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Long Range Frequency Tuning for QML

Michael Poppel, Jonas Stein, Sebastian Wölckert, Markus Baumann, Claudia Linnhoff-Popien

TL;DR

This work demonstrates through systematic experiments that frequency prefactors exhibit limited trainability: movement is constrained to approximately +/-1 units with typical learning rates, and proposes grid-based initialization using ternary encodings, which generate dense integer frequency spectra.

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).

Long Range Frequency Tuning for QML

TL;DR

This work demonstrates through systematic experiments that frequency prefactors exhibit limited trainability: movement is constrained to approximately +/-1 units with typical learning rates, and proposes grid-based initialization using ternary encodings, which generate dense integer frequency spectra.

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).
Paper Structure (28 sections, 9 theorems, 41 equations, 8 figures, 1 table)

This paper contains 28 sections, 9 theorems, 41 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Fixed-Frequency Variational Quantum Circuits
  4. Trainable-Frequency Variational Quantum Circuits
  5. Quantum Models as Universal Function Approximators
  6. Comparison of Encoding Gate and Ansatz Parameter Requirements
  7. Frequency Reachability Limitations in Trainable Models
  8. Experimental Setup
  9. Results
  10. Analysis
  11. Dense Grid Initialization via Ternary Trainable-Frequency VQCs
  12. Experimental Validation
  13. Discussion
  14. Conclusion
  15. Improved Encoding Gate Complexity with Ternary Encoding
  16. ...and 13 more sections

Key Result

Theorem 2.2

For a variational quantum circuit with angle encoding, the model output $f_{\boldsymbol{\theta}}(x)$ can be expressed as a Fourier series: where $\Omega$ is the frequency spectrum determined by the feature map $S(x)$ and observable $M$, and $c_{\omega}(\boldsymbol{\theta}) \in \mathbb{C}$ are parameter-dependent Fourier coefficients. For unary fixed-frequency circuits with $L$ layers, $|\Omega| =

Figures (8)

  • Figure 1: Target and trained predictions for the original Jaderberg experiment with unary initial prefactors (a), the frequency-shifted variant with unary prefactors showing poor fit (b), and the same shifted frequencies with ternary grid initialization demonstrating successful recovery (c).
  • Figure 2: Prefactor displacement and gradient analysis across 100 target functions. Initial prefactors were set to $\{1.01, 1.02, 1.03\}$ to ensure distinct gradient signals from the start of training.
  • Figure 3: Prefactor evolution for different learning rates with initial prefactors of $\{1.01, 1.02, 1.03\}$ for unary initialization on target functions with frequency spectrum $\Omega_2 = \{11, 11.2, 13\}$. Each panel shows the best-performing run (highest $R^2$ score) for that learning rate. Only the aggressive learning rate of 0.1 enables substantial prefactor movement, but this occurs unreliably as demonstrated by \ref{['fig:r2s_distorted_undistortedAlphas']}.
  • Figure 4: $R^2$ score interquartile ranges for frequency-shifted targets. Trainable ternary prefactors (LR = 0.001) achieve consistently high scores across all shifts, fixed ternary prefactors come close, while trainable unary and fixed unary prefactors fail.
  • Figure 5: Prefactor evolution for different learning rates with initial prefactors of $\{1.0, 1.0, 1.0\}$. For each learning rate, the evolution shows the prefactors during training for the experiment with the highest $R^2$ score. With learning rates of 0.001 and 0.01, prefactors remain coupled and evolve identically.
  • ...and 3 more figures

Theorems & Definitions (24)

  • Definition 2.1: Unary Fixed-Frequency VQC
  • Theorem 2.2: Fourier Representation of VQCs Schuld_2021
  • Corollary 2.3: Parameter Scaling Schuld_2021
  • Definition 2.4: Ternary Fixed-Frequency VQC
  • Definition 2.5: Unary Trainable-Frequency VQC
  • Theorem 2.6: Universal Approximation for Trainable-Frequency VQCs yu2022powerPhysRevA.104.012405
  • Remark 3.2: Dynamic Frequency Spectrum
  • Proposition 3.3: Unary Fixed-Frequency Requirements pérezsalinas2025universalapproximationcontinuousfunctions
  • proof
  • Proposition 3.4: Ternary Fixed-Frequency Requirements
  • ...and 14 more