Depth-Efficient Quantum Circuit Synthesis for Deterministic Dicke State Preparation

The $n$-qubit $k$-weight Dicke states $|D^n_k\rangle$, defined as the uniform superposition of all computational basis states with exactly $k$ qubits in state $|1\rangle$, form a basis of the symmetric subspace and represent an important class of entangled quantum states with broad applications in quantum computing. We propose deterministic quantum circuits for Dicke state preparation under two commonly seen qubit connectivity constraints: 1. All-to-all qubit connectivity: our circuit has depth $O(\log(k)\log(n/k)+k)$, which improves the previous best bound of $O(k\log(n/k))$. 2. Grid qubit connectivity ($(n_1\times n_2)$-grid, $n_1\le n_2$): (a) For $k\ge n_2/n_1$, we design a circuit with depth $O(k\log(n/k)+n_2)$, surpassing the prior $O(\sqrt{nk})$ bound. (b) For $k< n_2/n_1$, we design an optimal-depth circuit with depth $O(n_2)$. Furthermore, we establish the depth lower bounds of $\Omega(\log(n))$ for all-to-all qubit connectivity and $\Omega(n_2)$ for $(n_1\times n_2)$-grid connectivity constraints, demonstrating the near-optimality of our constructions.