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

TL;DR

This work establishes a concrete link between quantum learning theory and thermodynamics by showing that learning algorithms can be made fully reversible and, once a quantum state is learned, additional copies can be erased at the optimal Landauer cost. The energy cost of erasing a class of quantum states scales with natural complexity measures (circuit depth, entanglement, magic, degree) and can be achieved efficiently for structured classes, while cryptographic hardness implies no efficient protocol can reach the information-theoretic optimum for otherwise hard ensembles such as pseudorandom states. The results also extend to work extraction, where learning-based protocols can realize maximal yields when learning is efficient, and they reveal fundamental differences between classical and quantum erasure costs. Overall, the paper provides a unified framework connecting quantum learning, complexity, and thermodynamics, with implications for energy-efficient quantum technologies and cryptographic considerations in quantum thermodynamics.

Abstract

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.

Figures (1)

• Figure 1: (a) Reversible learning algorithm $\mathcal{L}$ can acquire knowledge of the unknown $n$-qubit state and erase $N$ copies of it at the optimal work cost, saturating Landauer's limit. Here, $S$ stores the copies used by the learning algorithm, $R$ stores the rest, $M$ is the memory of the learning algorithm, and $M'$ is an auxiliary memory. (b) The work cost of erasing physically relevant classes of states grows with the complexity of the states, as measured by circuit depth $d$, magic $t$, entanglement entropy $\mathcal{S}$, and degree $k$. When the complexity is bounded by a constant, Landauer's limit can be achieved efficiently by learning. In contrast, when the complexity grows poly-logarithmically, no polynomial-time quantum algorithm can erase the states without paying a nearly maximal amount of work.

Theorems & Definitions (17)

• Definition 1: Landauer erasure reebImprovedLandauerPrinciple2014
• Remark 1: System Hamiltonian
• Remark 2: System state
• Remark 3: Ancilla qubits
• Remark 4: Finite-size effect
• Lemma 1: One-shot Landauer's principle faistMinimalWorkCost2015
• Lemma 2: Standard Landauer erasure
• proof : Proof of \ref{['lem:standard-erasure']}
• Remark 5: Computational complexity
• Remark 6: Unknown pure states