Proper agnostic learning for periodic-boundary matrix product states

Establish proper agnostic learning for periodic-boundary matrix product states of bond dimension at most D, meaning an algorithm that outputs a periodic-boundary matrix product state in the same class whose score against an arbitrary unknown density operator is within a prescribed additive error of the optimum over that class.

Background

The paper establishes proper agnostic learning for open-boundary matrix product states and bounded-degree tree tensor networks. It also observes that a periodic-boundary matrix product state of bond dimension D can be represented as an open-boundary matrix product state of bond dimension at most D^2, which yields an improper guarantee for the periodic-boundary comparator class: the output is open-boundary rather than periodic-boundary and therefore does not satisfy the desired properness requirement.

The unresolved issue is to obtain an algorithm whose output itself is a periodic-boundary matrix product state of the prescribed bond dimension while retaining an agnostic near-optimality guarantee. This would extend the paper’s proper-learning results beyond open-boundary tensor-network geometries.