Using glyphosate-based herbicides, such as Roundup, can be effective when applied properly. It's best to apply the herbicide to the foliage late in the summer or early fall, as this timing allows for better absorption and movement into the roots. If you choose to cut the plants first, you can immediately apply the herbicide to the cut surfaces for maximum effectiveness. Another method is to use a "cut-stump" technique, where you slice through the stem and apply a concentrated herbicide directly onto the freshly cut surface. This targets the plant more directly and reduces the amount of chemical used.

Once the plants are dead (which may take several weeks), you will need to carefully dig them out to ensure that no root pieces remain in the soil. Since the barberry plants are in landscape rocks, you will need to move these rocks aside to access the roots effectively. Make sure to dig down deep enough to extract all roots, as even small pieces left in the ground can lead to regrowth.
If you prefer not to use chemicals, you could consider repeated mowing or cutting of the plants. This approach can weaken them over time but may require persistence as barberry can regrow.

OR Once you have removed the plants and any roots, cover the area with clear plastic and leave it in place all summer to cook/sterilize the ground.  

https://extension.umn.edu/planting-and-growing-guides/solarization-occultation

Dispose of any removed plants properly to prevent them from re-establishing elsewhere.
https://extension.umd.edu/resource/invasive-vine-and-groundcover-control/

https://ohioline.osu.edu/factsheet/anr-0106: