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arXiv:quant-ph/0308022

Quantum error correcting codes and one-way quantum computing: Towards a quantum memory

Dirk Schlingemann

Published Aug 4, 2003• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0308022

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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.
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.
The present paper is concerned with the concept of the one-way quantum computer, beyond binary-systems, and its relation to the concept of stabilizer quantum codes. This relation is exploited to analyze a particular class of quantum algorithms, called graph algorithms, which correspond in the binary case to the Clifford group part of a network and which can efficiently be implemented on a one-way quantum computer. These algorithms can ``completely be solved" in the sense that the manipulation of quantum states in each step can be computed explicitly. Graph algorithms are precisely those which implement encoding schemes for graph codes. Starting from a given initial graph, which represents the underlying resource of multipartite entanglement, each step of the algorithm is related to a explicit transformation on the graph.
We show how to explicitly construct an O(nd)O(nd)O(nd) size and constant quantum depth circuit which encodes any given nnn-qubit stabilizer code with ddd generators. Our construction is derived using the graphic description for stabilizer codes and the one-way quantum computation model. Our result demonstrates how to use cluster states as scalable resources for many multi-qubit entangled states and how to use the one-way quantum computation model to improve design of quantum algorithms.
In a previous paper a straight forward construction method for quantum error correcting codes, based on graphs, has been presented. These graph codes are directly related to cluster states which have been introduced by Briegel and Raussendorf. We show that the preparation of a cluster state as well as the coding operation for a graph code, can be implemented by a logical network by only using one type of two-qubit gate, e.g. CNOT or controlled Phase, and one type of one-qubit gate, e.g. the Hadamard transform. Concerning the qubit case each vertex corresponds to an Hadamard gate and each edge corresponds to a controlled not gate.
In a previous paper a straight forward construction method for quantum error correcting codes, based on graphs, has been presented. These graph codes are directly related to cluster states which have been introduced by Briegel and Raussendorf. We show that the preparation of a cluster state as well as the coding operation for a graph code, can be implemented by a logical network. Concerning the qubit case each vertex corresponds to an Hadamard gate and each edge corresponds to a controlled not gate.
We present an approach to one-way quantum computation (1WQC) that can compensate for single-qubit errors by encoding the logical information residing on physical qubits into five-qubit error-correcting code states. A logical two-qubit cluster state that is the fundamental resource for encoded quantum teleportation is then described by a graph state containing ten vertices with constant degree seven. Universal 1WQC that incorporates error correction requires only multiple copies of this logical two-qubit state and a logical four-qubit linear cluster state, which are prepared only just in advance of their use in order to minimize the accumulation of errors. We suggest how to implement this approach in systems characterized by qubits in regular two-dimensional lattices for which entangling gates are generically global operations, such as atoms in optical lattices, quantum dots, or superconducting qubits.