Quantum Constraint Problems can be complete for $\mathsf{BQP}$, $\mathsf{QCMA}$, and more

TL;DR

This paper introduces three quantum constraint problems that are complete for different complexity classes, demonstrating the rich diversity of quantum complexity and the realization of these problems on qubits.

Contribution

It presents the first natural BQP_1-complete problem and shows all quantum constraint problems can be implemented on qubits, expanding understanding of quantum complexity classes.

Findings

Three quantum constraint problems are complete for BQP_1, QCMA_1, and coRP.

All quantum constraint problems can be realized on qubits.

The results reveal a wide range of complexity classes within quantum constraint problems.

Abstract

A quantum constraint problem is a frustration-free Hamiltonian problem: given a collection of local operators, is there a state that is in the ground state of each operator simultaneously? It has previously been shown that these problems can be in P, NP-complete, MA-complete, or QMA_1-complete, but this list has not been shown to be exhaustive. We present three quantum constraint problems, that are (1) BQP_1-complete (also known as coRQP), (2) QCMA_1-complete and (3) coRP-complete. This provides the first natural complete problem for BQP_1. We also show that all quantum constraint problems can be realized on qubits, a trait not shared with classical constraint problems. These results suggest a significant diversity of complexity classes present in quantum constraint problems.