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

Adrian Harkness, Hamidreza Validi, Ramin Fakhimi, Illya V. Hicks, Tamás Terlaky, Luis F. Zuluaga

TL;DR

The paper tackles the challenge of solving constrained combinatorial problems on quantum devices by deriving tight, instance-specific penalty coefficients for QUBO reformulations of the max $k$-cut problem. It analyzes two formulations, the natural BQO and the reduced R-BQO, providing closed-form degree-based bounds that ensure reformulations under nonnegative edge weights and exploring gaps and conjectures for negative weights. Computational experiments with QAOA on Qiskit demonstrate that the reduced R-BQO yields significant qubit reductions and higher feasibility, guiding practical deployment on NISQ hardware. Overall, the results advance scalable QUBO-based approaches for max $k$-cut and offer guidance on penalty design and formulation choice for quantum and classical solvers alike.

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.

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

TL;DR

The paper tackles the challenge of solving constrained combinatorial problems on quantum devices by deriving tight, instance-specific penalty coefficients for QUBO reformulations of the max -cut problem. It analyzes two formulations, the natural BQO and the reduced R-BQO, providing closed-form degree-based bounds that ensure reformulations under nonnegative edge weights and exploring gaps and conjectures for negative weights. Computational experiments with QAOA on Qiskit demonstrate that the reduced R-BQO yields significant qubit reductions and higher feasibility, guiding practical deployment on NISQ hardware. Overall, the results advance scalable QUBO-based approaches for max -cut and offer guidance on penalty design and formulation choice for quantum and classical solvers alike.

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 -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 -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 -cut problem on a quantum computer simulator.

Paper Structure

This paper contains 9 sections, 6 theorems, 28 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. QUBO reformulations
  4. QUBO reformulation of BQO
  5. QUBO reformulation of R-BQO
  6. Computational Experiments
  7. Final Remarks
  8. Conjectures' Examples
  9. Feasibility characteristics of QUBO vs R-QUBO

Key Result

Lemma 1

Let $G(V, E)$ be a graph with edge weights $w_{uv}$ for all $\{u,v\} \in E$. Let $c \in \mathbb{R}^n_+$ be a penalty coefficients vector and $\hat{x} \in \{0,1\}^{n \times k}$ be an optimal solution of the $\operatorname{QUBO}$ formulation eq:maxKcut_qubo. If $c_v > \max\{\tfrac{d^+_v}{k}, -\tfrac{3

Figures (8)

  • Figure 1: Optimal solutions of the $\operatorname{QUBO}$ formulation \ref{['eq:maxKcut_qubo']} for the max 3-cut problem with different penalty coefficients. Left: optimal solution with $c_1 = c_2 = c_3 = c_4 = 1 + \epsilon$ that is optimal for the max $k$-cut problem. Right: optimal solution after changing $c_2$ from $1$ to $1 - \epsilon$ that is infeasible for the max $k$-cut problem.
  • Figure 2: Optimal solutions of the $\operatorname{QUBO}$ formulation \ref{['eq:maxKcut_qubo']} for the max 3-cut problem with different penalty coefficients. Left: optimal solution with $c_1 = c_2 = 1.5 + \epsilon$ and $c_3 = c_4 = 1 + \epsilon$ that is optimal for the max $k$-cut problem. Right: optimal solution after changing $c_2$ from $1.5 + \epsilon$ to $1.5 - \epsilon$ that is still optimal for the max $k$-cut problem.
  • Figure 3: Optimal solutions of the $\operatorname{R-QUBO}$ model for the max 3-cut problem with different penalty coefficients. Left: optimal solution with $c_1 = c_4 = c_5 = 2 + \epsilon$ and $c_2 = c_3 = 3 + \epsilon$ that is optimal for the max $k$-cut problem. Right: optimal solution after changing $c_2$ from $3 + \epsilon$ to $3 - \epsilon$ that is infeasible for the max $k$-cut problem.
  • Figure 4: Optimal solutions of the $\operatorname{R-QUBO}$ model for the max 3-cut problem with different penalty coefficients. Left: optimal solution with $c_1 = c_3 = 3 + \epsilon$, $c_4 = c_5 = 2 + \epsilon$, and $c_2 = 4 + \epsilon$ that is optimal for the max $k$-cut problem. Right: optimal solution after changing $c_2$ from $4 + \epsilon$ to $4 - \epsilon$ that is still an optimal solution for the max $k$-cut problem.
  • Figure 5: Approximation ratios of sample solutions of the BQO and R-BQO formulations of the max $3$-cut problem for a randomly generated graph with $n=6$ nodes that are obtained by directly solving these formulations using Qiskit's automatic QUBO reformulation approach and QAOA on a noiseless quantum simulator.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Remark 1
  • Lemma 1
  • Claim 1
  • Claim 2
  • Theorem 1
  • Remark 2
  • Remark 3
  • Example 1
  • Corollary 1
  • Example 2
  • ...and 9 more