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Evaluating Parameter-Based Training Performance of Neural Networks and Variational Quantum Circuits

Michael Kölle, Alexander Feist, Jonas Stein, Sebastian Wölckert, Claudia Linnhoff-Popien

TL;DR

This paper tackles the problem of parameter efficiency in machine learning by directly comparing classical neural networks (NNs) with variational quantum circuits (VQCs) on simple supervised and reinforcement learning tasks. The authors employ a grid-search approach to pair NN and VQC configurations with similar performance, predominantly using VQC simulations and estimating real-hardware training times by executing selected circuits on IBM hardware. They demonstrate that VQCs can match or slightly exceed NN performance while using significantly fewer parameters, albeit with substantially longer training times in the current setup, and they provide hardware-time estimates to gauge practical feasibility. The work highlights that, as quantum technology advances and VQC architectures improve, VQCs may offer distinct advantages for particular applications, and it provides a reproducible framework and openly available code for future benchmarking.

Abstract

In recent years, neural networks (NNs) have driven significant advances in machine learning. However, as tasks grow more complex, NNs often require large numbers of trainable parameters, which increases computational and energy demands. Variational quantum circuits (VQCs) offer a promising alternative: they leverage quantum mechanics to capture intricate relationships and typically need fewer parameters. In this work, we evaluate NNs and VQCs on simple supervised and reinforcement learning tasks, examining models with different parameter sizes. We simulate VQCs and execute selected parts of the training process on real quantum hardware to approximate actual training times. Our results show that VQCs can match NNs in performance while using significantly fewer parameters, despite longer training durations. As quantum technology and algorithms advance, and VQC architectures improve, we posit that VQCs could become advantageous for certain machine learning tasks.

Evaluating Parameter-Based Training Performance of Neural Networks and Variational Quantum Circuits

TL;DR

This paper tackles the problem of parameter efficiency in machine learning by directly comparing classical neural networks (NNs) with variational quantum circuits (VQCs) on simple supervised and reinforcement learning tasks. The authors employ a grid-search approach to pair NN and VQC configurations with similar performance, predominantly using VQC simulations and estimating real-hardware training times by executing selected circuits on IBM hardware. They demonstrate that VQCs can match or slightly exceed NN performance while using significantly fewer parameters, albeit with substantially longer training times in the current setup, and they provide hardware-time estimates to gauge practical feasibility. The work highlights that, as quantum technology advances and VQC architectures improve, VQCs may offer distinct advantages for particular applications, and it provides a reproducible framework and openly available code for future benchmarking.

Abstract

In recent years, neural networks (NNs) have driven significant advances in machine learning. However, as tasks grow more complex, NNs often require large numbers of trainable parameters, which increases computational and energy demands. Variational quantum circuits (VQCs) offer a promising alternative: they leverage quantum mechanics to capture intricate relationships and typically need fewer parameters. In this work, we evaluate NNs and VQCs on simple supervised and reinforcement learning tasks, examining models with different parameter sizes. We simulate VQCs and execute selected parts of the training process on real quantum hardware to approximate actual training times. Our results show that VQCs can match NNs in performance while using significantly fewer parameters, despite longer training durations. As quantum technology and algorithms advance, and VQC architectures improve, we posit that VQCs could become advantageous for certain machine learning tasks.

Paper Structure

This paper contains 28 sections, 4 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Quantum Supervised Learning
  4. Quantum Reinforcement Learning
  5. Approach
  6. Classical Neural Network Architecture
  7. Variational Quantum Circuit Architecture
  8. State Preparation
  9. Angle Embedding
  10. Amplitude Embedding
  11. Variational Layers
  12. Measurement
  13. Experimental Setup
  14. Supervised Learning Experiments
  15. Training and Hyperparameters
  16. ...and 13 more sections

Figures (4)

  • Figure 1: (a) The $l$-th variational layer with 4 qubits, where $\boldsymbol{\theta}_l = [\theta_{1,1}, \theta_{1,2}, \dots, \theta_{4,3}]$ are the trainable parameters for layer $l$. (b) A VQC with $L$ layers and repeated data embedding; $U(\mathbf{x})$ encodes the input $\mathbf{x}$, and $U_l(\boldsymbol{\theta}_l)$ represents the trainable operations in layer $l$cerezo2021variationalschuld2020circuit.
  • Figure 2: Accuracy curves for each chosen NN and VQC. Averaged across ten runs (seeds 0--9); shaded areas are 95% confidence intervals.
  • Figure 3: Training reward on Frozen Lake for the comparable NN (112 parameters) and VQC (41 parameters). Mean across ten runs (seeds 0--9); shaded areas are 95% confidence intervals. The dashed red line (0.95) indicates the environment is solved.
  • Figure 4: Mean training times (seeds 0--9) for comparable NNs and VQCs. Error bars indicate 95% confidence intervals. Note the logarithmic y-axis.