This work connects a wide range of fields: quantum thermodynamics, learning theory, complexity theory, and cryptography. We hope that our results can spark interest in this interdisciplinary frontier. Many thanks to @YuzhenZhang_ph and @preskill for this fun collaboration!.

The thermodynamic implications of quantum learning theory extend beyond erasure. We also discuss the task of work extraction, where learning theory gives similar (in)efficiency results. Maybe there're many more thermodynamic applications where learning algorithms provably helps!.

Conversely, using low-depth pseudorandom states developed in we show that the optimal energy cost (Landauer's limit) is actually NOT achievable efficiently in general, and a nearly maximal amount of work must be paid to erase these low-complexity states.

This allows us to relate the energy cost of erasing quantum states to their complexity, entanglement, and magic, with a simple counting argument. Furthermore, the constructed erasing algorithm is provably efficient when learning is efficient.

We rigorously study this process of "learning to erase" using tools in quantum learning theory. We show that learning algorithms can acquire such knowledge to erase at the optimal energy cost. This is proved by showing that learning has NO fundamental energy cost itself.

We take Landauer erasure as an example, which illustrates that the energy cost of erasing an unknown quantum system is proportional to our ignorance. But as we see more copies of it and learn more about the system, our ignorance decreases and the energy cost goes down.

Do abstract learning processes have tangible physical consequences?. Yes! In this work, we rigorously show that the (in)ability to learn dramatically impacts the amount of physical resources required to perform certain thermodynamic tasks.

RT @preskill: Students Haimeng Zhao and Yuzhen Zhang decided to revisit thermodynamics from the perspective of learning theory and computat….

RT @PRX_Quantum: On the cover of our latest issue: the complexity of learning quantum states and evolutions is related to the complexity of….

RT @jrrhuang: The classical shadows protocol predicts M properties from log(M) samples of a quantum state but crucially relies on observabl….

Glad to see that this work is published in @PRX_Quantum ! We have added some numerical experiments to showcase the relation between sample complexity and gate complexity. Check out the GitHub repo if you're interested!

RT @PRX_Quantum: The complexity of learning quantum states and evolutions is related to the complexity of creating them. @haimengzhao @Laur….

Code for reproducing the results are available at

This is my undergrad thesis at Tsinghua University and a joint work with my undergrad advisor @DonglingDeng . Huge thanks to his mentorship throughout the years!.

Our proof is information-theoretic and pinpoints the origin of this advantage: quantum entanglement can be used to reduce the communication complexity of non-local machine learning tasks. In other words, we exploit quantum pseudo-telepathy!

All these good properties enable us to demonstrate the quantum learning advantage with IonQ's trapped-ion platform. We are able to exhibit an exponential gap between quantum and classical accuracies using relatively small system sizes.

We show through numerical simulations that even though the classical models can have better performance as their sizes increase, they would suffer from overfitting. This fact bolsters the quantum advantage and suppresses the accuracy of classical models no matter their size.

We prove that the quantum model can be trained within constant time, and the advantage persists under constant strength depolarization noise. These nice properties enable demonstration on noisy quantum devices.

We achieve this by designing a sequence translation task called magic square translation task. It can be solved quantumly with constant parameter size using entanglement, whereas classical models must scale linearly to get an accuracy above an exponentially small value.

We rigorously establish a noise-robust, unconditional quantum advantage in machine learning, in terms of expressivity, inference speed, and training efficiency, compared to commonly used classical models. This includes all autoregressive and encoder-decoder models (e.g., GPTs)!