Generalization Metrics for Practical Quantum Advantage in Generative Models

Abstract: As the quantum computing community gravitates towards understanding the practical benefits of quantum computers, having a clear definition and evaluation scheme for assessing practical quantum advantage in the context of specific applications is paramount. Generative modeling, for example, is a widely accepted natural use case for quantum computers, and yet has lacked a concrete approach for quantifying success of quantum models over classical ones. In this work, we construct a simple and unambiguous approach to probe practical quantum advantage for generative modeling by measuring the algorithm's generalization performance. Using the sample-based approach proposed here, any generative model, from state-of-the-art classical generative models such as GANs to quantum models such as Quantum Circuit Born Machines, can be evaluated on the same ground on a concrete well-defined framework. In contrast to other sample-based metrics for probing practical generalization, we leverage constrained optimization problems (e.g., cardinality-constrained problems) and use these discrete datasets to define specific metrics capable of unambiguously measuring the quality of the samples and the model's generalization capabilities for generating data beyond the training set but still within the valid solution space. Additionally, our metrics can diagnose trainability issues such as mode collapse and overfitting, as we illustrate when comparing GANs to quantum-inspired models built out of tensor networks. Our simulation results show that our quantum-inspired models have up to a $68 \times$ enhancement in generating unseen unique and valid samples compared to GANs, and a ratio of 61:2 for generating samples with better quality than those observed in the training set. We foresee these metrics as valuable tools for rigorously defining practical quantum advantage in the domain of generative modeling.

• The paper introduces a framework with fidelity, rate, and coverage metrics to quantify how generative models generalize beyond training data.
• It demonstrates that the quantum-inspired TNBM significantly outperforms classical GANs, achieving up to 68× better coverage along with higher fidelity and rate.
• The study highlights the critical influence of bond dimensions in TNBMs and exposes training pitfalls like mode collapse, guiding future improvements.

Generalization Metrics for Practical Quantum Advantage in Generative Models

The pursuit of quantum advantage over classical computation has sparked considerable interest across various applications, including generative modeling. This paper introduces a comprehensive framework for assessing practical quantum advantage specifically in the domain of generative models. It emphasizes generalization, which is the ability of models to produce novel and high-quality outputs beyond the training set. The study introduces quantitative generalization metrics applicable to both classical models like Generative Adversarial Networks (GANs) and quantum models such as Tensor Network Born Machines (TNBMs).

Framework and Methodology

The framework proposed in this paper aims to measure the generalization capabilities of generative models. Key to this approach is the evaluation of the model's ability to generate unseen and high-quality samples that adhere to specified constraints. Three primary metrics are developed:

1. Fidelity (F): Assesses the model's precision in generating valid samples versus noise.
2. Rate (R): Evaluates the frequency at which valid and previously unseen samples are generated.
3. Coverage (C): Measures the diversity of unique solutions derived across the valid solution space.

The framework's strength lies in its ability to provide a standardized method to compare both classical and quantum-inspired generative models on an equal footing.

Key Findings

The study demonstrates the application of this framework using two types of generative models. The Tensor Network Born Machine (TNBM), renowned for its quantum-inspired tensor network architecture, contrasts with a more classical GAN. The findings revealed that:

• The TNBM exhibited superior generalization capabilities over the GAN across all metrics, illustrating promising quantum-inspired techniques.
• Specifically, the TNBM achieved a $68\times$ better coverage and a significantly higher fidelity and rate compared to its GAN counterpart.
• The TNBM's bond dimension, which affects its expressive power, played a critical role in determining its generalization capabilities, revealing a peak performance at an optimal bond dimension.

Additionally, the capabilities of these metrics to identify training pitfalls like mode collapse in GANs were highlighted. Such insights are pivotal for enhancing the robustness and interpretability of generative models in practice.

Implications and Future Directions

This work provides a new methodology for defining and measuring practical quantum advantage in generative models, emphasizing the importance of generational metric-based approaches. It opens avenues for further exploration of quantum models and their capabilities. Future endeavors could involve exploring hybrid quantum-classical models or extending this quantitative framework to purely quantum systems.

The emerging frontier in AI where quantum models may offer significant advantages will likely benefit from such rigorous evaluation frameworks. The potential for quantum technologies to redefine computational capabilities across domains signifies an exciting trajectory for ongoing research and development in this field.