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arXiv:1005.3968

A Scheme of Concatenated Quantum Code to Protect against both Computational Error and an Erasure

G. O. Santos, F. M. de Assis, A. F. Lima

Published May 21, 2010• 0 citations
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Abstract

We present a description of encoding/decoding for a concatenated quantum code that enables both protection against quantum computational errors and the occurrence of one quantum erasure. For this, it is presented how encoding and decoding for quantum graph codes are done, which will provide the protection against the occurrence of computational errors (external code). As internal code is used encoding and decoding via scheme of GHZ states for protection against the occurrence of one quantum erasure.

DOI
10.48550/arxiv.1005.3968

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We present a description of encoding/decoding for a concatenated quantum code that enables both protection against quantum computational errors and the occurrence of one quantum erasure. For this, it is presented how encoding and decoding for quantum graph codes are done, which will provide the protection against the occurrence of computational errors (external code). As internal code is used encoding and decoding via scheme of GHZ states for protection against the occurrence of one quantum erasure.
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.
For realizing a quantum memory we suggest to first encode quantum information via a quantum error correcting code and then concatenate combined decoding and re-encoding operations. This requires that the encoding and the decoding operation can be performed faster than the typical decoherence time of the underlying system. The computational model underlying the one-way quantum computer, which has been introduced by Hans Briegel and Robert Raussendorf, provides a suitable concept for a fast implementation of quantum error correcting codes. It is shown explicitly in this article is how encoding and decoding operations for stabilizer codes can be realized on a one-way quantum computer. This is based on the graph code representation for stabilizer codes, on the one hand, and the relation between cluster states and graph codes, on the other hand.
We present a scheme able to protect k >= 3 qubits of information against the occurrence of multiple erasures, based on the code proposed by Yang et al. (2004 JETP Letters 79 236). In this scheme redundant blocks are used and we restrict to the case that each erasure must occur in distinct blocks. We explicitly characterize the encoding operation and the restoring operation required to implement this scheme. The operators used in these operations can be adjusted to construct different quantum erasure-correcting codes. A special feature of this scheme is that no measurement is required. To illustrate our scheme, we present an example in which five-qubits of information are protected against the occurrence of two 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 hidded information code over each qubit. In addition, an encoding scheme for hiding nnn-qubit quantum information over each qubit is proposed.
For realizing a quantum memory the encoded quantum information can be protected against decoherence via repeated decoding and re-encoding operations. This requires us to perform fast encoding and decoding operations. The computational model underlying the one-way quantum computer provides a suitable concept for a fast implementation, which has been solved in a previous article for the encoding operation. In the present paper we show that the missing part, the decoding operation, can also be realized on a one-way quantum computer. This is based on the graph code representation for stabilizer codes, on the one hand, and the relation between cluster states and graph codes, on the other hand.