Error correction code for protecting three-qubit quantum information against erasures

We present a quantum error correction code which protects three quantum bits (qubits) of quantum information against one erasure, i.e., a single-qubit arbitrary error at a known position. To accomplish this, we encode the original state by distributing quantum information over six qubits which is the minimal number for the present task (see reference [1]). The encoding and error recovery operations for such a code are presented. It is noted that the present code is also a three-qubit quantum hidden information code over each qubit. In addition, an encoding scheme for hiding n-qubit quantum information over each qubit is proposed. PACS numbers: 03.67.Lx, 03.65.Bz Typeset using REVTEX ∗Email address: [email protected] †Email address: [email protected] ‡Email address: [email protected] 1 Quantum computing has become an active aspect of current research fields with the discovery of Shor’s algorithm for factorizing a large number [2-3]. It has become clear that quantum computer are in principle able to solve hard computational problems more efficiently than present classical computers [2-5]. However, the biggest difficulty inhibiting realizations is the fragility of quantum states. Decoherence of qubits caused by the interaction with environment will collapse the state of the quantum computer and thus lead to the loss of information. To solve this problem, Shor, and independently Stean, inspired by the theory of classical error correction, proposed the first two quantum error correction codes (QECCs), i.e., the nine-qubit code [6] and the seven-qubit code [7], which are able to correct errors that occur during the store of qubits. Following this work, many new QECCs have been discovered [8-21]. For the most general error model, Laflamme et al. have shown that the smallest quantum error correction code, for encoding one qubit of quantum information and correcting a single-qubit arbitrary error at an unknown position, is the five-qubit code [8]. On the other hand, apart from the QECCs, many alternative quantum codes have been proposed, such as the quantum error preventing codes (based on the quantum Zeno effect) [22-23] and the quantum error-avoiding codes (based on decoherence-free subspaces (DFSs) [24-26]. Moreover, dynamical suppression of decoherence [27-29] and noiseless subsystems [30-33] have been presented. In 1997 M. Grassl et al. [34] considered an error model where the position of the erroneous qubits is known. In accordance with classical coding theory, they called this model the quantum erasure channel. Some physical scenarios to determine the position of an error have been given [34]. In their work, they showed that only four-qubit error correction code is required to encode one qubit and correct one erasure ( i.e., a single-qubit arbitrary error for which the position of the “damaged” qubit is known). Also, they showed that two qubits of quantum information could be encoded and one erasure could be corrected by extending such four-qubit code, in a sense that only one additional qubit is required for encoding one “message” qubit on average. Clearly, this code is a very compact code for protecting one or two qubits of quantum information as long as the position of the “bad” qubit is known.