Width-one conservation conjecture

Prove that every measurement-free Clifford+T circuit of frontier width one with clean ancillas implementing a diagonal level-three gate satisfies \(\delta(\kappa)\le N_{\ker}\), and consequently uses at least \(\delta(U)\) T gates.

Background

For circuits with internal Hadamards, the paper reduces the desired lower bound to comparing the number Nker⁡N_{\ker} of odd branch-dependent parities with the T-count δ(κ)\delta(\kappa) of the interference correction phase.

The conjecture asserts this comparison for frontier width one, a structural condition on the temporal overlap of branch variables. It is supported by 2,153 realizable families but remains unproved; abstract fiber forms show that realizability constraints are essential.

References

Every measurement-free circuit of frontier width one with clean ancillas that implements a diagonal level-three gate satisfies $\delta(\kappa)\le N_{\ker}$, and hence $t\ge\delta(U)$ for every $h$.

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Conjecture 1, Appendix J, Section 10.5