Entanglement Induced Barren Plateaus (2010.15968v2)

Abstract: We argue that an excess in entanglement between the visible and hidden units in a Quantum Neural Network can hinder learning. In particular, we show that quantum neural networks that satisfy a volume-law in the entanglement entropy will give rise to models not suitable for learning with high probability. Using arguments from quantum thermodynamics, we then show that this volume law is typical and that there exists a barren plateau in the optimization landscape due to entanglement. More precisely, we show that for any bounded objective function on the visible layers, the Lipshitz constants of the expectation value of that objective function will scale inversely with the dimension of the hidden-subsystem with high probability. We show how this can cause both gradient descent and gradient-free methods to fail. We note that similar problems can happen with quantum Boltzmann machines, although stronger assumptions on the coupling between the hidden/visible subspaces are necessary. We highlight how pretraining such generative models may provide a way to navigate these barren plateaus.

• The paper shows that excessive entanglement between visible and hidden units creates barren plateaus that severely impede gradient-based optimization in QNNs.
• It employs rigorous numerical analysis on feedforward unitary QNNs and Quantum Boltzmann Machines to validate theoretical insights on entanglement scaling and Lipshitz constant behavior.
• The findings underscore the need for alternative training strategies and careful entanglement management to unlock quantum advantages in machine learning.

Entanglement Induced Barren Plateaus in Quantum Neural Networks

Quantum Neural Networks (QNNs) have shown promise in the burgeoning field of Quantum Machine Learning (QML), which leverages quantum mechanics to enhance computational capabilities beyond classical machine learning paradigms. However, a significant challenge that arises in training these systems is the phenomenon known as entanglement-induced barren plateaus. The paper by Ortiz Marrero, Kieferova, and Wiebe explores this issue, providing a comprehensive analysis of how entanglement can adversely affect the learning process in QNNs.

Fundamental Insights

The primary assertion of the paper is that excessive entanglement between the visible and hidden units in a QNN can lead to barren plateaus in the optimization landscape. A barren plateau refers to regions where the gradient of the cost function becomes exceedingly small, impeding the training process via gradient-based optimization methods such as gradient descent. The authors demonstrate that QNNs satisfying a volume-law in entanglement entropy are prone to such plateaus, rendering these models unsuitable for learning with high probability.

Their argument hinges on the scaling behavior of the Lipshitz constants of the expectation values of bounded objective functions on the visible layers. These constants, which determine how sensitive the function is to parameter changes, scale inversely with the dimension of the hidden subsystem. This inverse scaling suggests that, as the size of the hidden subsystem increases, the sensitivity of the model's performance to its parameters decreases, amplifying the problem.

Numerical Analysis and Claims

The authors support their theoretical framework with rigorous numerical analysis. They explore the impact of entanglement on training deep quantum models, specifically focusing on two configurations: feedforward unitary QNNs and Quantum Boltzmann Machines (QBMs). For both, they uncover that the typical states generated — those that adhere to a unitary design — often obey a volume-law scaling. Consequently, these setups minimize the effective influence of their parameters on the training objective, highlighting the entanglement as more of a hindrance than an advantage in quantum deep learning contexts.

The paper also derives from propositions that showcase how typical states in the quantum neural landscape, assumed to be similar to Haar-random states, exhibit volume-law entanglement. These typical states result in outputs that are statistically indistinguishable from the maximally mixed state, a state devoid of information that negates any potential quantum advantage.

Implications and Future Considerations

The implications of this research are twofold. Practically, it underscores the necessity for strategic deployment of entanglement when designing QNNs, pointing towards configurations less susceptible to volume-law entanglement. Theoretically, it bridges quantum thermodynamics and machine learning, elucidating how randomness and entanglement dictate the feasibility of QNN training.

Pretraining techniques that minimize entanglement-induced noise may offer a viable solution to navigate these barren plateaus. By adjusting the initial quantum state, or through non-gradient-based training methods, it might be possible to mitigate the adverse effects highlighted. Furthermore, exploiting different objective functions that are nonlinear with respect to the quantum state might provide alternative paths away from the detrimental impacts of entanglement.

As QML continues to progress, further research must explore the delicate balance between utilizing entanglement as a quantum resource and safeguarding the trainability of quantum models. The insights from this paper provide a foundational understanding that can guide future developments in constructing robust QNNs, offering a new lens through which to view the integration of quantum effects in machine learning algorithms.