Approximate Quantum Fourier Transform in Logarithmic Depth on a Line

The approximate quantum Fourier transform (AQFT) on $n$ qubits can be implemented in logarithmic depth using $8n$ qubits with all-to-all connectivity, as shown in [Hales, PhD Thesis Berkeley, 2002]. However, realizing the required all-to-all connectivity can be challenging in practice. In this work, we use dynamic circuits, i.e., mid-circuit measurements and feed-forward operations, to implement the AQFT in logarithmic depth using only $4n$ qubits arranged on a line with nearest-neighbor connectivity. Furthermore, for states with a specific structure, the number of qubits can be further reduced to $2n$ while keeping the logarithmic depth and line connectivity. As part of our construction, we introduce a new implementation of an adder with logarithmic depth on a line, which allows us to improve the AQFT construction of Hales.