American Journal of Applied Mathematics and Statistics, 2015, Vol. 3, No. 2, 76-79 Available online at http://pubs.sciepub.com/ajams/3/2/6 © Science and Education Publishing DOI:10.12691/ajams-3-2-6 Undetected Error Probability for Quantum Codes Manish Gupta 1 , R.K. Narula 2 , Divya Taneja 3,4,* 1 Baba Farid College of Engineering & Technology Bathinda, Punjab, India 2 PIT, Mansa, Punjab, India 3 Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo, Punjab, India 4 Research Scholar Punjab Technical University, Jalandhar, Punjab, India *Corresponding author: dtaneja25@yahoo.co.in Received April 02, 2015; Revised April 10, 2015; Accepted April 15, 2015 Abstract From last fourteen years the work on undetected error probability for quantum codes has been silent. The undetected error probability has been discussed by Ashikhmin [3] in which it was proved that the average probability of undetected error for a given code is essentially given by a function of its weight enumerators. In this paper, new upper bounds on undetected error probability for quantum codes used for error detection on depolarization channel are given. It has also been established that the probability of undetected errors for quantum codes over depolarization channel do satisfy the upper bound analogous to classical codes. Keywords: additive codes, stabilizer, pure and impure codes, weight enumerator, probability of undetected error Cite This Article: Manish Gupta, R.K. Narula, and Divya Taneja, “Undetected Error Probability for Quantum Codes.” American Journal of Applied Mathematics and Statistics, vol. 3, no. 2 (2015): 76-79. doi: 10.12691/ajams-3-2-6. 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 [[n,k,d]] code is an additive quantum code of minimum-distance d of length n encoding k quantum bits and an ( ) ( , , ) nKd 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 [ ] ,, nkd code is given by ( ) ( ) 1 1 (1 ) ,0 2 n i ni u i i P p A p p p − = = − ≤ ≤ ∑ where i A 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 1 , 2 p ≤ is upper bounded by ( ) 2 nk − − . 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 ( ) ( ) 0 , ( ) 1 3 i n ni ue i i i p P Qp B B p − ⊥ =   = − −     ∑ where 3 0 4 p ≤ ≤ and and i i B B ⊥ are the weight distributions of the quantum codes as defined in [10].