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 $n \ge 2$”