Learning beyond polylogarithmic depth

Identify structural conditions under which n-qubit quantum circuits of depth exceeding polylog(n) can be learned efficiently (e.g., in polynomial or quasipolynomial classical time), and develop corresponding learning algorithms with provable guarantees.

Background

The work establishes efficient learning for constant-depth circuits and shows hardness for general log-depth circuits in the non-geometric setting. In geometrically local settings, the authors obtain learning up to polylogarithmic depth (depending on dimension) with polynomial or quasipolynomial time.

The open question targets the boundary between tractable and intractable depths and seeks additional structural assumptions that would enable efficient learning beyond polylogarithmic depth.

References

What structural assumptions allow us to efficiently learn quantum circuits beyond polylog-depth remains an important open question.

— Learning shallow quantum circuits  (2401.10095 - Huang et al., 2024) in Discussion – Higher circuit depth

This identifies a clear computational obstruction for our algorithm to work in that case, albeit some more involved protocols could potentially overcome this. We conjecture however, that any such method necessarily runs into similar computational issues, rendering these states hard to learn.

— 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”