Operator quantum error correction for continuous dynamics

TL;DR

This paper establishes necessary and sufficient conditions for operator quantum error correction under continuous Markovian and Hamiltonian dynamics, extending decoherence-free subsystem criteria to time-dependent scenarios.

Contribution

It generalizes existing conditions for quantum error correction to continuous dynamics, including time-dependent and Hamiltonian cases, with new criteria for unitary and approximate correction.

Findings

Derived conditions for correctability during entire time intervals.

Extended decoherence-free subsystem conditions to time-dependent cases.

Provided criteria for correctability at specific moments and for unitary correction.

Abstract

We study the conditions under which a subsystem code is correctable in the presence of noise that results from continuous dynamics. We consider the case of Markovian dynamics as well as the general case of Hamiltonian dynamics of the system and the environment, and derive necessary and sufficient conditions on the Lindbladian and system-environment Hamiltonian, respectively. For the case when the encoded information is correctable during an entire time interval, the conditions we obtain can be thought of as generalizations of the previously derived conditions for decoherence-free subsystems to the case where the subsystem is time dependent. As a special case, we consider conditions for unitary correctability. In the case of Hamiltonian evolution, the conditions for unitary correctability concern only the effect of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-environment Hamiltonian. We also derive conditions on the Hamiltonian which depend on the initial state of the environment, as well as conditions for correctability at only a particular moment of time. We discuss possible implications of our results for approximate quantum error correction.