NP has log-space verifiers with fixed-size public quantum registers

In classical Arthur-Merlin games, the class of languages whose membership proofs can be verified by Arthur using logarithmic space (AM(log-space)) coincides with the class P [Con89]. In this note, we show that if Arthur has a fixed-size quantum register (the size of the register does not depend on the length of the input) instead of another source of random bits, membership in any language in NP can be verified with any desired error bound. In a public-coin interactive proof system, a resource-bounded verifier (Arthur) checks proofs presented to it in a sequence of messages by an all-powerful prover (Merlin), from which Arthur can hide no information. Arthur is supposed to accept all and only the correct proofs with high probability. Quantum versions of these systems, where the verifier is a quantum computer, and the messages consist of quantum bits, have also been examined, with the most general variant with polynomial bounds on the exchanged messages and the verifier runtime shown to be equivalent in computational power to the classical version [MW05]. More restricted scenarios have also been examined; for instance, the class QCMA [AN02,AK06] corresponds to single-message quantum Arthur-Merlin games, where the proof is a classical string. 1 In this paper, we restrict this model further, by imposing a logarithmic space bound on the verifier, so that Arthur cannot even hold the entire proof string in memory, and take most of his " quantumness " away, by limiting him to use a quantum register (with a state set of size just three) as his only source of randomness. Without that small register, Arthur is a deterministic logspace machine, and the class of languages that it can verify membership in is just NL. In the version where Arthur is allowed to 1 Watrous [Wat09] proposes the usage of the name MQA for this class. use a classical random number generator, this class is known to equal P [Con89]. We will show that our version of Arthur with the small quantum register can verify membership in every language in NP. We use the abbreviation qTM, with a lowercase q, for the quantum Turing machine model with which we represent the verifier, to stress that it uses only a fixed number of quantum bits. The classical state set of Arthur is partitioned to three subsets, the reading states, communication states, and halting states. Every computational step of the qTM consists of two stages: …