Accurate processing of information is crucial both in technology and in the natural world. To achieve it, any information processing system needs an initial supply of resources away from thermal equilibrium. Here we establish an in-principle limit on the maximum accuracy achievable with a given amount of nonequilibrium resources. The limit applies to arbitrary information processing tasks and to arbitrary information processing systems subject to the laws of quantum mechanics. It is expressed in terms of an entropic quantity, hereby called the thermodynamic complexity, which characterises the thermodynamic hardness of the given information processing task. When applied to the task of replicating quantum information, the limit reveals a thermodynamic advantage of copy machines that preserve quantum entanglement. Our results shed light on the interplay between thermodynamic resources and information-theoretic tasks, and in the long term can inform the design of quantum devices approaching the ultimate efficiency limits.
