Key management is how many comply with GDPR today. They encrypt the PII and associate it with the user. Then, when someone requests their info to be "deleted", they zero out the encryption key.
This should continue to work as long as you use systems that do not fall to pieces under quantum attacks.
AES is considered "resistant" in that quantum does an effective square-rooting of the brute forcing effort (or if you prefer, halving of the binary key length). So, do not use anything under AES 256.
Asymmetric algorithms fall apart though, which is why NIST has had a multi-year effort to select new standardized asymmetric algorithms.
There are select bits of info we should protect, but can't. If you're in the US, your SSN is one of those.
It never ceases to make me chuckle that it says that it's not a form of ID on front, and yet everyone considers it a form of ID. Even state governments. It's usually listed under one of the documents they accept to prove ID.
AES is considered "resistant" in that quantum does an effective square-rooting of the brute forcing effort (or if you prefer, halving of the binary key length). So, do not use anything under AES 256.
Asymmetric algorithms fall apart though, which is why NIST has had a multi-year effort to select new standardized asymmetric algorithms.