Dicke states serve as a critical resource in quantum metrology, communication, and computation. However, unitary preparation of Dicke states is limited to logarithmic depth in standard circuit models and existing constant-depth protocols require measurement and feed-forward. In this work, we present the first unitary, constant-depth protocols for exact Dicke state preparation. We overcome the logarithmic-depth barrier by moving beyond the standard circuit model and leveraging global interactions (native to architectures such as neutral atoms and trapped ions). Specifically, utilizing unbounded CZ gates (i.e. within the QAC$^0$ circuit class), we offer circuits for exact computation of constant-weight Dicke states, using polynomial ancillae, and approximation of weight-1 Dicke states (i.e. $W$ states), using only constant ancillae. Granted additional access to the quantum FAN-OUT operation (i.e. upgrading to the QAC$_f^0$ circuit class), we also achieve exact preparation of arbitrary-weight Dicke states, with polynomial ancillae. These protocols distinguish the constant-depth capabilities of quantum architectures based on connectivity and offer a novel path toward resolving a long-standing quantum complexity conjecture.

翻译：迪克态在量子计量学、通信和计算中作为关键资源。然而，在标准电路模型中，迪克态的幺正制备被限制在对数深度，而现有的恒定深度协议需要测量和前馈操作。在本工作中，我们首次提出了用于精确制备迪克态的幺正、恒定深度协议。我们通过超越标准电路模型并利用全局相互作用（天然存在于中性原子和囚禁离子等架构中），克服了对数深度壁垒。具体而言，利用无界CZ门（即在QAC$^0$电路类中），我们提供了精确计算恒定权重迪克态的电路（使用多项式辅助量子比特），以及近似权重为1的迪克态（即$W$态）的电路（仅使用常数个辅助量子比特）。若额外允许使用量子扇出操作（即升级到QAC$_f^0$电路类），我们还实现了任意权重迪克态的精确制备（使用多项式辅助量子比特）。这些协议基于连接性区分了量子架构的恒定深度能力，并为解决一个长期存在的量子复杂性猜想提供了一条新路径。