[Paper Review] Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming

F. Wagner, Jonas Nüßlein+1 more|arXiv (Cornell University)|Feb 10, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a hybrid quantum-classical heuristic for the weighted maximum-cut problem, combining linear programming relaxation with QAOA to reduce problem size and improve solution quality on near-term quantum hardware. By shrinking instances via optimal cycle relaxation solutions and deriving efficient QAOA parameters, the method enables solving larger MaxCut instances than current quantum hardware can handle directly, with experimental results showing improved approximation ratios on real devices.

ABSTRACT

To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing realizes, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem or, equivalently, for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of QAOA, a parameterized quantum algorithm, by deriving optimal parameters for special instances which motivates a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.

Motivation & Objective

• To bridge the gap between classical integer programming and near-term quantum computing for combinatorial optimization.
• To address the limitation of current quantum hardware in handling large-scale optimization problems.
• To enhance QAOA applicability by deriving efficient, problem-specific parameter estimates for weighted MaxCut.
• To integrate quantum heuristics into classical branch-and-cut frameworks by reducing problem size via optimal variable fixing.
• To demonstrate that problem shrinking via linear relaxation preserves solution quality while enabling quantum acceleration.

Proposed method

• The method uses a cycle relaxation of the MaxCut problem to compute optimal variable correlations for shrinking.
• It reduces large MaxCut instances by fixing variables based on the optimal solution of the cycle relaxation, preserving optimality.
• The reduced problem is solved using QAOA, with parameters derived analytically for regular and triangle-free graphs with binary weights.
• A parameter estimation strategy is proposed for general instances based on analytical results, reducing the need for iterative tuning.
• The quantum solution is then mapped back to the original problem by reversing the shrinking operations.
• The approach is designed to integrate seamlessly into classical branch-and-cut algorithms by replacing the enumeration phase with a quantum heuristic.

(a) Instance a.

Experimental results

Research questions

• RQ1 Can linear programming relaxations be used to shrink large MaxCut instances in a way that preserves optimality and enables quantum computation on limited hardware?
• RQ2 How can QAOA parameters be estimated efficiently without iterative classical optimization, especially for weighted MaxCut?
• RQ3 Does optimal shrinking via cycle relaxation improve the performance of QAOA on noisy intermediate-scale quantum (NISQ) devices?
• RQ4 To what extent does problem shrinking enhance the approximation ratio of QAOA compared to random or suboptimal shrinking?
• RQ5 Can the hybrid quantum-classical framework be integrated into classical exact solvers like branch-and-cut to accelerate solution of hard combinatorial problems?

Key findings

• The cycle relaxation yields optimal variable correlations for shrinking, preserving solution quality and enabling exact recovery of the original solution.
• QAOA on real quantum hardware outperforms the coin-flipping heuristic when shrinking is applied, especially with optimal shrinking strategies.
• Approximation ratios for QAOA improved with increasing numbers of shrunk vertices when using optimal correlations, indicating performance gains from problem reduction.
• Suboptimal shrinking revealed a trade-off between potential solution loss and increased approximability, suggesting a performance sweet spot.
• The proposed QAOA parameter estimate significantly reduces classical tuning overhead and performs well in practice, enabling faster integration into hybrid workflows.
• The method successfully solved MaxCut instances an order of magnitude larger than the native qubit capacity of current quantum devices, demonstrating proof-of-principle scalability.

(b) Instance b.