Thermodynamic Recycling of Algorithmic Failure Branches: Quantum-Computer Demonstration with Quantum Error Correction

Thermodynamic trade-off relations dictate fundamental limits on the performance of thermodynamic tasks through costs such as heat dissipation. Here, we propose a framework called thermodynamic recycling to circumvent these limits in quantum processors by exploiting failure branches of quantum algorithms, which are usually discarded. The key component is an athermal bath naturally generated during the resetting of a failure branch. By coupling this bath to a target system prior to relaxation, thermodynamic tasks can be performed beyond conventional thermodynamic limits. We apply this framework to information erasure and derive the reduction in heat dissipation analytically. As a demonstration, we implement our framework on IBM's superconducting quantum processor by combining the Harrow--Hassidim--Lloyd algorithm with three-qubit quantum error correction, thereby reducing the heat dissipated in erasing syndrome information. Despite substantial noise and errors in current hardware, our method achieves erasure with heat dissipation below the Landauer limit. This work establishes an operational connection between quantum computing and quantum thermodynamics for resource-efficient quantum computation.