Necessity of the S gate for QAC^0

Determine whether the S gate is necessary for QAC^0, or whether QAC^0 can be realized using only Hadamard and generalized Toffoli gates.

Background

The paper proves that every QAC0 circuit can be approximately compiled into a circuit using generalized Toffoli, Hadamard, and S gates. It therefore establishes a discrete gate set, but does not determine whether the phase gate S contributes essential computational power.

Circuits using only Hadamard and generalized Toffoli gates have real matrix entries, which creates a possible obstruction. Nevertheless, the authors note that existing QAC0 constructions for Boolean functions do not use S, and that S is unnecessary for universal computation when circuit depth is allowed to grow linearly. The bounded-depth case remains unresolved.

References

One immediate question left open by our work is whether the $S$ gate is still necessary for $\QAC0$.

— The Robustness of QAC0  (2610.02154 - Grier et al., 1 Oct 2026) in Section 3, “Future directions”