Private keys are random points of an elliptic curve, there are about 2^256 of them.
Publlic keys are also points on a curve, they generated from private keys using a complicated (bijective?) elliptic curve based function, there are also about 2^256 of them.
Addresses are generated by hashing the private key, there are about 2^160 of them.
Therefore, there are about 2^(256-160)=2^96 keys per address.
If you search keys randomly for one containing bitcoin (the hard part, but RawDog apparentlly has a quantum computer running Grover's algorithm in his basement...and probably a nuclear reactor to provide power), if you find one, it is very likely to be a different one than the one that was originaly used.

Therefore, RawDog can offer to return the coins in exchange for the orginal key, and publish both keys to prove the hash collision.
However, that would not prove that RawDog found a preimage, and collisions in hash160 actually do not actually impact the security of bitcoin if used properly because a preimage is needed to steal coins from an existing address.
In fact, it only takes about 2^80 time to find a hash160 collision, this can is barely in the realm of classical computing (as far as I know, it also requires 2^80 space, which is quite impractical, but there may be a time-space tradeoff I don't know about).
The simplest way to prove a preimage is to find something that hashes to 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000, then publish it.
So RawDog, if you want to convince people of the existance of your super-fast quantum computer, set it to work on finding a preimage to the zero string, and collect the 56.80944011 BTC on it:
https://blockchain.info/es/address/1111111111111111111114oLvT2