Short and useful quantum proofs for sublogarithmic-space verifiers

TL;DR

This paper introduces quantum Merlin-Arthur proof systems with sublogarithmic space bounds, demonstrating problems verifiable by small-space quantum machines with short proofs, surpassing classical and stand-alone quantum capabilities.

Contribution

The study pioneers the analysis of quantum Merlin-Arthur systems with space bounds between constant and logarithmic, showing they can verify problems beyond classical and quantum standalone methods.

Findings

Problems with short quantum proofs are verifiable in sublogarithmic space.

Classical verification and stand-alone quantum algorithms cannot solve these problems under standard assumptions.

Protocols require only subpolynomial-length quantum proofs.

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.