Computational complexity of the Rydberg blockade in two dimensions

Abstract: We discuss the computational complexity of finding the ground state of the two-dimensional array of quantum bits that interact via strong van der Waals interactions. Specifically, we focus on systems where the interaction strength between two spins depends only on their relative distance $x$ and decays as $1/x^6$ that have been realized with individually trapped homogeneously excited neutral atoms interacting via the so-called Rydberg blockade mechanism. We show that the solution to NP-complete problems can be encoded in ground state of such a many-body system by a proper geometrical arrangement of the atoms. We present a reduction from the NP-complete maximum independent set problem on planar graphs with maximum degree three. Our results demonstrate that computationally hard optimization problems can be naturally addressed with coherent quantum optimizers accessible in near term experiments.