Emperor Qin Shi Huang actually had a lot of success with the crossbow in his campaign to unify China, both on horseback and on foot. It reportedly fired an arrow more than 2,500 feet, and were twice as powerful as the modern assault rifle. If I recall correctly, they used crossbows for the efficiency. They could pierce through the heaviest armor of their enemies without a doubt. Furthermore, the Han Dynasty have attributed the success of several of their battles against the Xiongnu to volleys of crossbow bolts. Now, that was around 230 BC, which is far from this period, but the premise wouldn't drastically change, would it?