Are there schemes for encrypting classical ciphertext using quantum algorithms? Specifically, if a public key system exists for classical data, and classical ciphertexts, can this scheme be implemented by quantum algorithms (A quantum algorithm implementing classical public key algorithms)?
Yes, a quantum computer can implement any classical algorithm.
A universal Turing machine can be used to simulate any Turing machine and by extension the computational aspects of any possible real-world computer.
And that's - as far as I can see - QED.