Quantum Parameterized Complexity

TL;DR

This paper develops a quantum version of parameterized complexity theory, introducing quantum classes and analyzing the complexity of quantum problems, revealing new classifications and separations among quantum computational problems.

Contribution

It establishes the first quantum parameterized complexity classes and explores their relationships with classical classes and quantum problems.

Findings

Quantum classes extend classical parameterized complexity.

Clear separation between Quantum Circuit Satisfiability and Local Hamiltonian problems.

Framework enables classification of QMA-hard problems in quantum complexity.

Abstract

Parameterized complexity theory was developed in the 1990s to enrich the complexity-theoretic analysis of problems that depend on a range of parameters. In this paper we establish a quantum equivalent of classical parameterized complexity theory, motivated by the need for new tools for the classifications of the complexity of real-world problems. We introduce the quantum analogues of a range of parameterized complexity classes and examine the relationship between these classes, their classical counterparts, and well-studied problems. This framework exposes a rich classification of the complexity of parameterized versions of QMA-hard problems, demonstrating, for example, a clear separation between the Quantum Circuit Satisfiability problem and the Local Hamiltonian problem.