Computational complexity of average-case learning guarantees

Ascertain the classical computational complexity required to learn polynomial-size quantum circuits to small average-case distance (with respect to Haar-random input states), by developing efficient algorithms or proving hardness results that clarify whether such learning can be achieved in polynomial or quasipolynomial time.

Background

While the paper focuses primarily on worst-case (diamond-distance) guarantees, the authors discuss weaker, PAC-like average-case notions motivated by classical learning theory. Prior work has established polynomial sample complexity for average-case learning but has not resolved the computational complexity.

This open question seeks to determine whether achieving small average-case distance is computationally efficient, and under what assumptions, thus complementing the worst-case results presented in the paper.