Space-Bounded Communication Complexity of Unitaries
Longcheng Li, Xiaoming Sun, Jialin Zhang, Jiadong Zhu
TL;DR
This work studies space-bounded communication complexity for distributed quantum unitary implementation, where each processor is limited in qubit capacity. It defines the minimum number of nonlocal two-qubit gates required to realize an $n$-qubit unitary $U$ across $k$ processors with $m$ ancillas and analyzes two key families: general unitaries and special classes like QFT and Clifford circuits. The main results provide a tight bound $\mathcal{C}_m^{(k)}(U)=O\left(\max\{4^{(1-1/k)n-m},n\}\right)$ with matching lower bounds, plus linear upper bounds for exact QFT and Clifford implementations and logarithmic communication for approximate QFT. The findings extend to general network topologies with topology-dependent overheads and establish both exact and approximate regimes, highlighting significant improvements over naive partitions and advancing scalable distributed quantum computing.
Abstract
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 communication complexity of a unitary as the minimum number of such nonlocal gates required for its realization. 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 distributed quantum computing (DQC) systems.