On Protected Realizations of Quantum Information

Abstract

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. 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 justifies 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 involves 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.

17 pages

References in corpus (1)

• Theory of Initialization-Free Decoherence-Free Subspaces and Subsystems

Cited by in corpus (15)

• Subsystem fault tolerance with the Bacon-Shor code
• The structure of preserved information in quantum processes
• Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction
• On Subsystem Codes Beating the Hamming or Singleton Bound
• Holonomic quantum computation in subsystems
• Generalized Coherent States as Preferred States of Open Quantum Systems
• Scheme for fault-tolerant holonomic computation on stabilizer codes
• Operator quantum error correction for continuous dynamics
• Asymmetric and Symmetric Subsystem BCH Codes and Beyond
• Robustness of operator quantum error correction with respect to initialization errors
• Quantum stabilizer codes and beyond
• On Quantum and Classical Error Control Codes: Constructions and Applications
• Clifford Code Constructions of Operator Quantum Error Correcting Codes
• Constructions of Subsystem Codes over Finite Fields
• Asymptotics of the quantum Hamming bound for subsystem codes