quantum-block-encoding-difference-of-gaussian

name	quantum-block-encoding-difference-of-gaussian
description	Quantum block encoding methodology for Difference-of-Gaussian (DoG) operators on periodic grids. Implements Linear Combination of Unitaries (LCU) framework without black-box oracles. Activation: quantum block encoding, DoG operator, quantum machine learning, quantum signal processing.

Quantum Block Encoding of Difference-of-Gaussian Operators

Methodology for constructing explicit quantum block encodings of Difference-of-Gaussian (DoG) operators on periodic grids using Linear Combination of Unitaries (LCU) framework.

Overview

The DoG operator is widely used in:

• Image processing (feature and edge detection)
• Quantum machine learning
• Finite-difference methods (Laplacian-of-Gaussian approximations)

This skill implements an explicit block encoding that achieves constant subnormalization factor λ=2, independent of grid size N, spatial dimension D, and stencil width.

Core Technique

DoG Decomposition

The DoG admits natural decomposition to two normalized Gaussian distributions:

DoG = Gaussian(σ₁) - Gaussian(σ₂)

Each Gaussian is preparable by explicit efficient circuits, with negation encoded using a single Pauli-Z gate on a branch-indicator qubit.

LCU Framework Mapping

The block encoding maps directly to Linear Combination of Unitaries without requiring:

• Signed amplitude loading
• Quantum random-access memory
• Black-box oracles

Key Properties

Property	Value	Independence
Subnormalization λ	2	Grid size N, dimension D, stencil width
Success probability	Exact closed-form via power spectrum	Input signal dependent
Scaling	O(h⁴)	Grid spacing h (fine grid limit)

Activation Keywords

• quantum block encoding
• DoG operator
• difference-of-gaussian quantum
• quantum signal processing
• LCU framework
• quantum machine learning encoding

Tools Used

• exec: Run quantum circuit simulations
• write: Generate circuit descriptions
• read: Load operator specifications

Implementation

Circuit Structure

[Gaussian Prep 1] ──┐
                    ├── [Controlled-SWAP] ── [Pauli-Z branch] ── Output
[Gaussian Prep 2] ──┘

Step 1: Gaussian State Preparation

Prepare two normalized Gaussian distributions with different scale parameters σ₁ and σ₂.

Step 2: Branch Indicator Setup

Initialize branch qubit in |+⟩ state to enable superposition.

Step 3: Controlled Operation

Apply controlled-SWAP or multiplexed rotation based on branch qubit state.

Step 4: Phase Kickback

Apply Pauli-Z on branch indicator to encode the subtraction (difference).

Step 5: Uncomputation

Uncompute Gaussian preparations and measure branch qubit for success probability.

Mathematical Foundation

Diagonalization

The DoG operator is diagonalized by the discrete Fourier basis:

DoG = F⁻¹ · diag(DoĜ(k)) · F

where DoĜ(k) is the transfer function in frequency domain.

Success Probability

Exact closed-form expression:

P_success = Σₖ |signal̂(k)|² · |DoĜ(k)|² / ||DoG||²

Weighted by input signal's power spectrum.

Frequency Response

Tunable wide-stencil bandpass filter controlled by two Gaussian scale parameters:

• σ₁: Inner Gaussian width
• σ₂: Outer Gaussian width

Usage Patterns

Pattern 1: Basic DoG Encoding

# Prepare grid parameters
N = 2^n  # Grid points (power of 2)
D = 2    # Spatial dimensions
sigma1 = 1.0
sigma2 = 2.0

# Build block encoding
encoding = DoGBlockEncoding(N, D, sigma1, sigma2)
circuit = encoding.build_circuit()

Pattern 2: Quantum Machine Learning Feature Detection

# For QML applications
from quantum_block_encoding import DoGEncoder

encoder = DoGEncoder(grid_size=256, dimensions=2)
encoded_features = encoder.encode_image_quantum(image_data)

Pattern 3: Finite Difference Laplacian Approximation

# Approximate Laplacian-of-Gaussian
sigma_ratio = 1.6  # Optimal for edge detection
operator = DifferenceOfGaussian(sigma, sigma * sigma_ratio)
block_encoded_op = operator.to_block_encoding()

Configuration

Parameters

Parameter	Type	Default	Description
N	int	256	Grid points per dimension (power of 2)
D	int	2	Spatial dimensions
sigma1	float	1.0	Inner Gaussian scale
sigma2	float	2.0	Outer Gaussian scale
periodic	bool	True	Periodic boundary conditions

Scale Parameter Selection

For edge detection: σ₂/σ₁ ≈ 1.6 (Marr-Hildreth optimal) For feature detection: σ₂/σ₁ ≈ 2.0 to 4.0

References

• arXiv:2604.09538 - "Explicit Block Encoding of Difference-of-Gaussian Operators on a Periodic Grid"
• Childs et al. - "Quantum linear systems algorithm" (LCU framework)
• Low & Chuang - "Hamiltonian simulation by qubitization"

Related Skills

• quantum-signal-processing
• quantum-machine-learning
• quantum-circuit-optimization

Notes

• Requires discrete Fourier transform implementation
• Success probability scales with signal frequency content
• Optimal for smooth signals with localized features
• Compatible with quantum singular value transformation (QSVT)