Quantum Hartley Transform

• Quantum Hartley Transform is a real-valued, unitary quantum transform derived from the classical Hartley transform, enabling efficient encoding of real data in quantum circuits.
• It employs techniques such as recursive decomposition and the Linear Combination of Unitaries to reduce circuit depth and gate complexity compared to traditional approaches.
• Its applications span quantum generative modeling, secure quantum money, and serving as a subroutine in real quantum transforms, while offering unique advantages in operator theory and spectral analysis.

The Quantum Hartley Transform (QHT) is a unitary quantum algorithmic analogue of the discrete Hartley transform, engineered for quantum circuits to process and represent real-valued data efficiently through real amplitudes. Unlike the quantum Fourier transform, which operates in the regime of complex-valued bases, the QHT enables quantum algorithms to use a real-valued orthonormal basis constructed from the Hartley kernel (cas x = cos x + sin x). This capability makes the QHT a foundational component in quantum generative modeling, quantum money schemes, quantum signal processing, and as a subroutine for other quantum real transforms.

1. Mathematical Foundations and Classical Origins

The classical discrete Hartley transform (DHT) for a function $f: \mathbb{Z}_N \to \mathbb{R}$ is defined as

$$H_N(f)(a) = \frac{1}{\sqrt{N}} \sum_{y=0}^{N-1} \operatorname{cas}\left(\frac{2\pi a y}{N}\right) f(y), \quad \operatorname{cas}(x) = \cos(x) + \sin(x)$$

In the quantum setting, the action of the QHT on the computational basis is

$$\mathrm{QHT}_N\,|a\rangle = \frac{1}{\sqrt{N}} \sum_{y=0}^{N-1} \operatorname{cas}\left(\frac{2\pi a y}{N}\right) |y\rangle$$

The kernel’s real-valued structure ensures all output amplitudes are real, making the QHT involutory ($\mathrm{QHT}_N^2 = I$) and self-adjoint, properties used both in security proofs and in practical circuit design (Wu et al., 6 Jun 2024, Doliskani et al., 24 Mar 2025, Ahmadkhaniha et al., 18 Oct 2025).

2. Quantum Circuit Realizations and Algorithmic Techniques

Quantum circuits for the QHT have evolved significantly:

• Recursive Decomposition: Early approaches decompose QHT$_N$ recursively in analogy with the quantum fast Fourier transform, splitting registers by significance and applying Hadamard, conditional negation, and controlled-rotation gates (Doliskani et al., 24 Mar 2025). This method achieves depth $O(\log^2 N)$ with improvements over black-box approaches.
• Linear Combination of Unitaries (LCU): The QHT can be efficiently expressed as a linear combination of the QFT and its adjoint (Ahmadkhaniha et al., 18 Oct 2025):

$H_N = \frac{1-i}{2} F_N + \frac{1+i}{2} F_N^*$

The LCU method uses an ancilla, block-diagonal control, and amplitude amplification to implement $$V = \frac{1}{\sqrt{2}}[e^{-i\pi/4} I + e^{i\pi/4} T]$$, where $T$ is the two's complement unitary. This procedure achieves a fourfold reduction in circuit size compared to prior recursive constructions, with asymptotic gate complexity $O(\frac{1}{2}\log^2 N + \log N)$.

• Circuit-Level Optimizations: Notable optimizations include efficient implementation of the conditional two’s complement using ancilla-driven ripple-carry adders and avoiding large multi-controlled gates by or-tree (“binary tree of or-gates”) constructions. These reduce hardware overhead and error susceptibility, which is critical on near-term quantum devices (Ahmadkhaniha et al., 18 Oct 2025).

Approach	Gate Depth	Key Primitives
Recursive	$O(\log^2 N)$	Hadamard, cond. negation, $U_R$
LCU-based	$O(\frac{1}{2}\log^2 N)$	QFT, two's complement, amplitude amplification
Circuit-level optimization	Reduced depth	Inc$_n$ with carry, or-tree

3. Core Properties and Theoretical Advantages

• Real-Valued Amplitudes: By construction, the QHT outputs quantum superpositions with real amplitudes. This feature leads to new identities not satisfied by complex-amplitude transforms, such as

$$\sum_{j} |\phi_j\rangle|\phi_j\rangle = \sum_{j} |\psi_j\rangle|\psi_j\rangle$$

for any pair of orthonormal real bases, which does not generally hold over the complex numbers (Doliskani et al., 24 Mar 2025). This property is relevant for cryptographic and proof-related applications.

