Higher-order autonomous quantum error correction

Autonomous quantum error correction (AutoQEC) harnesses engineered coupling to an external reservoir to protect quantum information from decoherence. First-order codes provide protection against a single fault and have simple structures. However, as a counterpart they suffer from the drawback of requiring an engineered dissipation that is typically orders of magnitude stronger than natural dissipation. In this work, we investigate the higher-order case and identify Knill-Laflamme conditions which, if satisfied up to some order r, lead to the existence of an AutoQEC protocol providing protection against at most r consecutive errors. Furthermore, we develop a general theoretical framework for analyzing the effective dynamics in the protected code space. Within this framework we show that the effective decay rate decreases exponentially in r, exceeding the first-order code performance already for moderate engineered dissipation strengths. Finally, we demonstrate that the AutoQEC scheme can be combined with an error-transparent Hamiltonian so as to perform a unitary quantum computation over time scales which far exceed the physical qubits lifetime.