Characterizing multi-qubit $T$-count beyond denominator-exponent certificates

Characterize the minimal $T$-count of multi-qubit Clifford+$T$ elements by a closed, possibly grammar-dependent criterion—particularly one adapted to the structure of quantum-Shannon-decomposition output—rather than only by per-instance denominator-exponent certificates.

Background

For single-qubit Clifford+TT words, the paper obtains an exact syntactic characterization through phase linkage and membership in the ZZ-axis normalizer. At two or more qubits, the identity between minimal TT-count and the least denominator exponent fails, as illustrated by the controlled-SS gate. The surviving result is an unconditional lower-bound certificate, but the authors leave open whether a closed characterization—perhaps specialized to the grammar of quantum-Shannon-decomposition output—can replace that certificate.

References

Whether some closed characterization exists at $n \ge 2$, perhaps one that is grammar-dependent, keyed to the specific structure of QSD output rather than to arbitrary Clifford+$T$ words, is open, and it is the natural next target for anyone wishing to extend the single-qubit rigidity story to a genuine multi-qubit formula.

— An Exactness Barrier for ZX-Calculus Optimization of Synthesized Clifford+T Circuits  (2608.22801 - Kam et al., 24 Aug 2026) in Section 6, Discussion and Open Problems, subsection “A closed characterization at $n \ge 2$”

Both rest on sketches and numerics (Appendix~\ref{sec:layered}). Do they hold, and does $CCZ\,H{\otimes3}\,CCZ$ cost $14$ at three Hadamards, one above that floor, as the realizable parity sets suggest?

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Open Problem 2, Section 4.2; Appendix K, Section 11.2

Is the $\mathrm{GF}(8)$ oracle's $T$-count $20$, $23$, or strictly between?

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Open Problem 5, Section 4.2

The exact $T$-count of the multi-controlled $Z$ lies between the nullity $n$ (Corollary~\ref{cor:mctoffoli}) and the known $O(n)$ constructions, and we leave it open for three reasons.

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Appendix K, Section 11.4

The permutation's own $\delta_T(\pi)$ is open, since Proposition~\ref{thm:reduce} applies only to circuits that implement a diagonal gate.

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Appendix E, Section 9.2

On $CCZ\,H{\otimes3}\,CCZ$ the floor is $13$, while the minimum coset weight over the parity sets realizable by a prefix of the Hadamard order is $14$ for all six orders, which suggests a cost of $14$ at $h=n$. The prefix sets are not exhaustive, so this remains a conjecture.

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Appendix K, Remark 11.2