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Efficient Measure for the Expressivity of Variational Quantum Algorithms.

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We introduce a novel method using statistical learning theory to measure the expressivity of variational quantum algorithms (VQAs). This approach quantifies expressivity, aiding in understanding quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs).

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Area of Science:

  • Quantum Computing
  • Statistical Learning Theory
  • Machine Learning

Background:

  • The performance of variational quantum algorithms (VQAs), including quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs), is critically dependent on the expressivity of their chosen Ansätze.
  • A trade-off exists between Ansatz simplicity (leading to insufficient expressivity) and complexity (leading to trainability challenges).
  • Quantifying VQA expressivity is crucial but lacks effective strategies.

Purpose of the Study:

  • To develop an effective strategy for measuring the expressivity of variational quantum algorithms (VQAs).
  • To provide a quantitative understanding of VQA expressivity for improved algorithm design and analysis.

Main Methods:

  • Utilizing the covering number, a tool from statistical learning theory, to analyze VQA expressivity.
  • Deriving upper bounds for VQA expressivity based on the number of quantum gates and measurement observable.
  • Investigating the impact of system noise and circuit depth on VQA expressivity in near-term quantum devices.

Main Results:

  • Established an upper bound for the expressivity of arbitrary VQAs.
  • Observed an exponential decay in expressivity with increasing circuit depth, particularly in noisy near-term quantum systems.
  • Demonstrated the utility of the expressivity measure in analyzing QNN generalization and VQE accuracy through numerical verification.

Conclusions:

  • The covering number provides a powerful tool for quantitatively assessing VQA expressivity.
  • Understanding expressivity is key to optimizing Ansätze for specific quantum machine learning tasks.
  • This work lays the foundation for a more rigorous and quantitative analysis of variational quantum algorithms.