Public-qubits versus private-coins

We introduce a new public quantum interactive proof system, namely qAM, by augmenting the verifier with a fixed-size quantum register in Arthur-Merlin game. We focus on spacebounded verifiers, and compare our new public system with private-coin interactive proof (IP) system in the same space bounds. We show that qAM systems not only can simulate all known space-bounded private-coin protocols, but also implements some protocols that are either not implementable by IP systems or currently not known to be simulated by IP systems. More specifically, we show that for any Turing-recognizable language, there exists a constant-space weak-qAM system (the nonmembers do not need to be rejected with high probability), and, different from the classical case, our protocol has perfect-completeness (the members are accepted exactly). For strong proof system, where the nonmembers must be rejected with high probability, we show that the known space-bounded private-coin protocols can also be simulated by qAM systems with the same space bound. In case of constant-space and log-space IP systems, the best known lower bounds are ASPACE(n) and EXP, respectively. We obtain better lower bounds for the corresponding qAM systems: Each language in NP has a constant-space (exp-time) qAM system, there is an NEXP-complete language having a constant-space qAM system, and each language in NEXP has a log-space qAM system.

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