arXiv:2604.16906v2 Announce Type: replace Abstract: In modern large-scale networked systems, rapidly solving optimization problems while utilizing communication resources efficiently is critical for addressing complex tasks. In this paper, we consider an unconstrained distributed optimization problem in which information exchange among nodes is governed by a directed communication graph. In our setup we focus on two key challenges. The first is the zigzag phenomenon caused by the objective functions of individual nodes having significantly different curvature along different directions. The second is that the communication channels among nodes are subject to limited bandwidth, which motivates the use of compressed (quantized) messages. To address both challenges simultaneously, we propose QANM, a distributed optimization algorithm that combines Nesterov momentum with a distributed finite-time quantized consensus protocol. Under strong convexity and smoothness assumptions, we show that our proposed algorithm converges linearly to a neighborhood of the optimal solution. Finally, we evaluate QANM on a distributed sensor-fusion problem. With each method's step size tuned separately, QANM exhibits the fastest initial error reduction among the tested methods, including QAGD, Acc-DNGD-SC, and MSDA. Comparisons with QAGD under the same quantized averaging framework illustrate the empirical benefit of momentum, while experiments at two quantization levels demonstrate quantization-dependent error floors.