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|0n⟩ 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.

References

While ψ=U{0n} can be learned from polynomially many copies, the restricted access model makes the problem computationally more challenging, and the question of whether there exists a polynomial time algorithm remains open.

— Learning shallow quantum circuits  (2401.10095 - Huang et al., 2024) in Subsubsection: Learning output states of geometrically-local shallow quantum circuits

Finally, it remains an open question to what extent these ``structured'' circuit learning techniques can be applied beyond the query setting.

— Proper Learning of Shallow All-to-All Quantum Circuits  (2608.20162 - Kordonowy et al., 20 Aug 2026) in Section 5, Discussion

Whether or not our guarantees can be extended to a robust setting is one of the main open questions we leave for the future. As we already stated, in Ref., an efficient learning algorithm is given also for the case where the Cliffords showcase some amount of magic and the product structure is slightly perturbed to have some entanglement. This is only possible in their case given full oracle access to the unitary and its inverse. In our case, under these perturbations, namely for state vectors of the form \begin{equation} U_CtU_d\ket{0}{\otimes n}, \end{equation} with $U_Ct$ a $t$-doped Clifford, and $U_d$ an arbitrary depth-$d$ circuit, for $t,d$ small enough, the exact quadratic constraints become noisy.

— Efficient learning of Clifford disentanglers and typical $t$-doped unitaries with exponentially more $T$ gates  (2609.27565 - Aguilar et al., 23 Sep 2026) in Discussion and future work, paragraph “Open questions”

It is known that this class of states can be efficiently learned~\; however, it is unclear how to exploit such a structure for cloning.

— Approximate cloning of structured pure states  (2610.06723 - Herasymenko et al., 5 Oct 2026) in Section 7, subsection “Outlook”