Hardness reduction for the doped-chain gate set

Establish a hardness reduction for the doped-chain gate set whose dynamical Lie-algebra dimension is conjectured to be $4^{n-1}-2^{n-1}$.

Background

The doped-chain gate set is introduced as an extension of the off-diagonal nearest-neighbor construction. Its dynamical Lie algebra has been observed numerically to have exponential dimension, but the paper emphasizes that exponential algebra dimension alone does not establish computational hardness for the particular product inputs and readouts studied.

A formal complexity-theoretic hardness result for this specific gate set is therefore left unresolved. Such a reduction would clarify whether the exponentially large algebra corresponds to provable computational hardness rather than only to a loss of efficient full-algebra simulation guarantees.

References

A hardness reduction for this gate set remains open.

— Unflattening by Flattening -- How Input Distributions Shape Output Variance in Angle-Encoded Circuits  (2610.01446 - Strobl et al., 1 Oct 2026) in Section Background, paragraph “Circuits”