I assume that by "state" you mean "country". If so, the the Soviet Union lost the most in terms of raw numbers- estimated about 23 million people lost, 9 million of them soldiers lost fighting in the war, and 14 million people dead for other reasons, like starvation, execution by German death squads, and so on.
In terms of percentage of population, Poland suffered the most casualties- about one in six people died.
If by "state" you mean "state in the United States", the state that suffered the most dead was New York with over 31,000 casualties; second most was Pennsylvania with over 26,000. I'm not sure which state suffered the most by percentage, those numbers are harder to find.
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I believe the answer is Massachusetts. Some entire neighborhoods there lost their sons and husbands. You can check with the US Census Bureau site if you need an exact number and state.
In the US Civil War, since the North lost more men than the South did, and New York had the most men...probably New York lost the most men.
North
California lost over 5,000 men in Vietnam; the highest of any state.
California lost the most men; over 5,000 dead.