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arXiv:quant-ph/0012111 · Physical Review A

Quantum error-correcting codes associated with graphs

Dirk-M. Schlingemann, Reinhard F. Werner

Published Dec 11, 2001• 406 citations
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Abstract

We present a construction for quantum error correcting codes. The basic ingredients are a graph and a finite Abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the one-error correcting property of codes of length 5 in any dimension. As examples, we construct a large class of maximum distance separable codes, i.e. codes saturating the Singleton bound, as well as a code of length 10 detecting three errors.

DOI
10.1103/physreva.65.012308

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We present a construction for quantum error correcting codes. The basic ingredients are a graph and a finite Abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the one-error correcting property of codes of length 5 in any dimension. As examples, we construct a large class of maximum distance separable codes, i.e. codes saturating the Singleton bound, as well as a code of length 10 detecting three errors.
Graph states are generalized from qubits to collections of nnn qudits of arbitrary dimension DDD, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large nnn and DDD are constructed using simple graphs, except when nnn is odd and DDD is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general DDD, and shown to provide a dual representation of an additive graph code.
Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space.
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.
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The theory of quantum error correcting codes is a primary tool for fighting decoherence and other quantum noise in quantum communication and quantum computation. Recently, the theory of quantum error correcting codes has developed rapidly and been extended to protect quantum information over asymmetric quantum channels, in which phase-shift and qubit-flip errors occur with different probabilities. In this paper, we generalize the construction of symmetric quantum codes via graphs (or matrices) to the asymmetric case, converting the construction of asymmetric quantum codes to finding matrices with some special properties. We also propose some asymmetric quantum Maximal Distance Separable (MDS) codes as examples constructed in this way.
Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph states to be in TQO-1, which is a class of quantum error-correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. The results indicate that graph states provide a fruitful approach to the construction and characterization of topological stabilizer quantum error-correction codes.