We propose an additional modification of the local tensor method that increases the speed of the algorithm and increases the fraction of problems on which the algorithm finds the correct solution. The modification is based on recognizing that the local tensor method is closely related to gradient descent on a relaxation of maxcut to continuous variables. We therefore propose to apply the algorithm to the relaxed problem with the same hyperparameters as the local tensor method. We find that all of the instances tested on which the local tensor method fails or does not work are also instances for which the algorithm based on this relaxation succeeds. Hence, if a maxcut instance cannot be solved by the local tensor method we propose to attempt to solve it using the relaxation algorithm. This approach has the additional advantage that it allows us to use the same software to test the algorithm on both the original and relaxed problems. We also propose a heuristic for choosing the hyperparameters of the algorithm that give optimal performance. We also describe a software implementation of the algorithm.
Local tensor methods provide a new class of algorithms for classical optimization problems. While our focus here is on the MAXCUT problem, the class is more generally applicable to settings where a cost function can be represented as a Hamiltonian. We perform benchmarking experiments on a collection of instances of the MAXCUT problem, finding the local tensor method to outperform existing approaches. Our experiments also find the local tensor method to be closely related to gradient descent and gradient algorithms based on optimizing relaxation to continuous variables, but possessing key advantages when applied to discrete optimization problems. The classes of algorithms are proposed to be close analogues of the QAOA and we find the local tensor method to be a viable optimization approach for problem instances where gradient descent and other more traditional approaches fail. Finally, we argue that the local tensor method closely follows both imaginary-time dynamics of the system under the cost function and discretized, real-time dynamics of the system under a relaxation of the cost function to continuous variables.
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