Filter Function Formalism for Unitary Quantum Operations

Alan Jobil Skaria Skaria

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.

Hendra I. Nurdin, Naoki Yamamoto

Joanna Wiśniewska

Alan Jobil Skaria Skaria

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.

Tobias Hangleiter, Pascal Cerfontaine, Hendrik Bluhm

Correlated, non-Markovian noise is present in many solid-state systems employed as hosts for quantum information technologies, significantly complicating the realistic theoretical description of these systems. In this regime, the effects of noise on sequences of quantum gates cannot be described by concatenating isolated quantum operations if the environmental correlation times are on the scale of the typical gate durations. The filter-function formalism has been successful in characterizing the decay of coherence under the influence of such classical, non-Markovian environments, and here we show it can be applied to describe unital evolution within the quantum operations formalism. We find exact results for the quantum process and a simple composition rule for a sequence of operations. This enables the detailed study of effects of noise correlations on algorithms and periodically driven systems. Moreover, we point out the method's suitability for numerical applications and present the open-source Python software package filter_functions. Amongst other things, it facilitates computing the noise-averaged transfer matrix representation of a unital quantum operation in the presence of universal classical noise for arbitrary control sequences. We apply the presented methods to selected examples.

Yuan Li, Xiu-Hong Sun, Yan-Ni Dou