Space-Bounded Communication Complexity of Unitaries

TL;DR

This paper investigates the minimal communication required for implementing unitaries in distributed quantum systems, providing bounds and optimizations for general and specific unitaries like QFT and Clifford circuits.

Contribution

It introduces new bounds on communication complexity for general unitaries and shows linear bounds for specific unitaries, improving understanding of distributed quantum computation.

Findings

Improved upper bounds for general unitaries' communication complexity.

Linear upper bounds for QFT and Clifford circuits in exact models.

Logarithmic communication complexity for QFT in approximate models.

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.