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On the Design of Expressive and Trainable Pulse-based Quantum Machine Learning Models

Han-Xiao Tao, Xin Wang, Re-Bing Wu

TL;DR

This work addresses the expressivity–trainability tension in pulse-based QML under dynamical symmetry on NISQ hardware. It develops a Dyson-series (Fliess-series) polynomial expansion and a Lie-algebraic criterion that yields a necessary condition linking the initial state, measurement, and the dynamical Lie algebra to expressivity. Numerical experiments across multiple models with different symmetry demonstrate that constraining dynamics to symmetry subspaces can deliver expressive yet trainable QML models, with the output variance scaling inversely with the Lie algebra dimension via $\mathrm{Var}\left[f(\mathbf{x},\Theta)\right]=\sum_{j=1}^k \frac{P_{\mathfrak{g}_j}(\rho)P_{\mathfrak{g}_j}(M)}{\dim(\mathfrak{g}_j)}$, and that fully controllable systems suffer barren plateaus. The results provide a practical framework for designing hardware-efficient pulse-based QML systems and highlight open questions regarding sufficiency, training landscapes, and generalization for robust NISQ implementations.

Abstract

Pulse-based Quantum Machine Learning (QML) has emerged as a novel paradigm in quantum artificial intelligence due to its exceptional hardware efficiency. For practical applications, pulse-based models must be both expressive and trainable. Previous studies suggest that pulse-based models under dynamic symmetry can be effectively trained, thanks to a favorable loss landscape that avoids barren plateaus. However, the resulting uncontrollability may compromise expressivity when the model is inadequately designed. This paper investigates the requirements for pulse-based QML models to be expressive while preserving trainability. We establish a necessary condition pertaining to the system's initial state, the measurement observable, and the underlying dynamical symmetry Lie algebra, supported by numerical simulations. Our findings provide a framework for designing practical pulse-based QML models that balance expressivity and trainability.

On the Design of Expressive and Trainable Pulse-based Quantum Machine Learning Models

TL;DR

This work addresses the expressivity–trainability tension in pulse-based QML under dynamical symmetry on NISQ hardware. It develops a Dyson-series (Fliess-series) polynomial expansion and a Lie-algebraic criterion that yields a necessary condition linking the initial state, measurement, and the dynamical Lie algebra to expressivity. Numerical experiments across multiple models with different symmetry demonstrate that constraining dynamics to symmetry subspaces can deliver expressive yet trainable QML models, with the output variance scaling inversely with the Lie algebra dimension via , and that fully controllable systems suffer barren plateaus. The results provide a practical framework for designing hardware-efficient pulse-based QML systems and highlight open questions regarding sufficiency, training landscapes, and generalization for robust NISQ implementations.

Abstract

Pulse-based Quantum Machine Learning (QML) has emerged as a novel paradigm in quantum artificial intelligence due to its exceptional hardware efficiency. For practical applications, pulse-based models must be both expressive and trainable. Previous studies suggest that pulse-based models under dynamic symmetry can be effectively trained, thanks to a favorable loss landscape that avoids barren plateaus. However, the resulting uncontrollability may compromise expressivity when the model is inadequately designed. This paper investigates the requirements for pulse-based QML models to be expressive while preserving trainability. We establish a necessary condition pertaining to the system's initial state, the measurement observable, and the underlying dynamical symmetry Lie algebra, supported by numerical simulations. Our findings provide a framework for designing practical pulse-based QML models that balance expressivity and trainability.

Paper Structure

This paper contains 12 sections, 1 theorem, 49 equations, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PULSE-BASED QML MODELS WITH DYNAMICAL SYMMETRY
  3. EXPRESSIVITY OF QML MODELS IN PRESENCE OF DYNAMICAL SYMMETRY
  4. NUMERICAL SIMULATIONS
  5. Expressivity of pulse-based models with dynamical symmetry
  6. Balance between expressivity and trainability
  7. CONCLUSION AND OUTLOOK
  8. VALIDATION OF EXPRESSIVITY AND VARIANCE CALCULATIONS FOR THE FOUR MODELS
  9. Model 1
  10. Model 2
  11. Model 3
  12. Model 4

Key Result

Theorem 1

Given a sufficiently long duration $T$, the pulse-based QML model quantum system can approximate any function $f: \mathcal{X}\rightarrow \mathbb{R}$ only when $\bra{\psi_0}\mathcal{L}_{j_1}\cdots\mathcal{L}_{j_n}M\ket{\psi_0}\neq0$ for all $(j_1,\cdots,j_n)\in \mathbb{J}_{k_1,\cdots,k_m}\}$ and all

Figures (4)

  • Figure 1: Schematics of (a) a four-qubit gate-based data re-uploading QML model, (b) the pulse-based model compiled from the circuit, and (c) the generalized pulse-based model.
  • Figure 2: Approximation of the polynomial function \ref{['poly']} using the two-qubit pulse-based model \ref{['eq:simu_1']} under $\mathfrak{so}(4)$ dynamic symmetry. The inset displays the trained control pulses.
  • Figure 3: Approximation results of bivariate function by two-qubit models with (a) $\theta_1(t)=\theta_2(t)=1$ and (b) $\theta_1(t)$ and $\theta_2(t)$ freely tunable.
  • Figure 4: Comparison of expressivity and trainability between model classes under $\mathfrak{su}(2)$, $\mathfrak{su}(2)^{\otimes n}$, $\mathfrak{so}(n)$ and $\mathfrak{su}(2^n)$ dynamic symmetries: (a) Expressivity (evaluated by minimal pulse duration when training loss reaches $10^{-3}$) versus qubit number. (b) Sample variance of loss function versus qubit number. The inset shows how the minimal pulse duration is obtained.

Theorems & Definitions (1)

  • Theorem 1