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arXiv:1305.4591

Protecting Information Against Computational Errors and Quantum Erasures via Concatenation

Gilson Oliveira dos Santos, Francisco de Assis

Published May 20, 2013• 0 citations
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Abstract

In this work, we introduce a new concatenation scheme which aims at protecting information against the occurrence of both computational errors and quantum erasures. According to our scheme, the internal code must be a quantum loss-correcting code that does not perform measurements, while the external code must be a quantum error-correcting code. We illustrate the concatenation proposed with an example in which one qubit of information is protected against the occurrence of two erasures and one computational error.

DOI
10.48550/arxiv.1305.4591

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In this work, we introduce a new concatenation scheme which aims at protecting information against the occurrence of both computational errors and quantum erasures. According to our scheme, the internal code must be a quantum loss-correcting code that does not perform measurements, while the external code must be a quantum error-correcting code. We illustrate the concatenation proposed with an example in which one qubit of information is protected against the occurrence of two erasures and one computational error.
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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 hidded information code over each qubit. In addition, an encoding scheme for hiding nnn-qubit quantum information over each qubit is proposed.
We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.
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. The encoding and error recovery operations for such a code are presented. We also show that the present code is a three-qubit quantum hidded information code over each qubit. In addition, hiding nnn-qubit quantum information over each qubit is considered.
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. The present code has a high encoding efficiency, since only one auxiliary qubit is needed for one message qubit on average. In addition, we note that the code can also work even in a worse case that the interaction with the environment causes a leakage out of the qubit space. The code may have some applications in the storage of quantum information for small-scale quantum computing, quantum information processing, and quantum communication.