Quantum Fourier Transform

• Quantum Fourier Transform is a unitary operation that converts computational basis states into periodic frequency representations, forming the backbone of quantum algorithms.
• The standard QFT circuit uses Hadamard and controlled-phase gates with O(n²) complexity, while approximations and dynamic circuits reduce resource costs for fault tolerance.
• QFT underpins landmark protocols such as Shor’s algorithm and phase estimation, driving advances in quantum arithmetic, simulation, and overall quantum computing performance.

The Quantum Fourier Transform (QFT) is a fundamental unitary transformation that underpins a broad array of quantum algorithms, providing the mechanism for extracting periodicity and frequency-domain information from quantum registers. Its mathematical structure is tightly connected to the discrete Fourier transform (DFT), yet it is efficiently implementable on quantum hardware using a sequence of elementary gates. The QFT is a critical component of landmark quantum algorithms such as Shor’s factoring algorithm and phase estimation, and it serves as the core for a variety of quantum protocols, arithmetic subroutines, and Hamiltonian simulation primitives.

1. Definition and Mathematical Structure

For an $n$-qubit register, the QFT is defined as the unitary operator $\mathrm{QFT}_n$ acting on the computational basis $\{\ket{x}:x\in\{0,\dots,2^n-1\}\}$ according to

$$\mathrm{QFT}_n:\quad \ket{x}\longmapsto \frac{1}{\sqrt{2^n}}\sum_{k=0}^{2^n-1} e^{2\pi i\,xk/2^n}\ket{k}.$$

In matrix terms, $$[\mathrm{QFT}_n]_{k,x} = \frac{1}{\sqrt{2^n}}e^{2\pi i\,xk/2^n}$$. For an arbitrary input state $\sum_x\alpha_x\ket{x}$, the QFT maps this to $$\sum_{k}(\frac{1}{\sqrt{2^n}}\sum_x\alpha_x e^{2\pi i\,xk/2^n})\ket{k}$$, performing an exact discrete Fourier transform on the amplitudes (Bäumer et al., 2024).

2. Circuit Decompositions and Resource Analysis

Standard Circuit Construction: The canonical unitary QFT circuit consists of:

• For each qubit $j$ (from most to least significant), apply a Hadamard gate $H$, then for each $k > j$, apply a controlled-phase gate $R_{k-j+1}$ with angle $2\pi/2^{k-j+1}$.
• Finally, perform a reversal of the qubit register via swap gates.

The controlled-phase operation $R_m$ is implemented as

$$R_m = \begin{pmatrix} 1 & 0 \ 0 & e^{2\pi i/2^m} \end{pmatrix}.$$

For an $n$-qubit system, the standard decomposition requires $n$ Hadamards and $O(n^2)$ two-qubit controlled-phase gates. Exact gate count is $n(n-1)/2$; limited connectivity adds SWAP overhead (Bäumer et al., 2024, Camps et al., 2020). The circuit depth is also $O(n^2)$ in the naive sequential implementation.

Approximate QFT and Fault-Tolerance: In fault-tolerant settings, the QFT is typically approximated by dropping controlled-phase gates with angles below a certain threshold $2^{-\ell}$, yielding diamond-norm error $O(\epsilon)$. The resource-dominant cost is quantified in T gates for Clifford+T synthesis:

• Standard (banded) AQFT: $T$-count $\simeq 8n\log_2(n/\epsilon)$, T-depth $\simeq 2n\log_2(n/\epsilon)$.
• Optimized constructions remove Toffoli gates, halving leading T-count to $\sim4n\log_2(n/\epsilon)$, and using parallel adders to reduce T-depth to $n\log_2(n/\epsilon)$ (Park et al., 2022).

3. Dynamic Circuits and Semi-Classical QFT

The dynamic-circuit (Griffiths–Niu) implementation exploits mid-circuit measurements and classical feed-forward, drastically reducing hardware resource requirements for QFTs followed immediately by measurement:

• The algorithm sequentially applies Hadamards, measures each qubit, and uses the measurement result to classically control subsequent single-qubit phase rotations, bypassing all two-qubit entangling gates.
• The dynamic QFT needs only $n$ mid-circuit measurements, $n$ Hadamards, and $O(n)$ classical control operations—no two-qubit gates or connectivity constraints (Bäumer et al., 2024).
• Empirically, this approach on superconducting hardware (ibm_kyiv) enables QFT$+$Measurement with certified process fidelity $>50\%$ up to $n=16$ and $>1\%$ up to $n=37$ qubits; these results exceed all previous multi-qubit QFT reports on any hardware platform. Key to this advance is a tailored dynamical decoupling protocol—feed-forward-compensated dynamical decoupling (FC-DD)—which protects idle qubits during measurement and feed-forward (Bäumer et al., 2024).

4. Alternative Physical and Algorithmic Realizations

Continuous-Variable QFT

The continuous-variable QFT (cvQFT) generalizes the transformation to bosonic mode systems. Mathematically, the cvQFT is implemented as a rotation operator $R(\Phi_\mathrm{DFT})$, where $e^{i\Phi_\mathrm{DFT}}$ is the DFT matrix $W_N$: $$\mathrm{cvQFT}_N = R(\Phi_\mathrm{DFT}),\quad [W_N]_{jk} = \frac{1}{\sqrt{N}}e^{2\pi i\,jk/N}.$$ The decomposition leverages phase shifters and beam splitters, with dense implementations requiring $O(N^2)$ elements, but FFT-inspired designs reduce the cost to $O(N\log N)$ (Cariolaro et al., 14 Dec 2025).

