Doped-chain dynamical Lie-algebra dimension conjecture

Determine whether the dynamical Lie algebra of the doped-chain gate set has dimension $4^{n-1}-2^{n-1}$ for all numbers of qubits $n$, beyond the numerically verified range through $n=9$.

Background

The paper studies extensions of the off-diagonal ansatz obtained by adding generators such as {XkYk+2}k=1n−2\{X_kY_{k+2}\}_{k=1}^{n-2}. These extensions can produce exponentially large dynamical Lie algebras, unlike the polynomial-dimensional matchgate and off-diagonal baseline families.

For the doped-chain construction, the dimension formula 4n−1−2n−14^{n-1}-2^{n-1} has been checked numerically only through n=9n=9. The paper explicitly presents the same formula as a conjecture beyond that range, leaving its validity for general nn unresolved.

References

For the doped chain, the formula $\dim=4{n-1}-2{n-1}$ is verified numerically through $n=9$ and conjectured beyond that range.

— 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”