# Learning to erase quantum states: thermodynamic implications of quantum learning theory

arXiv: https://arxiv.org/abs/2504.07341 · HTML: https://mlsanity.com/paper/2504.07341/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 115 references that are in the map.
- First posted on arXiv 9 Apr 2025.

## Builds on (most cited first, 12 of 12)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [Optimal algorithms for learning quantum phase states](https://mlsanity.com/paper/2208.07851/) (2022)
- [Quantum hardness of learning shallow classical circuits](https://mlsanity.com/paper/1903.02840/) (2019)
- [Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates](https://mlsanity.com/paper/2305.13409/) (2023)
- [Learning quantum states prepared by shallow circuits in polynomial time](https://mlsanity.com/paper/2410.23618/) (2024)
- [How to Construct Random Unitaries](https://mlsanity.com/paper/2410.10116/) (2024)

## Abstract (arXiv)

The energy cost of erasing quantum states depends on our knowledge of the states. We show that learning algorithms can acquire such knowledge to erase many copies of an unknown state at the optimal energy cost. This is proved by showing that learning can be made fully reversible and has no fundamental energy cost itself. With simple counting arguments, we relate the energy cost of erasing quantum states to their complexity, entanglement, and magic. We further show that the constructed erasure protocol is computationally efficient when learning is efficient. Conversely, under standard cryptographic assumptions, we prove that the optimal energy cost cannot be achieved efficiently in general. These results also enable efficient work extraction based on learning. Together, our results establish a concrete connection between quantum learning theory and thermodynamics, highlighting the physical significance of learning processes and enabling provably-efficient learning-based protocols for thermodynamic tasks.