.2.1. Remark 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
[1]  Calderbank, A., Rains, E. M., Shor, P. W. and Sloane, N. J. A., “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 13691387,1998. 
 In article  CrossRef 

[2]  Cross, A., Smith, G., Smolin, J.A. and Zeng, B., “Codeword Stabilized Quantum Codes.” IEEE Trans. Inf. Theory,vol.55, pp. 433438, 2009. 
 In article  CrossRef 

[3]  Ashikhmin, A. E., Barg, A. M., Knill, E.and Litsyn, S. N., “Quantum Error Detection I: Statement of the Problem”, IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 778788, 2000. 
 In article  CrossRef 

[4]  Gottesman, D., “Stabilizer Codes and Quantum Error Correction,” Caltech Ph.D. dissertation, California Institute of Technology, Psadena, CA, 1997. 
 In article  

[5]  Rains, E. M., “Quantum codes of minimum distance two,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 266271, 1999. 
 In article  CrossRef 

[6]  Rains, E. M., “Quantum shadow enumerators”, IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2361 2366, 1999. 
 In article  CrossRef 

[7]  Rains, E. M., Hardin, R. H., Shor, P. W. and Sloane, N. J. A.,“A nonadditive quantum code,” Phys. Rev. Lett., vol. 79, pp. 953954,1997. 
 In article  CrossRef 

[8]  Smolin, J. A., Smith, G., and Wehner, S., “A simple family of nonadditive quantum codes” arXiv: quantph/0701065v2, 2007. 
 In article  

[9]  Feng, K. and Xing, C. P., “A new construction on quantum errorcorrecting codes,” Trans. Amer. Math. Soc., vol. 360, pp. 20072019, 2008. 
 In article  CrossRef 

[10]  Shor, P. W. and Laflamme, R., “Quantum analog of the MacWilliams identities in classical coding theory,” Phys. Rev. Lett., vol. 78, pp. 16001602, 1997. 
 In article  CrossRef 

[11]  Aggarwal, V. and Calderbank, R., “Boolean Functions, Projection Operators and Quantum Error Correcting Codes.”IEEE Trans. Inf. Theory, vol.54, pp. 17001707, 2008. 
 In article  CrossRef 

[12]  Cary Huffman, W. and Vera Pless, “Fundamentals of ErrorCorrecting Codes.” Cambridge University Press, 2003. 
 In article  

[13]  LeungYanCheong, S. K. and Hellman, M.E. “Concerning a bound on undetected error probability,” IEEE Trans. Inf. Theory, vol. IT 22. pp. 235231, 1976. 
 In article  

[14]  Gupta, M., Narula, R. K and Taneja, D., “On the Construction of Odd Length Quantum Codes,”British Journal of Mathematics & Computer Science 6(5), pp. 444450, 2015. 
 In article  CrossRef 
