Fermionic FFT in Quantum Systems

• Fermionic FFT is a generalized Fourier transform for many-body quantum systems that efficiently switches fermionic operators between real-space and momentum bases.
• It underpins scalable tensor network methods and low-depth quantum circuits by recursively decomposing operator transformations while preserving antisymmetry.
• The approach integrates classical FFT principles and group-theoretic decompositions to optimize simulation resources across free, interacting, and topological fermionic phases.

The Fermionic Fast Fourier Transform (Fermionic FFT or FFFT) generalizes the classical fast Fourier transform to the domain of many-body quantum systems with fermionic degrees of freedom. It encompasses not only operator-level transformations for efficiently switching between real-space and momentum representations of fermionic operators, but also extends to tensor networks, quantum circuits, and group-theoretic decompositions tailored to capture antisymmetry and many-body entanglement in computational and simulation tasks. The FFFT underpins several advances in the simulation and analysis of quantum systems, providing scalable, contractible, and symmetry-respecting representations essential for free, interacting, and topological fermionic phases.

1. Algebraic Definition and Operator Structure

The fundamental object of the FFFT is the transformation of fermionic creation and annihilation operators from a local (site) basis to a momentum basis. For a lattice of $n$ fermionic modes, the basis change implements

$$\tilde{c}^\dagger_k = \frac{1}{\sqrt{n}} \sum_{x=0}^{n-1} e^{\frac{2\pi i kx}{n}} \hat{c}^\dagger_x,$$

where $\hat{c}^\dagger_x$ is the creation operator at site $x$ and $k$ indexes the momentum mode. The FFFT decomposes this transformation recursively using even/odd index partitioning, in analogy to the Cooley–Tukey algorithm: $$\tilde{a}^\dagger_k = \frac{1}{\sqrt{n/2}} \sum_{x=0}^{n/2-1} e^{\frac{2\pi i kx}{n/2}} \hat{c}^\dagger_{2x}, \qquad \tilde{b}^\dagger_k = \frac{1}{\sqrt{n/2}} \sum_{x=0}^{n/2-1} e^{\frac{2\pi i kx}{n/2}} \hat{c}^\dagger_{2x+1}.$$ The full transform on $n$ sites is then

$$\tilde{c}^\dagger_k = \frac{1}{\sqrt{2}}\big[ \tilde{a}^\dagger_k + e^{2\pi i k/n} \tilde{b}^\dagger_k \big],$$

where the complex exponential factor is implemented as a phase-delay gate acting on the occupation number basis. Iterating this decomposition yields a circuit of depth $\log_2 n$ and $O(n\log n)$ local gates, corresponding to a log-depth tensor network (Ferris, 2013).

Careful management of fermionic statistics is crucial: all operator reorderings and unitary gates must preserve anticommutation relations. Jordan–Wigner or Bravyi–Kitaev mappings are typically used to represent fermionic operators as qubit operators in numerical and quantum circuit contexts (Maskara et al., 10 Sep 2025).

2. Spectral Tensor Networks and Hierarchical Representations

The spectral tensor network is a direct realization of the FFFT structure as a contractible tensor network. This network mirrors the data flow of the FFT: layers of two-site "beam-splitter" unitaries implement the recursive mixing of even and odd modes, while phase gates introduce the required complex phases per layer. The spectral tensor network efficiently captures both area-law and logarithmic violations of entanglement, representing translationally invariant free-fermion states and certain interacting states (Ferris, 2013).

A summary of key properties:

Feature	Spectral Tensor Network	Branching MERA
Tensors per Layer	Fewer (smaller overall network)	More
Contractibility	$O(n\log n)$ for local observables	Greater overhead
Long-range Entanglement	Encoded at network bottom	Unfolded by coarse-graining
Scaling of Contraction	$O(\chi^5 n)$ for one-site expec.	Higher in $\chi$

For interacting or paired (superconducting) systems, additional Bogoliubov layers can be incorporated at the top of the spectral tensor network to diagonalize anomalous or pairing terms (Ferris, 2013).

3. Quantum Circuit Models and Resource Scaling

Implementing the FFFT on quantum hardware involves recursive circuits composed of local mode-mixing unitaries and phase gates, with careful attention to fermionic parity and operator ordering. The basic computational primitive is the "fermionic permutation"—an operation that rearranges the mapping between logical fermionic modes and physical qubits while maintaining the correct antisymmetry.

Key circuit optimizations:

• Dynamic remapping of fermion-to-qubit correspondences, so fermionic two-mode gates can always be realized as local gates in the hardware layout.
• Fermionic permutations (including interleave/bit-reversal steps) are compiled using constant-depth Clifford circuits augmented by mid-circuit measurement and classical feedforward.
• With these primitives, the overall space-time overhead per FFFT layer is reduced from $O(N)$ (standard Jordan–Wigner) to $O(\log N)$ in the worst case, and $O(1)$ when the problem has further exploitable structure (e.g., lattice symmetries in 2D or 1D FFFT) (Maskara et al., 10 Sep 2025).

For a system with $N$ fermion modes, the FFFT circuit recurses via splitting into halves, performing FFFT subcircuits on each, and then applying "mixing" gates and fermionic permutations: $$c_k = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} e^{-i 2\pi k j / N} c_j.$$ Each permutation can be accomplished by an engineered combination of CNOT (Clifford) gates and is natively compatible with common error-correcting code operations.

