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<p>Multidimensional numerical integration is a central ingredient of theoretical predictions in high-energy physics, where multiloop Feynman diagrams and phase-space integrals are computationally demanding due to divergences and complex mathematical structures. Established Adaptive Importance Sampling methods for numerical integration, such as VEGAS, iteratively refine a grid in a separable way, dimension by dimension. This keeps the algorithm scalable but reduces performance when strong inter-variable correlations are present. In this work, we introduce a hybrid quantum-classical algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for multidimensional Monte Carlo integration. Our approach uses a Parametrized Quantum Circuit to encode a non-separable Probability Density Function on a multidimensional grid and allocate samples efficiently in the integration domain. We apply the method to a sharply peaked loop Feynman integral and to multi-modal benchmark integrals. Our results show that QAIS provides an efficient route for high-precision evaluation of multidimensional integrals.</p>

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Unlocking multidimensional integration with quantum adaptive importance sampling

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月22日

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  • Konstantinos Pyretzidis,
  • Jorge J. Martínez de Lejarza,
  • Germán Rodrigo

摘要

Multidimensional numerical integration is a central ingredient of theoretical predictions in high-energy physics, where multiloop Feynman diagrams and phase-space integrals are computationally demanding due to divergences and complex mathematical structures. Established Adaptive Importance Sampling methods for numerical integration, such as VEGAS, iteratively refine a grid in a separable way, dimension by dimension. This keeps the algorithm scalable but reduces performance when strong inter-variable correlations are present. In this work, we introduce a hybrid quantum-classical algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for multidimensional Monte Carlo integration. Our approach uses a Parametrized Quantum Circuit to encode a non-separable Probability Density Function on a multidimensional grid and allocate samples efficiently in the integration domain. We apply the method to a sharply peaked loop Feynman integral and to multi-modal benchmark integrals. Our results show that QAIS provides an efficient route for high-precision evaluation of multidimensional integrals.