We study space-bounded communication complexity for unitary implementation in distributed quantum processors, where we restrict the number of qubits per processor to ensure practical relevance and technical non-triviality. We model distributed quantum processors using distributed quantum circuits with nonlocal two-qubit gates, defining the distributed communication complexity of a unitary as the minimum number of such nonlocal gates required for its realization, up to permutations of data qubit positions. Our contributions are twofold. First, for general $n$-qubit unitaries, we improve upon the trivial $O(4^n)$ communication bound. Considering $k$ pairwise-connected processors (each with $n/k$ data qubits and $m$ ancillas), we prove the communication complexity satisfies $O\left(\max\{4^{(1-1/k)n - m}, n\}\right)$ -- for example, $O(2^n)$ when $m=0$ and $k=2$ -- and establish the tightness of this upper bound. We further extend the analysis to approximation models and general network topologies. Second, for special unitaries, we show that both the Quantum Fourier Transform (QFT) and Clifford circuits admit linear upper bounds on communication complexity in the exact model, outperforming the trivial quadratic bounds applicable to these cases. In the approximation model, QFT's communication complexity reduces drastically from linear to logarithmic, while Clifford circuits retain a linear lower bound. These results offer fundamental insights for optimizing communication in distributed quantum unitary implementation, advancing the feasibility of large-scale DQC systems.

翻译：我们研究了分布式量子处理器中酉实现的空间有界通信复杂度，通过限制每个处理器的量子比特数以确保实际相关性和技术非平凡性。我们使用具有非局部两量子比特门的分布式量子电路对分布式量子处理器建模，将酉的分布式通信复杂度定义为实现该酉所需的最小非局部门数量（允许数据量子比特位置的置换）。我们的贡献有两方面。首先，对于一般的$n$量子比特酉，我们改进了平凡的$O(4^n)$通信界。考虑$k$个两两互联的处理器（每个处理器包含$n/k$个数据量子比特和$m$个辅助量子比特），我们证明通信复杂度满足$O\left(\max\{4^{(1-1/k)n - m}, n\}\right)$——例如当$m=0$且$k=2$时为$O(2^n)$——并证明了该上界的紧致性。我们进一步将分析推广到近似模型和一般网络拓扑。其次，对于特殊酉，我们证明量子傅里叶变换（QFT）和克利福德电路在精确模型中的通信复杂度均具有线性上界，优于适用于这些情况的平凡二次界。在近似模型中，QFT的通信复杂度从线性急剧降至对数级，而克利福德电路仍保持线性下界。这些结果为优化分布式量子酉实现中的通信提供了基础性洞见，推动了大规模分布式量子计算系统可行性的发展。