Quantum bang-bang control has recently emerged as a general strategy for manipulating quantum evolutions by enforcing suitable time scale separations between the controller and the natural dynamics of the system [1]. For open quantum systems, this has lead to establishing quantum error suppression schemes, whereby active decoupling from environmental noise is achieved by continuously undoing system-bath correlations on time scales that are short compared to the typical memory time of the bath [2]. Decoupling techniques were shown to be consistent with efficient quantum information processing [3], thereby offering an alternative scenario compared to error-correcting [4] and error-avoiding quantum codes [5]. In contrast to the latter methods, no redundant encoding is necessary for preserving or manipulating quantum information provided that the required control operations can be implemented. However, one may ask whether quantum coding could be advantageous or necessary in situations where the available control options are limited. Answering the above question naturally connects the decoupling formalism with the notion of noiseless subsystem that has been identified as the most general route to noise-free information storage [6]. The basic philosophy is to envision the bang-bang control procedure as a tool for effectively endowing the system dynamics with a nontrivial group of symmetries. Such symmetries generate structures in the system’s state space which are in principle inaccessible to unwanted interactions and are therefore suited for encoding quantum information. Mathematically, the crucial requirement relates to the reducibility properties of operator algebras associated with the action of the decoupling group. Variants of the same basic idea have been argued to lie at the heart of all existing approaches for stabilizing quantum information in a recent work by Zanardi [7]. In this Letter we examine the implications of the above concept within the decoupling framework, by showing that the action of the control group allows for a complete classification of the choices available for both safe information encoding and universal control over coded states. At variance with the case where noiseless subsystems emerge by virtue of preexisting static symmetries in the overall Hamiltonian, the dynamical origin of the noise-protected structures also precisely constrains the admissible methods for implementing universal control in a way which simultaneously preserves the effect of decoupling as well as the selected coding space. Using coding methods has several attractive consequences. First, bang-bang operations are needed only for noise suppression. Additional manipulations on encoded subsystems become fully implementable via weak strength controls [3]. Second, for schemes where the relevant Hamiltonians are allowed to be turned on or off slowly, an advantage is that the corresponding pulses can be made more easily frequency-selective. Finally, coded states may be intrinsically more robust against imperfections in the decoupler operations. For a potentially large class of quantum information processors characterized by linear quantum noise, we outline a scheme where noisedecoupling involves a minimal set of two collective bangbang rotations and universal quantum computation on encoded qubits can be performed entirely through slow tuning of two-body bilinear interactions. Decoupling.− Let S be a finite-dimensional quantum system with self-Hamiltonian HS on HS , dim(HS) = d. S interacts with the environment B via a Hamiltonian HSB = ∑ α Eα ⊗ Bα, the Bα’s being linearly independent environment operators. The error operators Eα are assumed to belong to a linear space E that we call the interaction space. We require that tr(Eα) = 0, thereby removing from HSB the internal evolution of the environment. Let AE denote the algebra generated by the identity, HS , and E . AE is a subalgebra of the full operator algebra End(HS) closed under Hermitian transpose (†-closed). For n-qubit systems, HS ' C, End(HS) ' Mat(d × d, C), with d = 2. In its essence, decoupling via bang-bang (b.b.) control relies on the idea of exploiting full strength/fast switching control actions [1–3], meaning that a certain set of Hamiltonians can be (ideally) turned on/off instantaneously with arbitrarily large strength. Let G denote a finite group determining the realizable b.b. operations

@inproceedings{ViolaDynamicalGO,
title={Dynamical Generation of Noiseless Quantum Subsystems},
author={Lorenza Viola and Emanuel Knill and Seth Lloyd}
}