Quadratic Quantum Speedup for Finding Independent Set of a Graph

TL;DR

A quadratic speedup of the quantum adiabatic algorithm (QAA) for finding independent sets (ISs) in a graph is proven analytically and provides practical guidance for optimizing near-term Rydberg atom experiments by revealing the significant impact of detuning on blockade violations.

Abstract

A quadratic speedup of the quantum adiabatic algorithm (QAA) for finding independent sets (ISs) in a graph is proven analytically. In comparison to the best classical algorithm with $O(n^2)$ scaling, where $n$ is the number of vertexes, our quantum algorithm achieves a time complexity of $O(n^2)$ for finding a large IS, which reduces to $O(n)$ for identifying a size-2 IS. The complexity bounds we obtain are confirmed numerically for a specific case with the $O(n^2)$ quantum algorithm outperforming the classical greedy algorithm, that also runs in $O(n^2)$. The definitive analytical and numerical evidence for the quadratic quantum speedup benefited from an analytical framework based on the Magnus expansion in the interaction picture (MEIP), which overcomes the dependence on the ground state degeneracy encountered in conventional energy gap analysis. In addition, our analysis links the performance of QAA to the spectral structure of the median graph, bridging algorithmic complexity, graph theory, and experimentally realizable Rydberg Hamiltonians. The understanding gained provides practical guidance for optimizing near-term Rydberg atom experiments by revealing the significant impact of detuning on blockade violations.

Figures (4)

• Figure 1: $O(n^2)$ Quantum algorithm versus $O(n^2)$ classical algorithm. The circular data points represent the average size of the IS found by the $O(n^2)$ quantum algorithm. The triangular data points represent the average size of the IS found by the $O(n^2)$ classical greedy algorithm. The square data points represent the exact size of the MIS for the Erdős--Rényi graph with $p=0.8$ we tested.
• Figure 2: A graph and its corresponding median graph. (a) An original graph for the independent set problem; (b) the corresponding median graph. Each box (or vertex) in the median graph represents an independent set.
• Figure 3: The $O(n)$ quantum algorithm's performance for finding size-2 IS. The three lines from top to bottom respectively present: the algorithm's actual performance, the asymptotic lower bound of the success rate provided in Eq. \ref{['bound2']}: $(0.5-c)^2\Omega^2T^2\kappa^4/8$, and the theoretical lower bound of the success rate provided in Eq. \ref{['bound1']}: $\Omega^2T^2\kappa^4\left({n(n-1)}/{2}-m\right)^2/8n^4$. We select a class of graphs with appropriate density, namely the Erdős--Rényi graph with $p=0.5$, to run the quantum algorithm.
• Figure 4: Minimum-degree greedy for MIS.

Theorems & Definitions (2)

• Lemma 1
• proof