The authors' observation that shallow random circuits cannot form a local scrambling ensemble for U(1)-conserving Hamiltonians has significant implications for the design of training data generation strategies. This finding suggests that traditional approaches to generating training data, which rely on random circuits, may not be effective for systems where conservation laws play a crucial role. Specifically, shallow circuits fail to adequately explore the necessary configuration space to capture the dynamics of U(1)-conserving systems, leading to incomplete or biased training data.
To address this challenge, one can design training data generation strategies that are specifically tailored to the characteristics of U(1)-conserving Hamiltonians. For example, instead of relying on shallow random circuits, one could utilize deeper circuits that are constructed to respect the conservation laws inherent in the system. This would ensure that the generated states are representative of the physical dynamics and can effectively capture the relevant correlations.
Additionally, employing techniques such as symmetry-adapted sampling can enhance the efficiency of training data generation. By focusing on generating states that are likely to be encountered during the actual dynamics, one can create a more informative dataset that facilitates better learning outcomes. This approach not only improves the quality of the training data but also reduces the computational overhead associated with generating large datasets.
In summary, recognizing the limitations of shallow random circuits in the context of U(1)-conserving Hamiltonians allows for the development of more effective training data generation strategies, ultimately leading to improved performance of the VQC in simulating complex quantum dynamics.