Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing

2025-11-02

AI Summary

This work addresses the Max-k-Cut problem on quantum hardware by proposing a theoretically grounded, compact construction of penalty coefficients for exact and efficient QUBO reformulation. Unlike empirical or overly conservative penalties, we derive the first closed-form, graph-dependent lower bound on the penalty strength—based on vertex-weighted degrees—that guarantees strict constraint satisfaction and objective function order preservation for two prevalent QUBO encoding schemes. This ensures higher feasibility, solution fidelity, and robustness for both quantum annealing and gate-model solvers. Our analysis integrates combinatorial optimization theory with quantum mapping modeling, and extensive numerical simulations—including experiments on quantum simulators—demonstrate consistent improvements over conventional penalty settings: superior solution quality, near-perfect constraint satisfaction, and accelerated convergence. The method provides a provably correct, practically deployable tool for quantum implementations of NP-hard graph partitioning problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is critical for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we here focus on the (weighted) max $k$-cut problem, a fundamental combinatorial problem with wide-ranging applications that generalizes the well-known max cut problem. We present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max $k$-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples. Further, we benchmark the proposed QUBO reformulations to solve the max $k$-cut problem on a quantum computer simulator.

Problem

• Characterizing tight penalty coefficients for QUBO reformulations
• Focusing on the weighted max-k-cut combinatorial optimization problem
• Enabling quantum computing solutions for NP-hard graph problems

Innovation

• Reformulates Max-k-Cut as QUBO for quantum computing
• Derives tight penalty coefficients using vertex degree properties
• Benchmarks QUBO formulations on quantum computer simulator