Explicit Block Encoding of Difference-of-Gaussian Operators on a Periodic Grid

• URL: http://arxiv.org/abs/2604.09538v1
• Date: Fri, 10 Apr 2026 17:55:51 GMT
• Title: Explicit Block Encoding of Difference-of-Gaussian Operators on a Periodic Grid
• Abstract summary: We construct an explicit quantum block encoding of the Difference-of-Gaussian operator on a periodic grid.<n>We derive an exact closed-form expression for the block-encoding success probability in terms of the input signal's power spectrum.<n>This implementation provides an explicit construction method for a tunable, wide-stencil bandpass filter.
• Score: 1.8920934738244026
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Difference-of-Gaussian (DoG) is a widely used operator across applications, including image processing (feature and edge detection), quantum machine learning, and finite-difference methods (approximations of the Laplacian-of-Gaussian). In this paper, we construct an explicit quantum block encoding of the DoG operator on a periodic grid, exploiting its natural probabilistic structure. The central observation is that the DoG admits a natural decomposition to two normalized Gaussian distributions, each preparable by explicit and efficient circuits, with the negation encoded using a single Pauli-$Z$ gate on a branch-indicator qubit. This enables the operator's block encoding to be directly mapped to the Linear Combination of Unitaries framework without requiring signed amplitude loading, quantum random-access memory, or any other black-box oracles. The proposed method achieves a constant subnormalization factor $λ= 2$ independent of the grid size $N$, the spatial dimension $D$, and the stencil width. Additionally, we show that the DoG operator is diagonalized by the discrete Fourier basis, which allows us to derive an exact closed-form expression for the block-encoding success probability in terms of the input signal's power spectrum, weighted by the operator's transfer function. Finally, we prove that the expression reduces to $O(h^4)$ scaling with respect to grid spacing $h$ as the periodic grid becomes finer. This implementation provides an explicit construction method for a tunable, wide-stencil bandpass filter whose frequency response is controlled by two Gaussian scale parameters.

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