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Unleashing the Expressive Power of Pulse-Based Quantum Neural Networks

Han-Xiao Tao, Jiaqi Hu, Re-Bing Wu

TL;DR

It is proved that the pulse-based model can approximate arbitrary nonlinear functions when the underlying physical system is ensemble controllable, and it is shown that the pulse-based model can unleash more expressive power compared to the gate-based model.

Abstract

Quantum machine learning (QML) based on Noisy Intermediate-Scale Quantum (NISQ) devices hinges on the optimal utilization of limited quantum resources. While gate-based QML models are user-friendly for software engineers, their expressivity is restricted by the permissible circuit depth within a finite coherence time. In contrast, pulse-based models enable the construction of "infinitely" deep quantum neural networks within the same time, which may unleash greater expressive power for complex learning tasks. In this paper, this potential is investigated from the perspective of quantum control theory. We first indicate that the nonlinearity of pulse-based models comes from the encoding process that can be viewed as the continuous limit of data-reuploading in gate-based models. Subsequently, we prove that the pulse-based model can approximate arbitrary nonlinear functions when the underlying physical system is ensemble controllable. Under this condition, numerical simulations demonstrate the enhanced expressivity by either increasing the pulse length or the number of qubits. As anticipated, we show through numerical examples that the pulse-based model can unleash more expressive power compared to the gate-based model. These findings lay a theoretical foundation for understanding and designing expressive QML models using NISQ devices.

Unleashing the Expressive Power of Pulse-Based Quantum Neural Networks

TL;DR

It is proved that the pulse-based model can approximate arbitrary nonlinear functions when the underlying physical system is ensemble controllable, and it is shown that the pulse-based model can unleash more expressive power compared to the gate-based model.

Abstract

Quantum machine learning (QML) based on Noisy Intermediate-Scale Quantum (NISQ) devices hinges on the optimal utilization of limited quantum resources. While gate-based QML models are user-friendly for software engineers, their expressivity is restricted by the permissible circuit depth within a finite coherence time. In contrast, pulse-based models enable the construction of "infinitely" deep quantum neural networks within the same time, which may unleash greater expressive power for complex learning tasks. In this paper, this potential is investigated from the perspective of quantum control theory. We first indicate that the nonlinearity of pulse-based models comes from the encoding process that can be viewed as the continuous limit of data-reuploading in gate-based models. Subsequently, we prove that the pulse-based model can approximate arbitrary nonlinear functions when the underlying physical system is ensemble controllable. Under this condition, numerical simulations demonstrate the enhanced expressivity by either increasing the pulse length or the number of qubits. As anticipated, we show through numerical examples that the pulse-based model can unleash more expressive power compared to the gate-based model. These findings lay a theoretical foundation for understanding and designing expressive QML models using NISQ devices.
Paper Structure (8 sections, 2 theorems, 25 equations, 6 figures)

This paper contains 8 sections, 2 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. From gate-based to pulse-based QNN models
  3. Ensemble controllability for the expressivity of pulse-based models
  4. Numerical experiments
  5. The approximation of nonlinear functions
  6. The affection of pulse duration
  7. Unleashing the expressive power from gate-based models
  8. Concluding remarks

Key Result

Theorem 1

Let $\mathcal{X}\subseteq\mathbb{R}^m$ be a connected domain that contains $\textbf{x}=0$ as an interior point. The pulse-based model quantum system can approximate any analytic functions $f: \mathcal{X}\rightarrow[\lambda_{\min},\lambda_{\max}]$ if the Lie algebra generated by $iH_0(\textbf{x}),iH_

Figures (6)

  • Figure 1: Schematics of gate-based models: (a) a standard QNN model and (b) a data re-uploading QNN model.
  • Figure 2: Schematics of (a) a single-qubit gate-based data re-uploading QNN and (b) the corresponding superconducting quantum circuit implementation.
  • Figure 3: The approximation results of (a) univariate function and (b) bivariate function by single-qubit pulse-based models. On the left are the fitted curve and surface, and on the right are the resulting control fields and the training curves.
  • Figure 4: The training loss versus the pulse duration time $T$: (a) fitting the univariate function \ref{['function_2']} with different sampling periods; (b) the statistics of training losses over a class of sampled polynomial functions.
  • Figure 5: The dependence of training loss on the depth and width of the pulse-based models with circularly coupled qubits.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Remark 1