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Finding quantum noiseless subsystems: A linear-algebraic approach

Francesco Ticozzi, Augusto Ferrante

Published Jan 1, 2007• 2 citations
OpenAlex

Abstract

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.

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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.
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 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 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.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics accurately describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway for faithfully representing quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model parameter and initial state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov’s stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular,
The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.
Operator quantum error correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error correction, and provides a unified framework for topics such as quantum error correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions that characterize when operator quantum error correction is feasible; (b) a representation theorem for a class of noise processes that can be corrected using operator quantum error correction; and (c) generalizations of the coherent information and quantum data processing inequality to the setting of operator quantum error correction.