How can I preserve an active redstone circuit when I leave the chunk it's built in?

Solution 1:

Redstone state is not saved when a chunk unloads, however entity state is. This means minecarts will continue to move when a chunk is reloaded. This in turn means if you replace your redstone clock generators with minecart based clock generators, they'll resume ticking when a chunk is reloaded. Here's the most basic Minecart ticker:

You can adjust the lengths of track to get the length of tick you require, and you can combine with other tickers outputs ANDed together to fine tune the length of the tick.

Solution 2:

This is a known bug: Redstone Repeaters will not update properly across a reload. However, Redstone Torches will, so if you build your system's clocks with torches then they will continue working.