Converse characterization of computational hardness by one-way puzzles

Establish whether the existence of infinitely-often one-way puzzles is necessary for average-case computational hardness of universal quantum inductive inference with polynomial round complexity.

Background

The paper proves that the existence of infinitely-often one-way puzzles implies that no quantum polynomial-time learner can solve average-case universal quantum inductive inference with polynomial round complexity. This establishes one direction from a cryptographic assumption to inference hardness.

For the general quantum inference task, the paper does not prove the converse: namely, that average-case hardness of universal quantum inductive inference would imply the existence of infinitely-often one-way puzzles. The authors obtain such an equivalence only for the restricted POVM-based variant, in which the learner observes classical measurement outcomes and predicts the conditional distribution of the next outcome.