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arXiv:quant-ph/0303165 · Physical Review A

Verification procedures for quantum noiseless subsystems

Lorenza Viola, Emanuel H. Knill

Published Sep 22, 2003• 7 citations
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Abstract

We examine the problem of verifying the implementation of a desired noiseless subsystem from a quantum error-correcting perspective. General conditions are identified, under which the verification of error-correcting behavior against a linear set of errors $\mathcal{E}$ suffices for the verification of noiseless subsystems of an error algebra $\mathcal{A}$ contained in $\mathcal{E}.$ From a practical standpoint, our results imply that the verification of a noiseless subsystem need not require the explicit verification of noiseless behavior for all possible initial states of the syndrome subsystem.

DOI
10.1103/physreva.68.032311

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We examine the problem of verifying the implementation of a desired noiseless subsystem from a quantum error-correcting perspective. General conditions are identified, under which the verification of error-correcting behavior against a linear set of errors E\mathcal{E}E suffices for the verification of noiseless subsystems of an error algebra A\mathcal{A}A contained in E.\mathcal{E}.E. From a practical standpoint, our results imply that the verification of a noiseless subsystem need not require the explicit verification of noiseless behavior for all possible initial states of the syndrome subsystem.
There are two complementary approaches to realizing quantum information so that it is protected from a given set of error operators. Both involve encoding information by means of subsystems. One is initialization-based error protection, which involves a quantum operation that is applied before error events occur. The other is operator quantum error correction, which uses a recovery operation applied after the errors have occurred. Together, the two approaches make it clear how quantum information can be stored at all stages of a process involving alternating error and quantum operations. In particular, there is always a subsystem that faithfully represents the desired quantum information. We give a definition of faithful realization of quantum information and show that it always involves subsystems. This supports the ``subsystems principle'' for realizing quantum information. In the presence of errors, one can make use of noiseless, (initialization) protectable, or error-correcting subsystems. We give an explicit algorithm for finding optimal noiseless subsystems. Finding optimal protectable or error-correcting subsystems is in general difficult. Verifying that a subsystem is error correcting is known to involve only linear algebra. We discuss the verification problem for protectable subsystems and reduce it to a simpler version of the problem of finding error-detecting codes.
We propose linear algebraic techniques for finding maximum dimensional noiseless subspaces and pure subsystems for a given quantum dynamical model. Noiseless subsystem are the most general structure that support protected realization of quantum information. After recalling the relevant characterization, we develop simple strategies to tackle particular cases and we sketch a possible approach for the general one. The linear algebraic problem is that of choosing a suitable basis in a finite dimensional Hilbert space, such that the relevant operators attain the desired block-form.
We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations. We suggest that this work brings the fundamental passive technique for error correction in quantum computing an important step closer to practical realization.
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.
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''.