# Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

## TL;DR

This paper explores the complexity of nonuniform families of quantum finite automata with polynomial size, revealing their relationships with quantum space-bounded computation and advice, and establishing new inclusion and separation results.

## Contribution

It introduces nonuniform state complexity classes for quantum automata with garbage tapes and connects them to quantum logarithmic-space computation with advice.

## Key findings

- Relationships among various quantum finite automata classes
- Equivalence between polynomial-size quantum automata and quantum advice computation
- Separation results for automata with different state complexities

## Abstract

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275--286] were concerned with nonuniform families of finite automata and they discussed the behaviors of the nonuniform complexity classes defined by such families of finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using nonuniform families of quantum finite automata empowered by the flexible use of garbage tapes. We first present general inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata having polynomially many inner states. For two-way quantum finite automata equipped with flexible garbage tapes, we show a close relationship between the nonuniform state complexity of the family of such polynomial-size quantum finite automata and the parameterized complexity class induced by logarithmic-space quantum computation assisted by polynomial-size advice. We further establish a direct connection between space-bounded quantum computation with quantum advice and quantum finite automata whose transitions are dictated by superpositions of transition tables.