An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

We give an approximation algorithm for Quantum Max-Cut which works by rounding a semi-definite program (SDP) relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz [AGM20], and Parekh, Thompson [PT21a] achieved approximation ratios of 0.531 and 0.533 respectively. In addition we study the EPR Hamiltonian, whose terms project onto EPR states rather than singlet states. (EPR are initials Einstein, Podolsky and Rosen.) We argue this is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio 1/root 2 on all graphs.