Scalable Multi-QPU Circuit Design for Dicke State Preparation: Optimizing Communication Complexity and Local Circuit Costs

Preparing large-qubit Dicke states is of broad interest in quantum computing and quantum metrology. However, the number of qubits available on a single quantum processing unit (QPU) is limited - motivating the distributed preparation of such states across multiple QPUs as a practical approach to scalability. In this article, we investigate the distributed preparation of n-qubit k-excitation Dicke states D(n,k) across a general number p of QPUs, presenting a distributed quantum circuit each QPU hosting approximately $lceil n/p rceil$ qubits that prepares the state with communication complexity O\(p log k\), circuit size O(nk), and circuit depth Oleft\(p² k + log k log (n/k\)right). To the best of our knowledge, this is the first construction to simultaneously achieve logarithmic communication complexity and polynomial circuit size and depth. We also establish a lower bound on the communication complexity of p-QPU distributed state preparation for a general target state. This lower bound is formulated in terms of the canonical polyadic rank (CP-rank) of a tensor associated with the target state. For the special case p = 2, we explicitly compute the CP-rank corresponding to the Dicke state D(n,k) and derive a lower bound of lceillog (k + 1)rceil, which shows that the communication complexity of our construction matches this fundamental limit.