4. Group-theoretic and Symmetry-resolved Decompositions

Beyond operator or tensor implementations, the FFFT encompasses a group-theoretic approach based on Fourier transformation over the symmetric group $S_N$. When many-body amplitudes are indexed by particle permutations, the Fourier transform over $S_N$ resolves the amplitudes into sums over irreducible representations (irreps), clarifying the role of antisymmetry for fermions: $$\hat{a}(1,1,\dots,1) = \sum_{\sigma \in S_N} a(\sigma) \text{sign}(\sigma),$$ where this contraction isolates the contribution in the antisymmetric irrep, giving, for standard transition amplitudes, the determinant of the associated single-particle matrix. This framework further generalizes to partially distinguishable particles and symmetry-suppressed transitions (Dufour et al., 23 Sep 2024).

Probability formulas involving resolved symmetry sectors take the form: $$P_{\{n_m\}} = \frac{1}{|\text{Stab}(\mathcal{i})||\text{Stab}(\mathcal{o})|} \sum_\lambda \frac{d_\lambda}{N!} \text{Tr}\left[ \hat{a}(\lambda)^\dagger \hat{j}(\lambda) \hat{a}(\lambda) \right],$$ where $d_\lambda$ is the dimension of irrep $\lambda$ and $\hat{j}(\lambda)$ encodes partial distinguishability.

This decomposition is especially powerful for analyzing many-body interference, resolving which symmetry types dominate, and explaining the occurrence of destructive interference or selection rules based on permutation invariance.

5. Fast Algorithms: Classical, Quantum, and Hybrid Implementations

The FFFT algorithmic structure inherits the complexity improvements of the classical FFT but adapted to the Fock space or operator algebra of fermions:

• On the classical side, decompositions inspired by Cooley–Tukey and Chinese remainder theorem (CRT) allow the transformation of large system states (or operator coefficients) by breaking the total Fourier transform into a sequence of smaller ones, reducing computational cost from $O(D^2)$ to $O(D \log D)$, where $D$ is the Hilbert space dimension (Lei et al., 8 May 2024).
• In a quantum computing context, efficient quantum circuit constructions for the FFFT utilize basis-encoding of data instead of amplitude encoding, bypassing notorious amplitude extraction bottlenecks and supporting low-depth circuits for multiple data sets (Asaka et al., 2019).
• The Fast Mode-Fourier-Transform (FMFT) protocol encodes the FFFT via a logarithmic-depth circuit built from two-site (neighboring-mode) unitaries, with the number of two-site gates scaling as $(N \log_2 N)/2$ for $N$ modes (Reslen, 25 Oct 2024).

These algorithmic innovations enable resource-efficient simulation of many-fermion dynamics, state-preparation, and Hamiltonian diagonalization for both translationally invariant and more general systems.

6. Applications, Extensions, and Computational Implications

The FFFT is a central subroutine and structural principle in:

• State preparation for translationally invariant free-fermion ground and excited states, including systems with pairing (e.g., superconductors) via Bogoliubov transforms (Ferris, 2013).
• Efficient contraction and expectation calculation in tensor networks representing ground, excited, or variationally optimized states of free and interacting fermions.
• Hamiltonian simulation of strongly correlated and topological materials, including systems with periodic or sparse nonlocal couplings (e.g., Sachdev-Ye-Kitaev models, Bethe chains) (Maskara et al., 10 Sep 2025, Reslen, 25 Oct 2024).
• Analysis of many-body interference in interferometric setups or quantum circuits, resolving symmetry-induced selection rules through $S_N$ Fourier methods (Dufour et al., 23 Sep 2024).
• Rapid computation of phase-space Wigner and Weyl functions in high-dimensional Hilbert spaces, particularly for fermions, by preserving the algebraic parity and ring structure via the CRT method (Lei et al., 8 May 2024).
• Scalable simulation of large quantum systems on early fault-tolerant quantum devices, as the reduction of fermionic to qubit overheads to $O(\log N)$ (or $O(1)$ for structured cases) minimizes gate counts and error correction needs (Maskara et al., 10 Sep 2025).

Contractibility, natural encoding of long-range entanglement, and efficient permutation compilation collectively position the FFFT as a unifying computational tool in modern quantum many-body physics and quantum information processing.

7. Limitations and Cautions

The application of the FFFT in fermionic systems requires careful handling of anticommutation, parity, and sign conventions (e.g., Jordan–Wigner strings). For Hilbert spaces labeled as $D = d^n$ (with $d$ odd), difficulties may arise due to the lack of isomorphism between $[\mathbb{Z}(d)]^n$ and $\mathbb{Z}(D)$, leading to arithmetic carry issues that must be resolved for accurate phase-space function evaluation or operator transformations (Lei et al., 8 May 2024).

Similarly, adaptation to interacting systems requires variational optimization, and preserving efficient contraction or simulation properties may depend on the structural features of the underlying Hamiltonian or circuit. Integration with error correction and real hardware depends on the platform's capabilities for mid-circuit measurement, classical feedforward, and nonlocal gate implementation, though these requirements align with progress in leading quantum technologies (Maskara et al., 10 Sep 2025).

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The Fermionic Fast Fourier Transform thus provides a deeply integrated set of protocols and representations for the efficient manipulation, simulation, and analysis of fermionic quantum systems. Its manifestations in tensor networks, quantum circuits, classical fast algorithms, and group-theoretic decompositions collectively yield a robust and flexible computational framework across physics, chemistry, and quantum information science.