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.

Bulk and Boundary Quantum Phase Transitions in a Square Rydberg Atom Array

“…Moreover, Rydberg simulators allow for detailed studies of dynamics across quantum phase transitions (QPTs) and other quantum critical phenomena. Finally, they provide a natural many-body platform for exploring quantum advantage in solving combinatorial optimization problems [19,20]. These advances motivate detailed quantitative understanding of the QPTs between complex phases in such systems, with realistic interactions and geometries.…”

Simulation of many-body dynamics using Rydberg excitons

“…Maximum Independent Set (MIS) problem aims to find the largest subset of nodes in a graph where no two nodes are adjacent (linked by an edge). It has been shown that solution to NP-complete problems such as the MIS problem on planar graphs can be mapped onto the ground state of a multi-body Rydberg system with proper arrangement of Rydberg atoms (or in our case excitons); see [26]. This ground state is our objective state with maximum number of Rydberg excitations such that no two excitations occur within the Rydberg blockade radius.…”

“…3(b). The objective states (which are called 'ground states' in [26]) with the maximum possible five excitons each, |rggrgrrggr and |rggrgrrgrg , have the highest probabilities, 0.01187 and 0.01124, with a total probability of 0.02311. The number of Rydberg excitation in the final state is plotted as an histogram in Fig.…”

Programmable Quantum Annealing Architectures with Ising Quantum Wires

“…While a quantum annealer might not reduce the classical computation complexity of NP-hard problems O(exp(αN γ )) [with N the problem-size] to polynomial [5][6][7][8][9], an exponential speedup for BQP has been suggested [10,11], and one might gain significant improvement on coefficients α and γ for NP problems [1,4,12] compared to classical algorithms. Because of important implication for both science [4] and commercial applications [13], quantum annealing has received significant attention in recent years [12,[14][15][16][17][18].…”