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

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 $O(h<sup>4)$ 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.

• The paper introduces an explicit quantum circuit for block encoding the Difference-of-Gaussian operator, achieving constant subnormalization and efficient LCU integration.
• It employs Gaussian state-preparation circuits and controlled shift unitaries to discretize cyclic shifts on periodic grids, eliminating the need for QRAM and signed amplitude loading.
• Spectral analysis confirms the operator’s bandpass filtering properties with tunable frequency selectivity and a success probability scaling as O(h^4) in the fine-grid regime.

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

Introduction

This work proposes a quantum circuit construction for the Difference-of-Gaussian (DoG) operator on periodic grids via explicit block encoding. The DoG is a canonical bandpass filter with extensive usage in image processing, quantum algorithms for PDEs, and related applications. The essential innovation in this approach lies in decomposing the DoG stencil into a difference of two normalized discrete Gaussian distributions. This enables an efficient and explicit mapping to the Linear Combination of Unitaries (LCU) framework, eliminating the requirement for signed amplitude loading, quantum random-access memory (QRAM), or structure-agnostic oracle circuits.

Mathematical Foundations and DoG Structure

The periodic grid setup used is $\mathbb{Z}_N^D$ with $n$ qubits per dimension, for a total computational basis suitable for quantum simulation of discretized spatial domains. The DoG operator is formulated as

$A_h = \sum_{t \in T} (p_t - q_t) S_t$

where $S_t$ is the cyclic shift operator, and $p_t$, $q_t$ are samples of discrete normalized Gaussians at varying spatial scales ($\sigma_p < \sigma_q$), properly renormalized post-discretization. The operator is naturally Hermitian when constructed with symmetric kernels, and the 1-norm of its coefficients is universally bounded by 2, framing an inherent advantage for quantum LCU-based implementation.

Block-Encoding Circuit Design

The explicit quantum circuit leverages the structure of the DoG operator, dividing computation into three key register types: indicator (one qubit splitting between $p$ and $q$), shift-label (specifying the stencil offset), and data (encoding the state on the grid). Amplitude encoding for both Gaussians is achieved via Gaussian loader circuits $G_p$, $n$0. The circuit proceeds as follows:

• A Hadamard gate on the indicator qubit creates a superposition, preparing branches for $n$1 and $n$2.
• Controlled-Gaussian state-preparation routines load the distribution amplitudes into the shift-register.
• The sign needed for the difference is implemented by applying a Pauli-$n$3 gate to the indicator qubit.
• Controlled shift unitaries induce the respective $n$4 operation on the data register.

Explicit projection onto $n$5 registers realizes the desired non-unitary $n$6, postselected with subnormalization $n$7 that is independent of the stencil width, grid size, or spatial dimension.

The most expensive resources are the Gaussian state-preparation routines, which for small stencils can be implemented via $A_h = \sum_{t \in T} (p_t - q_t) S_t$0 rotations, but for larger stencils, an alternative $A_h = \sum_{t \in T} (p_t - q_t) S_t$1 routine with additional ancilla qubits can be used. Controlled shifts have polylogarithmic cost in grid size.

Spectral Analysis and Transfer Function

The paper provides a spectral characterization: $A_h = \sum_{t \in T} (p_t - q_t) S_t$2 is diagonalized by the discrete Fourier basis, with eigenvalues determined by the difference in the Fourier transforms of $A_h = \sum_{t \in T} (p_t - q_t) S_t$3 and $A_h = \sum_{t \in T} (p_t - q_t) S_t$4. This directly exposes the bandpass character of the DoG filter as implemented on the periodic quantum grid.

$A_h = \sum_{t \in T} (p_t - q_t) S_t$5

for each frequency $A_h = \sum_{t \in T} (p_t - q_t) S_t$6. For normalized inputs, DC response is always zero ($A_h = \sum_{t \in T} (p_t - q_t) S_t$7), and selectivity is tunable via the variance parameters $A_h = \sum_{t \in T} (p_t - q_t) S_t$8.

This spectral structure allows direct application of quantum linear algebra primitives, notably QSVT and qubitization transformations, without additional Hermitian dilations, provided $A_h = \sum_{t \in T} (p_t - q_t) S_t$9 symmetry is maintained.

Success Probability: Exact and Asymptotic Results

The probability of successful postselection (i.e., the amplitude squared in the $S_t$0 ancillas after applying $S_t$1 to a normalized state), is given by:

$S_t$2

where $S_t$3 is the spectral power of the input state. This quantifies the exact filtering action of the DoG block encoding: output amplitude is maximal when the input is concentrated within the DoG passband.

In the continuum limit ($S_t$4), for smooth input functions, the success probability scales as $S_t$5. This scaling originates from fundamental error and normalization constraints for second-order finite-difference approximations on quantum circuits. Unlike Laplacian encodings where the $S_t$6 factor appears as a diverging subnormalization, here the normalization is kept constant ($S_t$7) and the $S_t$8 appears in the amplitude instead, which has ramifications for QSVT-based transformations.

Figure 1: Scaling of the success probability with grid size $S_t$9, evidencing the theoretical $p_t$0 behavior and convergence between the exact and asymptotic analytic predictions as resolution increases.

Numerical Examples

The explicit bandpass filtering and exact postselection probability are validated numerically on one-dimensional grids for various sizes, input states, and Gaussian parameters. The empirical results confirm that the transfer function matches analytic predictions and that the success probability aligns tightly with both the exact formula and $p_t$1 scaling in the fine-grid regime.

Implications and Future Directions

The explicit block encoding presented here adapts naturally to any operator expressible as a signed combination of normalized distributions over spatial shifts. Key theoretical implications include:

• The approach achieves a constant subnormalization and eliminates the scaling bottlenecks present in structure-agnostic block encodings, which is crucial for practical, deep circuits.
• It provides an explicitly tunable, bandpass quantum filter—by varying $p_t$2 and $p_t$3, the frequency selectivity of the operator is finely controlled.
• The Hermitian structure and exact spectral form make this encoding directly compatible with the full spectrum of quantum linear algebra primitives, including time evolution for Hamiltonian simulation, polynomial eigenvalue filtering, and (bandpass) thresholding via QSVT.

On the practical side, this operator can serve as a direct quantum primitive in algorithms for quantum image processing, quantum PDE solvers, and quantum spectral filtering, providing a more resource-efficient path than approaches that rely on black-box data loading or general multi-controlled arithmetic. The design avoids QRAM, signed amplitude preparation, and expensive NEQR-style arithmetic circuits used in other quantum image processing schemes.

Potential extensions include adaptation to non-periodic boundary conditions (Dirichlet, Neumann), resource estimation and synthesis via toolkits such as Qualtran, and experimental deployment within quantum pipelines for edge detection or feature extraction in images.

Conclusion

The construction developed provides an explicit, efficient quantum block encoding for the discrete DoG operator, mapping its probabilistic structure directly to the LCU framework with constant subnormalization and well-characterized success probability. This enables practical quantum implementations of bandpass filters with direct spectral control, avoids generic data-loading overhead, and integrates with downstream QSVT-based quantum linear algebra methods. Future work can extend these techniques to broader classes of structured finite-difference operators and explore their utility in quantum algorithms for computational science and machine intelligence.