Encodings of the weighted MAX k-CUT on qubit systems

TL;DR

This work addresses encodings of the weighted MAX $k$-CUT problem on qubit systems within the QAOA framework, focusing on non-power-of-two values of $k$. It develops systematic binary- and subspace-encoding strategies, including balanced colorings and constrained LX- and Grover-mixers, to realize efficient phase-separation circuits and feasible-state preservation. The study provides detailed circuit constructions, resource analyses, and simulations on small graphs, showing that balanced color partitions and subspace mixers can reduce circuit depth while maintaining strong approximation ratios. Overall, the results demonstrate practical pathways to implement MAX $k$-CUT on near-term quantum devices, highlighting trade-offs between full-space and subspace encodings and guiding design choices for scalable quantum optimization.

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.

Figures (5)

• Figure 1: An example of an optimal solution for a MAX $3$-CUT problem.
• Figure 2: Overview of different ways to encode the problem.
• Figure 3: The resource comparison shows that the circuits for realizing the phase separating operator $U_P$ as proposed in this work require fewer resources in terms of entangling gates as previous work fuchs2021efficient. Another advantage of the proposed method is that it does not require ancilla qubits. Whereas we realize the diagonal operator directly, previous work fuchs2021efficient utilized the circuit shown in (b) between each edge $(i,j)$. For each element $\{y_1,y_2\} \in \operatorname{clr}^k\setminus \cup_{i,j} \{i,i\}$, one such circuit is necessary with $N_{y_i} = X^{\operatorname{bin}(y_i)}$.
• Figure 4: Initial state preparation and mixer for $k=3,5,6,7$ when restricting to a suitably chosen subspace. We use the notation $R_{P}(t) = \exp \left(-i\frac{t}{2}p\right)$, $P\in\{X,Y,Z\}$, and $R_{XY}(t) = \exp \left(-i\frac{t}{4}(X\otimes X+Y\otimes Y)\right)$.
• Figure 5: Graph instances used in this paper.