Complexity of Fermionic 2-SAT

TL;DR

This paper introduces Fermionic 2-SAT, a quantum fermionic satisfiability problem, and demonstrates its classical solvability for k=2, while also establishing NP-completeness and QMA$_1$-hardness results for related problems.

Contribution

It defines Fermionic 2-SAT, proves its efficient classical solvability for k=2, and explores complexity boundaries including NP-completeness and QMA$_1$-hardness.

Findings

Fermionic 2-SAT can be solved efficiently classically for k=2.

Deciding fixed particle number parity solutions is also efficiently solvable.

Fermionic 9-SAT is QMA$_1$-hard.

Abstract

We introduce the fermionic satisfiability problem, Fermionic $k$-SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on $n$ fermionic modes, where each fermionic projector involves at most $k$ fermionic modes. We prove that this problem can be solved efficiently classically for $k=2$. In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA$_1$-hard.