Unconditional proofs of quantumness between small-space machines

TL;DR

This paper introduces a new protocol for unconditionally proving a machine's quantumness using small-space resources, avoiding reliance on unproven assumptions.

Contribution

It formulates a new class of problems solvable by quantum but not classical small-space machines, enabling unconditional proofs of quantumness.

Findings

Quantum machines solve the new problems efficiently.

Classical small-space machines cannot solve these problems.

The protocol provides unconditional verification of quantumness.

Abstract

A proof of quantumness is a protocol through which a classical machine can test whether a purportedly quantum device, with comparable time and memory resources, is performing a computation that is impossible for classical computers. Existing approaches to provide proofs of quantumness depend on unproven assumptions about some task being impossible for machines of a particular model under certain resource restrictions. We study a setup where both devices have space bounds $\mathit{o}(\log \log n)$. Under such memory budgets, it has been unconditionally proven that probabilistic Turing machines are unable to solve certain computational problems. We formulate a new class of problems, and show that these problems are polynomial-time solvable for quantum machines, impossible for classical machines, and have the property that their solutions can be "proved" by a small-space quantum machine to a classical machine with the same space bound. These problems form the basis of our newly defined protocol, where the polynomial-time verifier's verdict about the tested machine's quantumness is not conditional on an unproven weakness assumption.