Thermodynamic optimization of quantum algorithms: On-the-go erasure of qubit registers

We consider two bottlenecks in quantum computing: limited memory size and noise caused by heat dissipation. Trying to optimize both, we investigate ``on-the-go erasure'' of registers that are no longer needed for a given algorithm: freeing up auxiliary qubits as they stop being useful would facilitate the parallelization computations. study minimal thermodynamic cost erasure these scenarios, applying results on Landauer entangled registers. For class algorithms solving Abelian hidden subgroup problem, find optimal on-the-go protocols. conclude there is trade-off: if have enough partial information about problem build efficient erasure, can use it instead simplify algorithm, so fewer run computation first place. provide explicit protocols approaches.