Implementing Quantum Finite Automata Algorithms on Noisy Devices

Quantum finite automata (QFAs) literature offers an alternative mathematical model for studying quantum systems with finite memory. As a superiority of quantum computing, QFAs have been shown exponentially more succinct on certain problems such as recognizing the language $ MOD_p = \{a^j \mid j \equiv 0 \mod p\} $ with bounded error, where $p$ is a prime number. In this paper we present improved circuit based implementations for QFA algorithms recognizing the $ MOD_p $ problem using the Qiskit framework. We focus on the case $p=11$ and provide a 3 qubit implementation for the $MOD_{11}$ problem reducing the total number of required gates using alternative approaches. We run the circuits on real IBM quantum devices but due to the limitation of the real quantum devices in the NISQ era, the results are heavily affected by the noise. This limitation reveals once again the need for algorithms using less amount of resources. Consequently, we consider an alternative 3 qubit implementation which works better in practice and obtain promising results even for the problem $ MOD_{31} $.