Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., "noiseless," "unitarily correctible," etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.

We introduce a general operational characterization of information-preserving structures -encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes-by demonstrating that they are isometric to fixed points of unital quantum processes. Using this, we show that every information-preserving structure is a matrix algebra. We further establish a structure theorem for the fixed states and observables of an arbitrary process, which unifies the Schrödinger and Heisenberg pictures, places restrictions on physically allowed kinds of information, and provides an efficient algorithm for finding all noiseless and unitarily noiseless subsystems of the process. DOI: 10.1103/PhysRevLett.100.030501 PACS numbers: 03.67.Pp, 03.65.Yz, 03.67.Lx, 89.70.ÿa Quantum processes, also known as quantum channels, quantum operations, or completely positive (CP) maps [1,2], are central to the theory and practice of quantum information processing (QIP). They describe how quantum states evolve over a period of time in the presence of noise, or how a device's output depends on its input. They are also complex and unwieldy: to fully specify a quantum process on a d-dimensional system requires d 4 real numbers. Most of these data are irrelevant to what one really wants to know: What information can pass unharmed through the process? Besides being central to QIP, a general answer is broadly relevant to both fundamental physics and quantum technologies, for the information-preserving degrees of freedom are precisely those that may be reliably characterized and exploited. Information-preserving structures (IPS) in quantum processes-what they are and how to find them -are the subject of this Letter.The quest for such structures has a long history in quantum physics. Pointer states (PS), defined in the context of quantum measurement theory, are ''most classical'' states that resist decoherence [3]. QIP science has spurred interest in the preservation of quantum information, leading to the notion of noiseless subsystems (NS) [4] as passive IPS that emerge from the existence of symmetries in the noise, and recover both decoherence-free subspaces (DFS) [5] and PS in special limits. Processes admitting no NS may still preserve information, which can be actively protected using quantum error correction (QEC) [6,7] to create an effective NS. Rapid experimental progress in implementing DFS [8], NS [9], and QEC [10] heightens the need for a complete and constructive characterization of preserved information.In this Letter, we formulate a general operational theory of IPS. The key insight is to identify preserved information with sets of states (or codes) whose mutual distinguishability is left unchanged. We prove that every preserved code can, through error correction, be made noiseless, then show that every maximal noiseless code is isometric [11] to the fixed-point set of the dynamics. This set, in turn, is isometric to a matrix algebra; thus, we conclude that every IPS is an algebra. Finally, we provide an ex...

We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise and have a lower overhead cost than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer and can be expressed either in terms of the operator norm of the bath's Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.