Quantum State Complexity of Formal Languages

In this extended abstract, our notion of state complexity concerns the minimal amount of descriptive information necessary for a finite automaton to determine whether given fixed-length strings belong to a target language. This serves as a descriptional complexity measure for languages with respect to input length. In particular, we study the minimal number of inner states of quantum finite automata, whose tape heads may move freely in all directions and which conduct a projective measurement at every step, to recognize given languages. Such a complexity measure is referred to as the quantum state complexity of languages. We demonstrate upper and lower bounds on the quantum state complexity of languages on various types of quantum finite automata. By inventing a notion of timed crossing sequence, we also establish a general lower-bound on quantum state complexity in terms of approximate matrix rank. As a consequence, we show that bounded-error 2qfa’s running in expected subexponential time cannot, in general, simulate logarithmic-space deterministic Turing machines.