A. C. Cem Say

M. Utkan Gezer, A. C. Cem Say

We study the capabilities of probabilistic finite-state machines that act as verifiers for certificates of language membership for input strings, in the regime where the verifiers are restricted to toss some fixed nonzero number of coins regardless of the input size. Say and Yakaryılmaz showed that the class of languages that could be verified by these machines within an error bound strictly less than $1/2$ is precisely NL, but their construction yields verifiers with error bounds that are very close to $1/2$ for most languages in that class when the definition of "error" is strengthened to include looping forever without giving a response. We characterize a subset of NL for which verification with arbitrarily low error is possible by these extremely weak machines. It turns out that, for any $\varepsilon>0$, one can construct a constant-coin, constant-space verifier operating within error $\varepsilon$ for every language that is recognizable by a linear-time multi-head nondeterministic finite automaton (2nfa($k$)). We discuss why it is difficult to generalize this method to all of NL, and give a reasonably tight way to relate the power of linear-time 2nfa($k$)'s to simultaneous time-space complexity classes defined in terms of Turing machines.

A. C. Cem Say, M. Utkan Gezer

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.