FFT-Optimized Quantum Fourier Transform

• FFT-optimized QFT is a method that leverages classical FFT divide-and-conquer strategies to reduce circuit depth and gate count in quantum algorithms.
• It employs recursive matrix factorizations, butterfly network decompositions, and garbage-free arithmetic to optimize quantum circuit implementations.
• This approach is crucial for both discrete and continuous-variable quantum computing, improving resource efficiency in quantum signal processing and fault tolerance.

An FFT-optimized Quantum Fourier Transform (QFT) refers to a suite of circuit constructions, complexity improvements, and encoding strategies that exploit classical fast Fourier transform (FFT) recursion and block factorizations to accelerate or optimize the implementation of QFT in quantum algorithms. This includes the direct application of “divide-and-conquer” decompositions, butterfly networks, garbage-free basis representations, and resource-aware approximation protocols, spanning discrete and continuous-variable quantum information settings. FFT-optimized QFTs yield significant reductions in circuit depth, gate count, and overhead compared with traditional “textbook” QFT circuits, and are foundational in fault-tolerant quantum algorithms and quantum signal processing.

1. Mathematical Foundations and Circuit Decomposition

The standard QFT is a linear transformation on quantum state amplitudes, mapping $\sum_{j=0}^{N-1} x_j |j\rangle$ to $\sum_{k=0}^{N-1} X_k |k\rangle$ with $$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$. FFT-optimized approaches recast this transform via recursive matrix factorizations, mirroring classical FFT decimation-in-time schemes. The DFT matrix $F_N$ can be factorized into block-diagonal and "twiddle factor" components:

$F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$

Here, $P_N$ is the bit-reversal permutation, and each $A_N^{(k)}$ consists of tensor products of Hadamard gates and diagonal phase matrices, enabling circuit implementations with layers of controlled-phase gates and single-qubit rotations (Camps et al., 2020, Marquezino et al., 2010).

The QR decomposition of each FFT "butterfly" step $P^{(s)}$ produces a sequence of Hadamard and controlled-phase operations, precisely matching the gate-level definitions of QFT subroutines:

• Hadamard: $H = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\1&-1\end{pmatrix}$
• Controlled-phase: $R_m = \operatorname{diag}(1, e^{2\pi i / 2^m})$

This explicit mapping from classical FFT recursion to quantum gate networks not only demonstrates the mathematical equivalence but also lays the foundation for parallel execution and resource-efficient circuit layouts (Marquezino et al., 2010).

2. FFT-Based Resource Optimization

Classical FFT recursion enables $\sum_{k=0}^{N-1} X_k |k\rangle$0 complexity for DFT computation. In the quantum regime, standard QFT circuits (without FFT optimization) require $\sum_{k=0}^{N-1} X_k |k\rangle$1 gates for $\sum_{k=0}^{N-1} X_k |k\rangle$2 qubits (one Hadamard per qubit, $\sum_{k=0}^{N-1} X_k |k\rangle$3 controlled-phase gates, $\sum_{k=0}^{N-1} X_k |k\rangle$4 swaps). FFT optimization restructures QFT circuits to mirror FFT’s butterfly stages:

• All $\sum_{k=0}^{N-1} X_k |k\rangle$5-separated controlled-phase gates at each stage $\sum_{k=0}^{N-1} X_k |k\rangle$6 commute and can be executed in parallel.
• Circuit depth compresses from $\sum_{k=0}^{N-1} X_k |k\rangle$7 to $\sum_{k=0}^{N-1} X_k |k\rangle$8 on fully connected hardware (Roy et al., 2024, Marquezino et al., 2010).
• Gate complexity for continuous-variable QFT (cvQFT) via FFT is reduced from $\sum_{k=0}^{N-1} X_k |k\rangle$9 (Murnaghan synthesis) to $$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$0: $$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$1 beam splitters, $$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$2 phase shifters (Cariolaro et al., 14 Dec 2025).

For approximate QFT, FFT-inspired grouping of phase rotations achieves a $$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$3 T-count, with explicit layer architectures and ancilla reuse via phase-gradient registers (Nam et al., 2018). The following table contrasts gate complexity for representative FFT-optimized circuits:

Approach	Gate Complexity	Depth
Textbook QFT	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$4	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$5
FFT-optimized QFT	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$6	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$7
Fault-tolerant (approx.)	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$8	$$X_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{2\pi ijk/N} x_j$$9
FFT-based AQFT	$F_N$0 (T gates)	$F_N$1

3. Garbage-Free Quantum FFT Networks

Quantum FFT networks operating on basis-encoded data leverage tensor-product states: $F_N$2. Here, arithmetic operations—addition, subtraction, left/right shift—are implemented as reversible gate sets (Toffoli, Peres, CNOT), specifically engineered to avoid garbage bit accumulation (Asaka et al., 2019).

