Encodings of the weighted MAX k-CUT on qubit systems

TL;DR

This paper investigates various encoding strategies for the weighted MAX k-CUT problem on qubit systems, optimizing quantum algorithms' efficiency and resource usage through systematic encoding and simulation.

Contribution

It introduces a systematic, resource-efficient encoding method for MAX k-CUT on quantum devices, including balanced color encoding and subspace encoding with tailored mixers.

Findings

Encoding schemes improve circuit depth and approximation ratios.

Balanced encoding of colors enhances quantum optimization performance.

Simulations demonstrate effectiveness of proposed encoding strategies.

Abstract

The weighted MAX k-CUT problem involves partitioning a weighted undirected graph into k subsets, or colors, to maximize the sum of the weights of edges between vertices in different subsets. This problem has significant applications across multiple domains. This paper explores encoding methods for MAX k-CUT on qubit systems, utilizing quantum approximate optimization algorithms (QAOA) and addressing the challenge of encoding integer values on quantum devices with binary variables. We examine various encoding schemes and evaluate the efficiency of these approaches. The paper presents a systematic and resource efficient method to implement phase separation for diagonal square binary matrices. When encoding the problem into the full Hilbert space, we show the importance of encoding the colors in a balanced way. We also explore the option to encode the problem into a suitable subspace, by designing suitable state preparations and constrained mixers (LX- and Grover-mixer). Numerical simulations on weighted and unweighted graph instances demonstrate the effectiveness of these encoding schemes, particularly in optimizing circuit depth, approximation ratios, and computational efficiency.