This paper presents a theoretical classification framework for phase-only digital signal processing filters interpreted as Quantum Fourier Transform diagonal quantum operators. It builds on an earlier DSP-to-QFT filtering framework and focuses on identifying which DSP-inspired filter responses can be directly implemented as valid quantum operations.The main result is that only phase-only sampled filter responses are directly compatible with unitary quantum gates. This means that ordinary magnitude-changing filters, such as classical low-pass or high-pass filters, cannot be implemented directly as quantum gates without additional methods such as ancilla systems, block encoding, post-selection, or unitary dilation.The paper connects the framework to circulant operator theory, showing that direct QFT-diagonal quantum filters correspond to circulant unitary operators. This gives the framework a clearer mathematical foundation and separates valid unitary quantum filters from non-unitary classical filtering operations.Several structured examples are analysed, including delay filters, linear phase filters, polynomial phase filters, and chirp-like filters. Delay filters are shown to produce cyclic shifts in the computational basis, making them useful as simple building blocks for modular shifts and quantum-walk-style operations. The paper also discusses how structured phase filters can have compact descriptions, while arbitrary phase filters may require exponentially many parameters.The central design principle is that direct QFT-diagonal quantum filters must be phase-only, while practical or efficient implementations require compact phase structure. The paper does not claim a new quantum speedup, but instead provides a DSP-inspired classification and design perspective for valid unitary QFT-diagonal quantum filters.
This paper presents a theoretical classification framework for phase-only digital signal processing filters interpreted as Quantum Fourier Transform diagonal quantum operators. It builds on an earlier DSP-to-QFT filtering framework and focuses on identifying which DSP-inspired filter responses can be directly implemented as valid quantum operations.The main result is that only phase-only sampled filter responses are directly compatible with unitary quantum gates. This means that ordinary magnitude-changing filters, such as classical low-pass or high-pass filters, cannot be implemented directly as quantum gates without additional methods such as ancilla systems, block encoding, post-selection, or unitary dilation.The paper connects the framework to circulant operator theory, showing that direct QFT-diagonal quantum filters correspond to circulant unitary operators. This gives the framework a clearer mathematical foundation and separates valid unitary quantum filters from non-unitary classical filtering operations.Several structured examples are analysed, including delay filters, linear phase filters, polynomial phase filters, and chirp-like filters. Delay filters are shown to produce cyclic shifts in the computational basis, making them useful as simple building blocks for modular shifts and quantum-walk-style operations. The paper also discusses how structured phase filters can have compact descriptions, while arbitrary phase filters may require exponentially many parameters.The central design principle is that direct QFT-diagonal quantum filters must be phase-only, while practical or efficient implementations require compact phase structure. The paper does not claim a new quantum speedup, but instead provides a DSP-inspired classification and design perspective for valid unitary QFT-diagonal quantum filters.
This paper presents a theoretical extension of the DSP-to-QFT quantum filtering framework by studying magnitude-changing digital signal processing filters in a quantum setting. Earlier work introduced a DSP-inspired framework for QFT-based filtering and classified direct QFT-diagonal quantum filters as phase-only unitary operators. This paper addresses the next natural case: classical DSP filters that change magnitude, such as attenuation, low-pass, high-pass, smoothing, and band-selective filters.Magnitude-changing filters are generally non-unitary, meaning they cannot be implemented directly as ordinary quantum gates. This paper shows how bounded magnitude-changing responses can instead be represented inside larger unitary quantum operations using ancilla-assisted constructions, unitary dilation, post-selection, and block-encoding ideas.The paper explains how a sampled DSP response can be embedded into a larger unitary system so that the desired non-unitary filter appears as a post-selected block of the operation. It also discusses how responses with gain greater than one must first be scaled before embedding, and clarifies that the embedded operation represents the scaled response rather than the original unscaled response.A controlled-rotation interpretation is provided, showing how the magnitude of each sampled response determines an ancilla rotation, while the phase component can be treated as a unitary phase operation. The paper also derives the success probability of post-selection, gives bounds on that probability, and explains the expected number of repetitions required.A concrete low-pass filtering example is included to demonstrate how attenuation can be represented probabilistically and how the final quantum state is renormalised after successful post-selection. The paper also discusses implementation limitations, including the cost of unstructured frequency-dependent rotations, the effect of scaling, and the fact that post-selection is not deterministic.The central message is that magnitude-changing DSP filters are not direct quantum gates, but bounded versions can be embedded into larger unitary quantum operations. This completes a three-stage development of the framework: defining DSP-to-QFT filtering, classifying direct phase-only filters, and extending the framework to non-unitary magnitude-changing filters through block encoding and post-selection.
This preprint presents a digital signal processing (DSP) framework for interpreting Quantum Fourier Transform (QFT)-based quantum operations. The main contribution is the DSP-to-QFT filtering map, which translates selected z-domain transfer responses into sampled QFT-domain quantum operators. The paper shows that the QFT can be interpreted as sampling a formal z-transform-style representation of a quantum amplitude vector. Under this framework, DSP delay elements become QFT-domain phase ramps and produce cyclic shifts in the computational basis. Circular convolution is interpreted through sampled z-domain multiplication, while ordinary linear convolution is recovered through zero padding into a larger Hilbert space. The purpose of the paper is not to claim a new quantum speedup, but to provide a unified conceptual and mathematical framework connecting DSP ideas such as z-plane sampling, delay, convolution, and filtering with QFT-based quantum operations.
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations.
A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
Many signal processing tasks require intensive computations. Quantum computing promises to accelerate certain tasks, but algorithms must be designed around the limitations of quantum mechanics. This paper provides a quantum implementation of finite impulse response (FIR) filters, which are a widely used tool in classical signal processing. The filter can be parallelized and composed as part of larger quantum algorithms, with potential for speedups using quantum amplitude estimation and future fault-tolerant quantum hardware. To accomplish these properties, we introduce a state-space framework wherein sample-based signal processing is performed with unitary quantum circuits. We present the quantum delay gate as the quantum analog of the delay line in discrete-time signal processing. Unitary circuits intrinsically describe lossless systems, but we also implement lowpass and other bounded filters by emulating a projection operator. The FIR filter implementation is tested and demonstrated with a quantum circuit simulator.
Spectral filtering has been proposed as a route to regularization in quantum generative models: the quantum Fourier transform exposes the amplitude spectrum of a quantum circuit Born machine, and a diagonal filter suppresses the high frequencies associated with finite-sample noise, an operation whose classical counterpart seemingly requires manipulating an exponentially long amplitude vector. We examine whether this coherent operation produces anything that classical post-processing of samples from the unfiltered model cannot match. Measuring the filter against convolution with a symmetric probability kernel at matched sampling cost, which accounts for the post-selection overhead of attenuation, we derive necessary and sufficient conditions for the gap between the two to vanish. Magnitude (attenuating) filters obey a dichotomy: at a fixed affordability threshold, the filtered output is either a constant-size Fourier object with an efficient classical sampler, or the passband must widen until no fixed frequency is attenuated and the filter no longer smooths. In neither case does the filter create a quantum-classical separation. Whatever separation survives is inherited from the spectral phase of the input state. Numerical experiments on trained circuit Born machines confirm the classification and show that the deciding phases are invisible to the Born-rule training loss and set by the initialization. Within the diagonal family, pure phase filters remain the only spectral operations exempt from these constraints.