Fig. 3
From: Reducing T-count and T-depth in approximate quantum Fourier transform circuits
AQFT Circuit 1: \(n\)-qubit AQFT with an error of \(O(\varepsilon )\) optimized for T-count, where \(b=\lceil{\text{log}}_{2}(n/\varepsilon )\rceil\). This circuit is decomposed into single-qubit \(H\) and \({Z}^{\theta }\) gates, and standard two-qubit CNOT gates. To improve readability, the \({Z}^{\theta }\) gates are aligned vertically to represent a gate layer; this does not imply a multi-qubit control structure. Each \({Z}^{\theta }\) gate layer in the blue boxes performs an inverse PGT with the help of a \({Z}^{-1}\) gate. The effect of inserting \({Z}^{-1}\) gates can be canceled by inserting \(Z\) gates. Note that this circuit contains \((n+1)\) inverse PGTs and these are constructed without using additional non-Clifford gates. We implement each inverse PGT using a quantum adder from Ref.37. Since the circuit is constructed entirely from unitary gates, it is inherently reversible.
