# A Quantum Query Complexity Trichotomy for Regular Languages

## TL;DR

This paper establishes a three-way classification of the quantum query complexity for regular languages, revealing that they are either constant, square root, or linear in complexity, with implications for quantum algorithms and complexity measures.

## Contribution

It provides the first explicit quantum algorithm for star-free languages with ~O(sqrt n) complexity and characterizes the quantum query complexity trichotomy for all regular languages.

## Key findings

- Regular languages have quantum query complexity Theta(1), ~Theta(sqrt n), or Theta(n).
- Star-free languages can be decided with ~O(sqrt n) quantum query complexity.
- Applications include new quantum algorithms for various computational problems.

## Abstract

We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Theta(1), ~Theta(sqrt n), or Theta(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Theta(n^c) for all computable c in [1/2,1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy---either Theta(1) or Theta(n)---for sensitivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity. The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language in ~O(sqrt n) time. This well-studied family of the regular languages admits many interesting characterizations, for instance, as those languages expressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also express functions such as "there exist a pair of 2's such that everything between them is a 0." Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known. We show a variety of applications---new quantum algorithms for dynamic constant-depth Boolean formulas, balanced parentheses nested constantly many levels deep, binary addition, a restricted word break problem, and path-discovery in narrow grids---all obtained as immediate consequences of our classification theorem.