Complexity Bounds of Constant-Space Quantum Computation

TL;DR

This paper investigates the computational power and complexity bounds of constant-space quantum automata, revealing their capabilities, limitations, and classical simulations under various acceptance criteria and error models.

Contribution

It provides a detailed analysis of the language recognition power of measure-many two-way quantum finite automata and explores their structural properties and complexity bounds.

Findings

Automata with finite-step halting must terminate in linear time.

Acceptance depends on paths terminating within exponential steps under bounded error.

Quantum automata with complex amplitudes can recognize non-recursive languages.

Abstract

We realize constant-space quantum computation by measure-many two-way quantum finite automata and evaluate their language recognition power by analyzing patterns of their exotic behaviors and by exploring their structural properties. In particular, we show that, when the automata halt "in finite steps" along all computation paths, they must terminate in worst-case liner time. In the bounded-error probability case, the acceptance of the automata depends only on the computation paths that terminate within exponentially many steps even if not all computation paths may terminate. We also present a classical simulation of those automata on two-way multi-head probabilistic finite automata with cut points. Moreover, we discuss how the recognition power of the automata varies as the automata's acceptance criteria change to error free, one-sided error, bounded error, and unbounded error by comparing the complexity of their computational powers. We further note that, with the use of arbitrary complex transition amplitudes, two-way unbounded-error quantum finite automata and two-way bounded-error 2-head quantum finite automata can recognize certain non-recursive languages, whereas two-way error-free quantum finite automata recognize only recursive languages.