Parameterized quantum circuits (PQCs) play a crucial role in variational quantum algorithms. While it is commonly believed that the optimal PQC can be used to reproduce the target state, here we reveal that the optimal PQC can also provide valuable insights into the state's properties. We propose variational toolboxes to identify the $k$-separability of pure states, with or without preparation noise, by checking the structure within the optimal PQCs. Additionally, we introduce adaptive optimization strategies to detect the $k$-separability of mixed states. Compared to fixed PQCs, our approach controls fewer parameters for low-rank states. Finally, we validate our methods through numerical demonstrations for various states.

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• Quantum algorithms
• Quantum circuits
• Quantum correlations in quantum information
• Quantum entanglement