Second order cone relaxations for quantum Max Cut

TL;DR

This paper introduces a second order cone relaxation for Quantum Max Cut, enabling efficient approximation of large quantum systems' ground state energies by combining it with the quantum Lasserre hierarchy.

Contribution

It presents a novel second order cone relaxation for QMC that is computationally feasible for large systems, improving approximation methods for quantum many-body problems.

Findings

Achieves a 0.526 approximation ratio to the ground state energy.

Solvable on systems with hundreds of qubits.

Provides a scalable approach for bounds on large quantum spin systems.

Abstract

Quantum Max Cut (QMC), also known as the quantum anti-ferromagnetic Heisenberg model, is a QMA-complete problem relevant to quantum many-body physics and computer science. Semidefinite programming relaxations have been fruitful in designing theoretical approximation algorithms for QMC, but are computationally expensive for systems beyond tens of qubits. We give a second order cone relaxation for QMC, which optimizes over the set of mutually consistent three-qubit reduced density matrices. In combination with Pauli level-$1$ of the quantum Lasserre hierarchy, the relaxation achieves an approximation ratio of $0.526$ to the ground state energy. Our relaxation is solvable on systems with hundreds of qubits and paves the way to computationally efficient lower and upper bounds on the ground state energy of large-scale quantum spin systems.