This is not poison ivy, but the groundcover known by many names like bishop's weed, snow-on-the-mountain, or goutweed (Aegopodium podagraria). Below is a write-up I have on this. Please read it through and let me know if you have any questions. 
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Goutweed (Aegopodium podagraria, aka bishop’s weed or snow-on-the-mountain) is a perennial plant in the Apiaceae (Carrot) family. This plant has the ability to form large colonies from underground rhizomes but it can also reproduce by seed. Leaves are alternate, compound, with toothed-margined leaflets. Some plants may contain white-edged leaflets, but others will be completely green in color. It is usually the variegated form that is cultivated as an ornamental ground cover.

Control of goutweed can be very difficult, depending on its location. If there is a neighboring property that does not also work on eradicating the goutweed, it will continue to enter your property.

Manual removal. Digging up the root system and chopping the stems are certainly options for control, but regrowth should be expected. Persistence will deplete the root and rhizome system over time. I would suggest wearing gloves and protecting your skin during any physical removal efforts. The sap in members of the Carrot family, such as goutweed, can cause blistering when the contacted skin is subsequently exposed to UV light from the sun. It is the same substance found on the exterior (and interior) of giant hogweed. People are not as likely to encounter it on other Carrot members as they are on the inside and would need to be broken open, whereas brushing against giant hogweed can expose you. Again, I would wear gloves just to be safe and then wash your hands afterward

Smothering/mulch. Impermeable mulch (plastic mulch) may help, but if there are any holes in the mulch for desirable vegetation, it is likely goutweed will exploit those. Organic mulches (e.g. wood chips, bark, etc. ) are not effective.

Tarping beds or covering with clear plastic for solarization over several months can help. You can read a bit more on this from the University of Minnesota- https://extension.umn.edu/planting-and-growing-guides/solarization-occultation

Ornamental Bed Herbicides. Herbicides are often the "fastest" option to gain control over a goutweed population, but this is nearly impossible when the plants are mixed in with a desirable species. One approach you could try is to cut the plants at the base and paint concentrated herbicide on the cut surface within a few minutes. A non-selective concentrated herbicide, such as Roundup Weed & Grass Killer Concentrate- Exclusive Formula (a.i. fluazifop, triclopyr, and diquat) or Ortho Weed & Grass B-Gon Super Concentrate (a.i. diquat, fluazifop, and dicamba), though there are likely other products available, may be effective. You can mix this herbicide as directed as well for a foliar application; however, anytime you spray, you risk off-target movement to your desirable plants, so I do not recommend that in an ornamental bed unless you’re renovating the whole thing. Treatment will be required multiple times over multiple seasons, persistence will be the key to success.

When using non-selective herbicides there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. The label is the law. Second, non-selective herbicides will injure or kill other plants contacted during application, so care is needed to avoid direct or drifted contact with green plant material, exposed roots, and injured bark of desired plants. As we move away from using glyphosate it is important to note that many of these products will prevent the replanting of the area for some period. Look for this time period to be stated on the label as it varies by product. Third, these products usually have an ingredient with fast-acting activity that will show burn quickly (e.g. the active ingredients diquat or pelargonic acid), but it may take 2 weeks or more to show the total effect. Herbicides should not be re-applied without waiting to see this total activity and reapplication may be prohibited. There is often a maximum number of annual applications allowed to protect the environment and your property (see label). Fourth, control of perennial species is most effective in the fall, but applications can be applied anytime the plants are actively growing (temperatures consistently above 50F). Finally, be sure that the product you choose is labeled for the area in which you intend to apply it. While products that promise long lasting activity may seem more attractive, often the added active ingredients or increased dose of active ingredients can be hazardous to trees and shrubs and/or may delay the planting of other plants in the coming season(s).