Subtleties in the trainability of quantum machine learning models (2110.14753v1)

Abstract: A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called Quantum Machine Learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called Variational Quantum Algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.

• The paper connects barren plateaus known in Variational Quantum Algorithms to trainability issues in Quantum Machine Learning, demonstrating similar challenges.
• A novel finding shows that specific dataset features and classical data embedding strategies can induce barren plateaus in QML models, impacting trainability.
• The study shows that barren plateau conditions cause elements of the empirical Fisher Information Matrix to vanish exponentially, requiring significant resources for efficient training.

Subtleties in the Trainability of Quantum Machine Learning Models

The paper by Thanaslip et al. titled "Subtleties in the trainability of quantum machine learning models" presents an in-depth analysis of the trainability challenges faced in Quantum Machine Learning (QML). Quantum Machine Learning, an emerging field, aims to leverage quantum data, models, and devices to achieve computational speed-ups compared to classical machine learning approaches. The paper addresses the critical issue of efficiently training Quantum Neural Networks (QNNs), which are central to QML.

Overview of Key Concepts

Quantum Machine Learning represents a paradigm shift where both quantum and classical datasets are processed using parametrized quantum circuits, known as Quantum Neural Networks (QNNs). These models apply quantum effects such as superposition and entanglement within the exponentially large Hilbert space, holding the promise of outperforming traditional neural networks. However, the field lacks robust theoretical results regarding the scalability and trainability of these models.

In this context, the paper draws parallels between QNNs used in QML and Variational Quantum Algorithms (VQAs). VQAs, widely studied, involve optimizing quantum circuits to perform tasks such as ground state estimation or quantum compiling. Common in VQAs is the challenge posed by barren plateaus (BPs), where gradients vanish exponentially, rendering the training inefficient. Thanaslip et al. investigate if similar issues could apply universally to QML settings.

Main Findings and Analytical Results

• Gradient Scaling and Barren Plateaus: The authors bridge VQA and QML frameworks by connecting BP conditions known for VQAs to those in QML. Specifically, gradient scaling results for VQAs are proved applicable to QML, meaning that QML models will also exhibit BPs in settings where VQAs do. Consequently, features that impede VQA trainability, such as global measurements and deep circuits, may also lead to barren plateaus in QML.
• Dataset-Induced Barren Plateaus: A novel insight relates to the datasets used in QML. The authors provide theoretical evidence that features of the dataset, accompanied by certain embedding schemes, can exacerbate trainability issues. This is especially relevant for classical data embedding strategies, where poorly chosen embeddings can induce barren plateaus due to large amounts of entanglement in the quantum states produced.
• Fisher Information Matrix Implications: The paper extends to quantitative aspects concerning the Fisher Information (FI) matrix. Under barren plateau conditions, matrix elements of the empirical FI matrix are shown to vanish exponentially, impacting methods like natural gradient descent, thus demanding exponential resources for efficient training.

Numerical Evidence

The paper corroborates its theoretical findings through numerical simulations. These simulations emphasize the role of global versus local measurements and the choice of embedding schemes, illustrating how these impact the trainability and performance of QML models. Particularly, global observables are a primary source of untrainability, validating the theoretical conclusions about barren plateaus.

Implications and Future Directions

The work prompts the re-evaluation of current QML models to avoid trainability pitfalls suggested by insights from VQAs. Moreover, the notion of dataset-induced barren plateaus opens a new dimension in the design and choice of embedding strategies for classical data in quantum settings. As such, future research should focus on embedding strategies that offer trainability-awareness to enhance the efficacy of QML models.

The exploration of the trainability landscape in QML has profound implications for the advancement of quantum computing applications in data science, encouraging theoretical researchers to explore understanding the scalability and efficiency of QNNs across diverse quantum computing platforms. Potential pathways include refining loss functions, optimizing architecture designs, and developing robust algorithms to circumvent trainability issues pervasive in current QML models.

In conclusion, Thanaslip et al.'s paper provides a rigorous examination of trainability challenges in quantum machine learning, offering valuable insights into the optimization landscapes of QML models and setting the stage for future developments in the field.