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Quantum error correction (QEC) is a procedure by which the quantum state of asystem is protected against a known type of noise, by preemptively addingredundancy to that state. Such a procedure is commonly used in quantumcomputing when thermal noise is present. Interestingly, thermal noise has alsobeen known to play a central role in quantum thermodynamics (QTD). This facthints at the applicability of certain QTD statements in the QEC of thermalnoise, which has been discussed previously in the context of Maxwell's demon.In this article, we view QEC as a quantum heat engine with a feedbackcontroller (i.e., a demon). We derive an upper bound on the measurement heatdissipated during the error-identification stage in terms of the Groenewoldinformation gain, thereby providing the latter with a physical meaning alsowhen it is negative. Further, we derive the second law of thermodynamics in thecontext of this QEC engine, operating with general quantum measurements.Finally, we show that, under a set of physically motivated assumptions, thisleads to a fundamental triple trade-off relation, which implies a trade-offbetween the maximum achievable fidelity of QEC and the super-Carnot efficiencythat heat engines with feedback controllers have been known to possess. Asimilar trade-off relation occurs for the thermodynamic efficiency of the QECengine and the efficacy of the quantum measurement used for erroridentification.

Thermodynamic Constraints on Quantum Information Gain and Error Correction: A Triple Trade-Off