Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks (2406.18316v3)

Abstract: Quantum neural networks (QNNs) require an efficient training algorithm to achieve practical quantum advantages. A promising approach is gradient-based optimization, where gradients are estimated by quantum measurements. However, QNNs currently lack general quantum algorithms for efficiently measuring gradients, which limits their scalability. To elucidate the fundamental limits and potentials of efficient gradient estimation, we rigorously prove a trade-off between gradient measurement efficiency (the mean number of simultaneously measurable gradient components) and expressivity in deep QNNs. This trade-off indicates that more expressive QNNs require higher measurement costs per parameter for gradient estimation, while reducing QNN expressivity to suit a given task can increase gradient measurement efficiency. We further propose a general QNN ansatz called the stabilizer-logical product ansatz (SLPA), which achieves the trade-off upper bound by exploiting the symmetric structure of the quantum circuit. Numerical experiments show that the SLPA drastically reduces the sample complexity needed for training while maintaining accuracy and trainability compared to well-designed circuits based on the parameter-shift method.

• The paper proves a trade-off theorem showing that increased QNN expressivity requires higher gradient measurement costs, defined by inequalities linking expressivity (X) and efficiency (F).
• It introduces the Stabilizer-logical Product Ansatz (SLPA) that leverages symmetry to optimize gradient measurement efficiency while maintaining model expressivity.
• Numerical experiments reveal that the SLPA achieves rapid training convergence and significant reduction in measurement shots compared to symmetric and non-symmetric circuits.

Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks

The paper "Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks" explores the inherent trade-offs in the design and training of Quantum Neural Networks (QNNs) for achieving practical quantum advantages. QNNs are a subset of Variational Quantum Algorithms (VQAs) which have gained prominence for their potential to solve complex problems in quantum chemistry, physics, and machine learning. The primary challenge addressed in this paper is the difficulty in efficiently estimating gradients necessary for training QNNs due to the quantum nature of measurements, where the quantum state collapses upon observation.

Core Contributions

Trade-off Theorem

The authors prove a fundamental trade-off between the gradient measurement efficiency and the expressivity of the QNNs. The crux of their theorem is encapsulated in two major inequalities:

• $X \leq \frac{4^n}{F} - F$
• $X \geq F$

Here, $X$ denotes the expressivity of the QNN, measured by the dimension of the dynamical Lie algebra (DLA) spanned by the QNN generators, indicating the diversity of quantum operations the QNN can perform. $F$ represents the gradient measurement efficiency, quantified as the mean number of simultaneously measurable gradient components. These inequalities indicate that highly expressive QNNs require a proportional increase in the complexity and cost of gradient measurements, while the efficiency in gradient measurement is intrinsically limited by the model's expressivity.

Stabilizer-logical Product Ansatz (SLPA)

The authors propose a new QNN architecture called the Stabilizer-logical Product Ansatz (SLPA), which aims to optimize this trade-off. The SLPA construction involves:

• Stabilizer Group $S$: A set of commuting Pauli operators providing a symmetry backdrop.
• Logical Operators $L$: Operators commuting with stabilizers, enabling the realization of various quantum operations.
• Product Generators: Combining stabilizers and logical operators, forming generators as $G_j^a = S_j L_a$.

The SLPA leverages the inherent symmetry of the problem to maximize gradient measurement efficiency while maintaining sufficient expressivity. The symmetry ensures that gradient components within each block can be measured simultaneously, and thus the SLPA can potentially reach the theoretical upper limit of the trade-off inequality.

Numerical Demonstrations

The practical efficacy of the SLPA is demonstrated through a task of learning a symmetric function. The models compared include a symmetric circuit (SC), the SLPA, and a non-symmetric circuit. Numerical simulations reveal that the SLPA not only achieves higher gradient measurement efficiency (approaching the upper limit of $F=4$ in the deep circuit limit) but also demonstrates rapid convergence in training and robust generalization. This efficiency is particularly significant when considering the cumulative number of measurement shots required, showcasing the SLPA's significant reduction compared to traditional methods.

Implications and Future Directions

The paper's findings have notable implications for the field of Quantum Machine Learning (QML):

1. Design of Quantum Models: By highlighting the trade-off between expressivity and gradient measurement efficiency, the paper guides the design of quantum models to balance these attributes according to the problem's requirements.
2. Gradient-free Optimization: The high measurement costs associated with gradient-based optimization in QNNs motivate the exploration of gradient-free optimization algorithms.
3. Multi-copy Settings: Investigating efficient gradient estimation algorithms in multi-copy settings, where multiple copies of the input state are available simultaneously, could potentially circumvent the limitations observed in the single-copy setting.

Conclusion

This research provides a comprehensive theoretical and practical framework that underscores the limits and possibilities of training QNNs efficiently. The introduction of the SLPA as a general and effective ansatz for QNNs signifies a step towards more practical quantum advantages in QML, leveraging problem symmetries to enhance training efficiency. Future research will likely explore extending these principles to broader classes of quantum circuits and further optimizing QNN architectures for large-scale quantum computations.