Verifiable Quantum Advantage without Structure
Abstract
We show the following hold, unconditionally unless otherwise stated, relative to a random oracle with probability 1:  There are NP search problems solvable by BQP machines but not BPP machines.  There exist functions that are oneway, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar separations hold for digital signatures and CPAsecure public key encryption (the latter requiring the assumption of a classically CPAsecure encryption scheme). Interestingly, the separation does not necessarily extend to the case of other cryptographic objects such as PRGs.  There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are noninteractive, whereas for nonuniform adversaries the proofs are two message public coin.  Our results do not appear to contradict the AaronsonAmbanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction. By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.02063
 arXiv:
 arXiv:2204.02063
 Bibcode:
 2022arXiv220402063Y
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 Computer Science  Cryptography and Security
 EPrint:
 50 pages, added a variant with worstcase completeness at the end of Section 6