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 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 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 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.
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.
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.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by
Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.