Quantum error-correcting subsystems are unitarily recoverable subsystems (7 pages)

Thomas Fösel, Petru Tighineanu, Talitha Weiß +1 more

Frank Gaitan, Ran Li

David W. Kribs, Robert W. Spekkens

We show that every correctable subsystem for an arbitrary noise operation can be recovered by a unitary operation, where the notion of recovery is more relaxed than the notion of correction insofar as it does not protect the subsystem from subsequent iterations of the noise. We also demonstrate that in the case of unital noise operations one can identify a subset of all correctable subsystems---those that can be corrected by a single unitary operation---as the noiseless subsystems for the composition of the noise operation with its dual. Using the recently developed structure theory for noiseless subsystems, the identification of such unitarily correctable subsystems is reduced to an algebraic exercise.

Ivan B. Djordjević

Cédric Bény, David W. Kribs, Aron Pasieka

We give a brief introduction to the algebraic formulation of error correction in quantum computing called operator algebra quantum error correction (OAQEC). Then we extend one of the basic results for subsystem codes in operator quantum error correction (OQEC) to the OAQEC setting: Every hybrid classical-quantum code is shown to be unitarily recoverable in an appropriate sense. The algebraic approach of the proof yields a new, less technical proof for the OQEC case.

Aron Pasieka

We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. The Lindblad master equation is presented as an example of the dynamics of a system thusly described, in the case of a Markovian system. We review noiseless subsystems and decoherence-free subspaces. Building on the generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems, given a specific, not necessarily unital, channel. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to find unitarily correctable subsystems, in the case of a unital noise model.