Polynomial-time learning of shallow-circuit states without geometric restrictions

Determine whether there exists a polynomial-time classical algorithm that, given copies of an unknown n-qubit pure state ψ = U |0^n⟩ prepared by a shallow quantum circuit U with no geometric restriction, learns a description of ψ within small trace distance under the restricted access model that only provides copies of ψ (as opposed to black-box access to U).

Background

The paper distinguishes between learning a circuit U from input–output access and learning an output state ψ = U|0^n⟩ from copies of ψ alone. Although sample-efficient procedures exist for learning ψ in certain settings, the lack of query access to U makes the computational problem more challenging.

The authors provide an efficient algorithm for the 2D geometrically local case, leaving the geometry-free case unresolved. This open problem asks for a computationally efficient method in the more general setting without geometric constraints.