Short and useful quantum proofs for sublogarithmic-space verifiers

Abstract

Quantum Merlin-Arthur proof systems are believed to be stronger than both their classical counterparts and ``stand-alone'' quantum computers when Arthur is assumed to operate in $\Omega(\log n)$ space. No hint of such an advantage over classical computation had emerged from research on smaller space bounds, which had so far concentrated on constant-space verifiers. We initiate the study of quantum Merlin-Arthur systems with space bounds in $\omega(1) \cap o(\log n)$, and exhibit a problem family $\mathcal{F}$, whose yes-instances have proofs that are verifiable by polynomial-time quantum Turing machines operating in this regime. We show that no problem in $\mathcal{F}$ has proofs that can be verified classically or is solvable by a stand-alone quantum machine in polynomial time if standard complexity assumptions hold. Unlike previous examples of small-space verifiers, our protocols require only subpolynomial-length quantum proofs.