Butterfly decomposition translates into “controlled subtraction,” “arithmetic shifts,” and “twiddle-shear” matrices, each realized with fixed-gate reversible blocks. Sign-extension ancillas are initialized and recycled layer-wise, further minimizing ancilla footprint. This explicit resource management enables efficient quantum image processing, convolution, and batch FFTs in parallel without extraction bottlenecks that afflict amplitude-encoded QFTs.

4. FFT Optimization in Continuous-Variable Quantum Fourier Transform (cvQFT)

For bosonic continuous-variable systems, cvQFT is defined by the rotation operator $F_N$3 with $F_N$4 corresponding to the DFT matrix $F_N$5. Implementation via Murnaghan procedure decomposes the unitary into single-mode rotations and beam splitters; when $F_N$6 is a power of $F_N$7, FFT-based decimation enables parallelization of $F_N$8-mode gates intermixed with “twiddle factor” phase-shifters.

Recursively, each level of the FFT structure applies independent cvQFTs to even and odd mode subspaces, recombines via 50:50 beam splitters, and iterates for $F_N$9 layers, yielding asymptotic resource scaling matching classical FFT (Cariolaro et al., 14 Dec 2025).

Transformation of Gaussian states under cvQFT is analytically tractable: the displacement vector, squeeze matrix, and rotation matrix undergo DFT, two-dimensional DFT, and Fourier-like similarity transformations, respectively. Covariance matrices transform as $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$0 with $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$1.

5. Approximate and Fault-Tolerant FFT-Optimized QFTs

Optimized QFT implementations for fault-tolerant quantum computing leverage FFT-inspired grouping and approximation:

• Small-angle phase rotations (controlled-$F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$2 with large $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$3) are truncated, bounding the spectral-norm error by choosing $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$4 layers.
• The circuit architecture employs phase-gradient ancilla registers, mid-circuit measurements, and classical feedforward, achieving $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$5 T gates for error $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$6 (Nam et al., 2018).
• Practical circuits demonstrate 3–10$F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$7 reduction in $F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$8-count over prior fully-coherent AQFTs, directly reducing the overhead of magic-state distillation in quantum algorithms based on QFT (e.g., Shor’s, phase estimation).

6. Extensions and Versatility of FFT-Optimized QFTs

FFT-optimizations transpose to quantum inverse fast Fourier transforms (QIFFT), discrete cosine/sine/wavelet transforms, and qudit-based systems:

• QIFFT employs the same decimation-in-time structure, with depth and gate counts matching forward FFT-optimized QFTs ($F_N = P_N \prod_{k=0}^{n-1} A_N^{(k)}$9 gates, $P_N$0 circuit depth), outperforming the naive inversion of QFT (Roy et al., 2024).
• Radix-$P_N$1 generalizations for qudits extend FFT-inspired decompositions, with Kronecker-sum factorization adapting controlled-phase gates and single-qudit DFT blocks, maintaining $P_N$2 gate complexity for $P_N$3-dimension systems (Camps et al., 2020).
• FFT-optimized quantum circuits underpin quantum image processing and convolution, offering direct parallel evaluation across superposed registers or images, seamless integration with basis-encoded data, and efficient output measurement schemes (Asaka et al., 2019).

7. Comparison with Standard QFT and Practical Implications

FFT-optimized QFTs offer several advantages compared to textbook (amplitude-encoding) QFT circuits:

• The resource scaling lowers gate count from $P_N$4 to $P_N$5.
• Circuit depth compresses from $P_N$6 to $P_N$7 under parallelization.
• Garbage-free arithmetic circuits are available for basis-encoded QFFT, beneficial for data-intensive applications and batch processing.
• Output can be directly measured from individual data registers (no tomography), facilitating practical integration in quantum signal and image processing, and mitigating classical extraction overhead.

A plausible implication is that FFT-optimized QFT constructions and their variants will underlie the design of scalable quantum subroutines for arithmetic, simulation, data analysis, and high-fidelity computation as quantum hardware matures. The class of FFT-inspired circuit architectures continues to bridge the gap between quantum advantage and classical algorithmic efficiency.