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<p>A major obstacle for quantum optimizers is the reformulation of constraints as a quadratic unconstrained binary optimization (QUBO). Current QUBO translators exaggerate the weight <i>M</i> of the penalty terms. Classically known as the “Big-<i>M</i>” problem, the issue becomes even more daunting for quantum solvers, since it affects the physical energy scale. We take a systematic, encompassing look at the quantum big-<i>M</i> problem, revealing NP-hardness in finding the optimal <i>M</i> and establishing bounds on the Hamiltonian spectral gap Δ as a function of the weight <i>M</i>, inversely related to the expected run-time of quantum solvers. We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks. Our algorithm gives values of Δ orders of magnitude greater, e.g. for portfolio optimization instances. Solving such instances with an adiabatic algorithm on 6-qubits of an IonQ device, we observe significant advantages in time to solution and average solution quality. Our findings are relevant to quantum and quantum-inspired solvers alike.</p>

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Alleviating the quantum Big-M problem

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年7月26日

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  • Edoardo Alessandroni,
  • Sergi Ramos-Calderer,
  • Ingo Roth,
  • Emiliano Traversi,
  • Leandro Aolita

摘要

A major obstacle for quantum optimizers is the reformulation of constraints as a quadratic unconstrained binary optimization (QUBO). Current QUBO translators exaggerate the weight M of the penalty terms. Classically known as the “Big-M” problem, the issue becomes even more daunting for quantum solvers, since it affects the physical energy scale. We take a systematic, encompassing look at the quantum big-M problem, revealing NP-hardness in finding the optimal M and establishing bounds on the Hamiltonian spectral gap Δ as a function of the weight M, inversely related to the expected run-time of quantum solvers. We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks. Our algorithm gives values of Δ orders of magnitude greater, e.g. for portfolio optimization instances. Solving such instances with an adiabatic algorithm on 6-qubits of an IonQ device, we observe significant advantages in time to solution and average solution quality. Our findings are relevant to quantum and quantum-inspired solvers alike.