Through a set of some essential reforms that redefined the relationship between the Qin state and its populace.
Mass conscription of peasant levies (as opposed to aristocratic chariot armies), which greatly increased the number of soldiers the Qin could have at its disposal.
During the conquest of other political entities, the land was redistributed away from the previous political nobility to Qin's own population, thus breaking "feudal" power in that region and tightening up the more direct link between subject and sovereign.
Ranks, rewards, and proscriptive laws regarding other forms of social advancement which flowed out of the central government, thus tying status primarily to the state, rather than to alternate and more local sources of power.
A simplification and systemization of farming land, which allowed the state to employ a maximum amount of peasants to work on the lands as individual households which could render individual military service, as opposed to later on in China's history where large landlords controlled much of the territory, obscuring the requirements for individual "household" tax and military obligations.
Geography. The region around the Qin state was ringed by mountains, with only one way into the central Yellow River plains regions that held most of the other Chinese states via the Hangu Pass. This easily defensible region meant that from the start, they had a secure base and only had to deal with one state level competitor to the east, allowing easy expansion. As opposed to other Chinese states on the Yellow River plains, which usually had multiple threats from multiple axises.
In a nutshell, a series of reforms that took out the middle men of the country (the local aristocrats) and tied everyone directly to the emperor and the state for prestige and advancement, a conscripted peasant army on the national scale of what you probably saw during the French Revolution, and geographic luck to be based in a region that could only be threatened from only one direction, which was easily defended.
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