Quadratic QFT and Analog Simulation

The quadratic QFT (QQFT), defined on cold-atom lattices via second-quantized bosonic operators, is realized by concatenating local unitaries (on-site potentials and nearest-neighbor tunnelings) programmable by digital micromirror devices. QQFT generalizes QFT to the many-body sector (particle-number conserving), reproducing QFT in the single-particle sector. It enables programmable Hamiltonian engineering, realizing nontrivial models such as 1D Poincaré spacetime crystals and 2D perfectly flat Chern bands in $O(L\log L)$ steps, robust against both white and colored noise (Wang et al., 2022).

Photonic and Resonator Implementations

QFT protocols have been adapted to photonic systems (OAM encoding, cavity-QED mediated transformations) and to circuit QED via coupled oscillators and engineered cross-Kerr interactions:

• Photonic OAM QFT achieves an efficient $O(\sqrt{d}\log d)$ scaling in optical elements for high-dimensional Hilbert spaces.
• Cavity QED can mediate QFT among photonic qubits using a single atom as the ancilla and adjustable time-delay feedback, with tunable phase gates via Stark shifting (Shi et al., 2021).
• Superconducting resonator-based QFTs use state transfer and cross-Kerr evolution between two harmonic oscillators to perform the $q$-dimensional transform, yielding a fully quantum result without mid-circuit measurements or feed-forward (Chen et al., 2019, Kysela, 2021).

5. Applications and Algorithmic Impact

The QFT is central to the following classes of quantum algorithms and primitives:

• Period-finding, order-finding, and integer factoring: Shor’s algorithm is built around the QFT to extract hidden periodicities efficiently (Kashani et al., 2022).
• Quantum phase estimation: QFT translates eigenvalue information from the quantum register to measurable bit outcomes.
• Quantum arithmetic: Addition, multiplication, and computation of weighted averages can be efficiently implemented in the QFT basis. For example, the Draper adder achieves modular addition with $O(n^2)$ gates and requires no ancilla carries, a substantial shift from classical reversible-arithmetic strategies (Ruiz-Perez et al., 2014).
• Simulation of circulant Hamiltonians: QFT diagonalizes circulant matrices, permitting linear-system and time-evolution simulations (Zhou et al., 2015).

6. Entanglement Properties and Classical Simulability

Although the operator QFT is maximally entangling due to the bit-reversal layer, the "core" QFT (without bit-reversal) produces operator Schmidt coefficients that decay exponentially, imposing a constant upper bound on the entanglement generated across any partition, regardless of system size (Chen et al., 2022). This property enables classical simulation of the QFT on matrix product states (MPS) of low bond-dimension in $O(n)$ time, permitting compressive speedups over the FFT for data that are MPS-compressible. However, this advantage does not extend to arbitrary ("maximally entangled") data.

7. Generalizations and Advanced Techniques

Generalized QFTs (GQFT) parameterized by bilinear phase matrices $\Phi$ have been developed for nontrivial group structures, notably extending the QFT to applications in non-Abelian hidden subgroup problems. The quantum Haar transform, required for quantum wavelet transforms and Schur duality, can be generated with explicit recursive formulas and implemented with $O(n^2)$ gates (Shao, 2017).

Alternative circuit factorizations—such as composite CZ$+$SWAP blocks—have yielded constant-factor improvements in two-qubit-gate resource for the QFT. These approaches preserve the $O(n^2)$ scaling but optimize practical performance in quantum pipeline settings (e.g., as subcircuits in Harrow-Hassidim-Lloyd solvers) (Romero et al., 11 Jul 2025).

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Summary Table: Quantum Fourier Transform—Key Properties

Aspect	Standard QFT Circuit	Dynamic (Semi-Classical) QFT	Fault-Tolerant AQFT (Optimized)
Two-qubit gate count	$O(n^2)$	$0$	$O(n\log n)$ or $O(n\log\log n)$
Mid-circuit measurements	None	$n$	$O(n)$ classical; single-shot
Classical feedforward	None	$O(n)$ operations	Optional
Max. tested qubits (2024)	$11$ ($>1\%$ fidelity, unitary)	$37$ ($>1\%$ fidelity, dynamic)	$>1000$ in asymptotic analysis
T-count (AQFT, error $\epsilon$)	$8n\log_2(n/\epsilon)$	N/A	$4n\log_2(n/\epsilon)$
T-depth (AQFT, error $\epsilon$)	$2n\log_2(n/\epsilon)$	N/A	$n\log_2(n/\epsilon)$ (minimized)

• "Quantum Fourier Transform using Dynamic Circuits" (Bäumer et al., 2024)
• "T-count optimization of approximate quantum Fourier transform" (Park et al., 2022)
• "Quantum Fourier Transform Revisited" (Camps et al., 2020)
• "The Quantum Fourier Transform for Continuous Variables" (Cariolaro et al., 14 Dec 2025)
• "Quantum arithmetic with the Quantum Fourier Transform" (Ruiz-Perez et al., 2014)
• "Programmable Hamiltonian engineering with quadratic quantum Fourier transform" (Wang et al., 2022)
• "Quantum Fourier Transform Has Small Entanglement" (Chen et al., 2022)
• "A novel quantum circuit for the quantum Fourier transform" (Romero et al., 11 Jul 2025)
• "High-dimensional quantum Fourier transform of twisted light" (Kysela, 2021)
• "Quantum Fourier Transform in Oscillating Modes" (Chen et al., 2019)
• "Quantum Fourier Transform in Computational Basis" (Zhou et al., 2015)
• "Generalization of Quantum Fourier Transformation" (Shao, 2017)