Unconditional proofs of quantumness between small-space machines
A. C. Cem Say, M. Utkan Gezer
TL;DR
The paper addresses verifying quantumness when both devices are extremely memory-bounded, removing reliance on unproven hardness assumptions. It develops an unconditional framework using DS-FK separation problems and very small provers (constant-/space) paired with a space-bounded verifier, leveraging 2qcfa capabilities and padded language constructions to certify quantum behavior. The main contributions include constructing DS-FK-based proofs of quantumness for languages recognizable by quantum small-space machines, and extending verification to beyond 2DFA(2) with robust interaction protocols that maintain a tunable acceptance gap in the presence of cheating provers. This work advances practical and theoretically grounded approaches for verifying quantum capabilities in ultra-scarce memory regimes, with implications for near-term quantum devices and foundational quantum-classical separation results.
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.