Gate-library-dependent reflection costs

Let $g_U,D_U$ price the supplied $U^{\ast }$ and $U^T$ circuits. Let

have a declared ancilla-free implementation of gate count $z(n)$ and depth $d_0(n)$.

This is an additional implementation assumption. It is not a constant-cost elementary gate for arbitrary $n$.

A Bell-subspace reflection on two $n$-qubit registers consists of inverse Bell preparation, $Z_0(2n)$, and Bell preparation. It has count $4n+z(2n)$ and depth at most $4+d_0(2n)$.