Constrained local Hamiltonians: quantum generalizations of Vertex Cover

TL;DR

This paper introduces quantum generalizations of the classical Vertex Cover problem, develops approximation algorithms inspired by classical methods, and demonstrates their effectiveness on constrained and unconstrained quantum local Hamiltonian problems.

Contribution

It defines the Transverse Vertex Cover problem, proves its complexity, and provides a simple, efficient approximation algorithm using a quantum local ratio method.

Findings

TVC is StoqMA-hard.

The quantum local ratio algorithm achieves a linear-time approximation.

The method applies to both constrained and unconstrained quantum Hamiltonian problems.

Abstract

Recent successes in producing rigorous approximation algorithms for local Hamiltonian problems such as Quantum Max Cut have exploited connections to unconstrained classical discrete optimization problems. We initiate the study of approximation algorithms for constrained local Hamiltonian problems, using the well-studied classical Vertex Cover problem as inspiration. We consider natural quantum generalizations of Vertex Cover, and one of them, called Transverse Vertex Cover (TVC), is equivalent to the PXP model with additional 1-local Pauli-Z terms. We show TVC is StoqMA-hard and develop an approximation algorithm for it based on a quantum generalization of the classical local ratio method. This results in a simple linear-time classical approximation algorithm that does not depend on solving a convex relaxation. We also demonstrate our quantum local ratio method on a traditional unconstrained quantum local Hamiltonian version of Vertex Cover which is equivalent to the anti-ferromagnetic transverse field Ising model.