Physically natural metric-measure Lindbladian ensembles and their learning hardness

TL;DR

This paper investigates the difficulty of learning random open quantum system dynamics generated by Lindblad equations, establishing exponential lower bounds on query complexity and proposing cryptographic applications through Lindbladian-PUF protocols.

Contribution

It introduces physically motivated ensembles of random Lindbladians, extends statistical query frameworks to open systems, and proves exponential hardness results for learning these dynamics.

Findings

Proved exponential lower bounds on learning random Lindbladian dynamics.

Derived a linear-response expression for total variation distance in these ensembles.

Designed Lindbladian-PUF protocols with cryptographic security guarantees.

Abstract

In open quantum systems, a basic question at the interface of quantum information, statistical physics, and many-body dynamics is how well can one infer the structure of noise and dissipation generators from finite-time measurement statistics alone. Motivated by this question, we study the learnability and cryptographic applications of random open-system dynamics generated by Lindblad-Gorini-Kossakowski-Sudarshan (GKSL) master equations. Working in the affine hull of the GKSL cone, we introduce physically motivated ensembles of random local Lindbladians via a linear parametrisation around a reference generator. On top of this geometric structure, we extend statistical query (SQ) and quantum-process statistical query (QPStat) frameworks to the open-system setting and prove exponential (in the parameter dimension $M$) lower bounds on the number of queries required to learn random Lindbladian dynamics. In particular, we establish average-case SQ-hardness for learning output distributions in total variation distance and average-case QPStat-hardness for learning Lindbladian channels in diamond norm. To support these results physically, we derive a linear-response expression for the ensemble-averaged total variation distance and verify the required nonvanishing scaling in a random local amplitude-damping chain. Finally, we design two Lindbladian physically unclonable function (Lindbladian-PUF) protocols based on random Lindbladian ensembles with distribution-level and tomography-based verification, thereby providing open-system examples where learning hardness can be translated into cryptographic security guarantees.