Simpler constructions for the required CNOT-group moment bound

Determine whether the spectral bound for the representation \(P\mapsto P^{\otimes 2}\) required in the CNOT-to-quantum-expander augmentation theorem admits a simpler proof than the uniform finite-dimensional representation argument based on Kassabov’s generators.

Background

The augmentation theorem only requires a spectral estimate for the particular second-tensor-power representation of the CNOT group. The paper proves a substantially stronger statement: Kassabov’s generators have a uniform gap for every finite-dimensional unitary representation. The authors explicitly do not know whether the weaker, representation-specific estimate has a simpler proof.

References

Thus, \cref{thm:gap_from_kas} is stronger than necessary for \cref{thm:expander_CNOTs_to_unitary}, and we do not know whether the required bound for P\mapsto P{\otimes 2} admits a simpler proof.

— Depth-1 expanders on the unitary group and applications  (2609.01605 - Anshu et al., 1 Sep 2026) in Remark following Theorem 2, Section 2.3, “Augment generating sets of the group of CNOTs to quantum expanders”