• Self-Inverse: The involutive nature ($\mathrm{QHT}_N^2 = I$) facilitates straightforward design of uncompute and sample-back steps, and simplifies state preparation and sampling in generative and cryptographic protocols.
• Unitary Structure: The QHT remains unitary (hence information-preserving and reversible) and adapts cleanly to the constraints imposed by quantum circuits.

4. Applications in Quantum Algorithms and Cryptography

• Quantum Generative Modeling: The QHT circuit, with its Hartley feature map, is used for encoding data via real-valued amplitudes, enabling gradient-based optimizations for regression, generative modeling, and solutions to stochastic differential equations. These models exploit the differentiability and structure of the Hartley kernel for inductive bias and regularization (Wu et al., 6 Jun 2024).
• Quantum Money: In public-key quantum money schemes, substitution of the QFT with the QHT produces “real banknotes” with real amplitudes. The group action “twist” is used in verification algorithms to resolve sign ambiguities induced by the symmetry of the Hartley basis, with acceptance probabilities matching those of QFT-based constructions but offering real-amplitude advantages (Doliskani et al., 24 Mar 2025).
• Subroutines for Real Transforms: The QHT serves as a platform for constructing other real quantum transforms, e.g., quantum sine and cosine transforms, via base change unitaries and ancilla management. Efficient implementation via QHT achieves logarithmic scaling in circuit complexity.
• Efficient Sampling and State Preparation: The self-inverse and orthonormal properties of the Hartley basis facilitate efficient mapping between latent and computational bases. This supports advanced generative sampling, including multidimensional and correlated distributions, using extended registers and differentiable circuits (Wu et al., 6 Jun 2024).

5. Connections to Functional Analysis, Algebra, and Mathematical Physics

• Operator Theory and Convolution Algebras: The QHT inherits structure from classical convolution algebras (e.g., Mellin-based convolutions for half-Hartley transform), preserving invertibility and diagonalization properties that are leveraged in quantum circuits for filtering and transformation (Yakubovich, 2014).
• Finite Field and Algebraic Generalizations: The Hartley kernel admits natural analogues over finite fields, constructed using $k$-trigonometric functions and “cask” kernels, supporting error-correction and quantum coding applications in finite-dimensional Hilbert spaces (Souza et al., 2015).
• Supersymmetry and Spectral Theory: Analytical eigenvectors for both finite and quantum Hartley transforms can be constructed via intertwining operators drawn from $N=1/2$ supersymmetric quantum mechanics, leading to overcomplete eigenbases formed from supersymmetric Hermite polynomials. The commutation relations and Poisson summation establish spectral decompositions with applications in continuous and discrete quantum systems (Bouzeffour, 15 Jun 2024).

6. Software and Implementation Ecosystem

QRTlib provides a library with Qiskit implementations for quantum real transforms, including QHT, cosine, and sine transforms of multiple types. Key contributions include a QHT built on the LCU framework, recursive and circuit-level circuit optimizations (e.g., two’s complement, or-tree detection), and practical templates for near-term devices (Ahmadkhaniha et al., 18 Oct 2025). The simulation and training workflows exploit quantum software stacks such as PennyLane, Qadence, PyTorch, and JAX, ensuring compatibility with prevalent quantum programming environments (Wu et al., 6 Jun 2024).

7. Comparative Analysis, Challenges, and Future Prospects

QHTs offer resource and circuit complexity savings (up to fourfold reduction in gate count in the LCU approach versus naive recursion (Ahmadkhaniha et al., 18 Oct 2025)), and foster opportunities for real-amplitude quantum computation in machine learning and cryptography. Major challenges persist in scalable circuit synthesis—particularly for arithmetic operations required in recursive decompositions, maintaining coherence under circuit depth constraints, and optimizing amplitude amplification and reflection steps. Current trends suggest ongoing integration of QHT-based subroutines into quantum algorithm design, with strong prospects for further reductions in hardware requirements and broader adoption in both foundational and applied quantum information science.