Long Range Frequency Tuning for QML

2026-02-26

AI Summary

This work addresses the challenge in quantum machine learning that trainable frequency encoding often fails to effectively fit high-frequency target functions due to limited gradient-based optimizability. To overcome this limitation, the authors propose a grid initialization strategy based on ternary angular encoding, which positions target frequencies within locally tunable ranges. By integrating this approach with quantum Fourier series modeling and gradient-based optimization, the method substantially enhances the capacity for high-frequency function approximation. Experimental results demonstrate its effectiveness: on synthetic high-frequency data, the model achieves a median R² of 0.9969, and on the Flight Passengers dataset, it attains a median R² of 0.9671—representing a 22.8% improvement over baseline methods—thereby validating both its efficacy and superiority.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Quantum machine learning models using angle encoding naturally represent truncated Fourier series, providing universal function approximation capabilities with sufficient circuit depth. For unary fixed-frequency encodings, circuit depth scales as O(omega_max * (omega_max + epsilon^{-2})) with target frequency magnitude omega_max and precision epsilon. Trainable-frequency approaches theoretically reduce this to match the target spectrum size, requiring only as many encoding gates as frequencies in the target spectrum. Despite this compelling efficiency, their practical effectiveness hinges on a key assumption: that gradient-based optimization can drive prefactors to arbitrary target values. We demonstrate through systematic experiments that frequency prefactors exhibit limited trainability: movement is constrained to approximately +/-1 units with typical learning rates. When target frequencies lie outside this reachable range, optimization frequently fails. To overcome this frequency reachability limitation, we propose grid-based initialization using ternary encodings, which generate dense integer frequency spectra. While this approach requires O(log_3(omega_max)) encoding gates -- more than the theoretical optimum but exponentially fewer than fixed-frequency methods -- it ensures target frequencies lie within the locally reachable range. On synthetic targets with three shifted high frequencies, ternary grid initialization achieves a median R^2 score of 0.9969, compared to 0.1841 for the trainable-frequency baseline. For the real-world Flight Passengers dataset, ternary grid initialization achieves a median R^2 score of 0.9671, representing a 22.8% improvement over trainable-frequency initialization (median R^2 = 0.7876).

Problem

• quantum machine learning
• frequency tunability
• trainable encoding
• gradient optimization
• function approximation

Innovation

• quantum machine learning
• frequency encoding
• trainable frequencies
• ternary encoding
• grid initialization