QFT ideal and finite-set gate-cost models

The ideal binary-QFT gate-count model treats each Hadamard $H$ and each exact controlled phase ${\mathrm{C}\mathrm{P}}_s$ as one elementary gate, for integer $s\ge 2$, where

$H=2^{-1\mathrm{/}2}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right),{\mathrm{C}\mathrm{P}}_s=\mathrm{diag}(1,1,1,e^{2\pi i\mathrm{/}{2}^s})\mathrm{.}$

A named SWAP, the wire permutation from Ordered tensor registers, their inner product, and wire permutations, counts as one two-qubit gate. If the implementation expands each SWAP into three exact CNOT gates of CNOT and Control-Target Order, count those three CNOTs in place of that SWAP. These conventions refine the one-qubit alphabet, controlled-phase rotation and elementary circuit-resource interfaces of Exact one-qubit gates and rotations, Controlled Phase Rotation and Finite circuit resource accounting.

The finite-set synthesis model keeps ideal gate count separate from compiled gate count. A one- or two-qubit target block is assigned a projective operator-error tolerance $\delta >0$; a scalar phase returned by the compiler is retained as metadata for the whole standalone circuit. Each CNOT expanded from a SWAP is exact in this model. Any compiled-count theorem must state the fixed-arity decomposition, finite alphabet, inverse-closure, density, exact-arithmetic and compiler-threshold hypotheses it uses. In particular, it makes no claim about classical bit-time for computing an arbitrary target-angle description. This is a cost convention; correctness and compiler resource bounds are separate theorems.

The adjoint ${\mathrm{C}\mathrm{P}}_s^{†}=\mathrm{diag}(1,1,1,e^{-2\pi i\mathrm{/}{2}^s})$ also counts as one ideal gate when it occurs in an inverse circuit.