Fundamental runtime barrier for continuous gate families

Determine whether the quasi-polynomial runtime required to control near-identity gates when properly learning logarithmic-depth circuits over continuous gate families is a fundamental limitation of local inversion methods or can be avoided by alternative techniques; the authors conjecture that it is a technical rather than fundamental barrier.

Background

The rigorous learning theorem requires a well-spaced discrete gate set, ensuring that an incorrect trial gate is separated from a valid factorization by a positive gap. Continuous gate families, such as the full Haar unitary group, do not satisfy this condition, so approximate inversions and near-identity residual factors must be controlled through more precise tomography.

The paper notes that existing analysis for Haar-random gates leads to quasi-polynomial runtimes for logarithmic-depth circuits. It remains unresolved whether this overhead reflects an inherent limitation of local factorization learning or merely the limitations of current error-control techniques.

References

At present, it remains unclear if this is a fundamental limitation to the method, or if there are clever workarounds that allow for controlling the effects of near-identity gates during the learning process. We conjecture that this is in fact a technical barrier and not fundamental to the practicality of these methods for continuous gate families.

— Proper Learning of Shallow All-to-All Quantum Circuits  (2608.20162 - Kordonowy et al., 20 Aug 2026) in Section 4, subsection “Broadening the setting”