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arXiv:quant-ph/0412076 · Physical Review Letters

Unified and Generalized Approach to Quantum Error Correction

David W. Kribs, Raymond Laflamme, David Poulin

Published May 9, 2005• 283 citations
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Abstract

We present a unified approach to quantum error correction, called operator quantum error correction. Our scheme relies on a generalized notion of a noiseless subsystem that is investigated here. By combining the active error correction with this generalized noiseless subsystems method, we arrive at a unified approach which incorporates the known techniques--i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Moreover, we demonstrate that the quantum error correction condition from the standard model is a necessary condition for all known methods of quantum error correction.

DOI
10.1103/physrevlett.94.180501

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We present a unified approach to quantum error correction, called operator quantum error correction. Our scheme relies on a generalized notion of a noiseless subsystem that is investigated here. By combining the active error correction with this generalized noiseless subsystems method, we arrive at a unified approach which incorporates the known techniques--i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Moreover, we demonstrate that the quantum error correction condition from the standard model is a necessary condition for all known methods of quantum error correction.
Operator quantum error correction provides a unified framework for the known techniques of quantum error correction such as the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method. We first show an example of a new quantum error correction scheme which can not be described by operation quantum error correction. Then we introduce a different notion of noiseless subsystems according to the example. Base on this notion, we present a more unified approach which incorporates operator quantum error correction as a special case. Moreover, we also give a sufficient and necessary condition of quantum error correction using this approach. We show that this approach provides more recovery operations than operator quantum error correction, which possibly leads to simpler decoding procedures.
Abstract. This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques — i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method — as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of “unitarily noiseless subsystems”. A unified and generalized approach to quantum error correction, called Operator Quantum Error Correction (OQEC), was recently introduced in [1]. This model unifies all of the known techniques for the error correction of quantum operations – i.e. the standard model [2, 3, 4, 5], the method of decoherence-free subspaces [6, 7, 8, 9]
This paper is an expanded and more detailed version of our recent work in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques - i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method - as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of ``unitarily noiseless subsystems''.
This paper is an expanded and more detailed version of the work \cite{KLP04} in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques --- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method --- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems''.
This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced.This is a new scheme for the error correction of quantum operations that incorporates the known techniques -i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem methodas special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations.We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".
In this work, it is shown that many quantum error-manipulating techniques, such as distillation, error mitigation, and dynamical decoupling, are special cases of the most general framework for quantum error correction. This unifying perspective is achieved by extending quantum error correction to include state-adaptive and channel-adaptive settings, as well as multi-stage coding scenarios. Based on this insight, a model of self-correcting quantum memory is also proposed. This work clarifies the relationship among these techniques and illustrates, through explicit constructions, how the unified perspective can guide the design of reliable quantum information systems.