Sidechains (at least the two-way pegged sidechains, as described in our paper), are about creating a new blockchain that uses the same currency as another blockchain.

Fiat isn't native to any blockchain, so it doesn't fit that category. You can't prove to the chain that fiat tokens are actually exchangable for real fiat. Fiat just doesn't live in the same world: it's defined by real world banks, not by a cryptographically verifiable ledger.

There are however many other ways to have fiat-representing tokens in an auditable ledger of transactions (a "blockchain"). It doesn't need pegging, or proof-of-work, or a sidechain (but isn't incompatible with them either). Still, a lot of the ledger technology itself can be reused. However, all of these solutions inherently require trusting an issuer, who promises to exchange the tokens for real fiat, forever - presumably by holding as much fiat in reserve as assets are issued.