Almost optimal classical algorithms for a quantum generalization of Max Cut.

Καράς, Αλέξανδρος Α., Καράς, Αλέξανδρος Α.

John Kallaugher, Ojas Parekh

Sevag Gharibian, Ojas Parekh

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively.

Matthew B. Hastings

WANG Fumin, NI Ming, ZHOU Ming, WU Yongzheng

When using classical approximation algorithm to solve the max-cut problem,the time complexity increases with the complexity of graph.In order to improve the solution efficiency,this paper uses quantum adiabatic approximation algorithm to solve the ground state for Hamiltonian of max-cut problem,and the ground state value is the optimal solution.The time complexity is independent of the number of vertices and edges of graph using quantum adiabatic approximation algorithm,and the max-cut problem can be calculated within finite steps.The computing process is based on the ProjectQ software developed.By establishing the evolutionary path from initial Hamiltonian to max-cut problem Hamiltonian,the change of the expected value is analyzed to determine whether the algorithm can find the optimal solution of max-cut problem or not.Numerical analysis results show that,this algorithm can effectively improve the efficiency of solving the max-cut problem,the accuracy of solution is 0.999 9 for 3-vertex undirected graph and 6-vertex undirected sparse graph,and the results for the 6-vertex undirected complete graph is 0.969 6.

Nongmeikapam Brajabidhu Singh, Arnab Roy, Anish Kumar Saha