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European Physical Society Conference on High Energy Physics (EPS-HEP)

A Quantum Adaptive Importance Sampling Algorithm for Multidimensional Integration

Konstantinos Pyretzidis

Published Dec 8, 2025• 0 citations
Publisher

Abstract

This work presents Quantum Adaptive Importance Sampling (QAIS), a hybrid quantum-classical algorithmic workflow that performs Monte Carlo numerical integration of multivariate functions. This workflow extends standard Importance Sampling to a quantum computational system. Its primary purpose is to produce an unbiased estimate of integrals, with a limited number of measurements. By using the exponentially sized Hilbert space of a Parameterized Quantum Circuit (PQC), we manipulate the Probability Density Function (PDF) over a grid in its entirety. With adequate expressivity and entanglement, we effectively capture correlations among different integrand variables, thus bypassing the separable PDF assumption, as implemented in VEGAS. By optimizing the PQC, we adapt and ultimately load a PDF that approximates the integrand's behavior. Direct sampling from it, allocates samples to the important regions, allowing a highly accurate integral estimation. As an application, we study a benchmark two-dimensional integral, and compare the precision of the best grid of VEGAS against that of the quantum-generated proposal PDF, as a function of the number of samples.

DOI
10.22323/1.485.0638

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This work presents Quantum Adaptive Importance Sampling (QAIS), a hybrid quantum-classical algorithmic workflow that performs Monte Carlo numerical integration of multivariate functions. This workflow extends standard Importance Sampling to a quantum computational system. Its primary purpose is to produce an unbiased estimate of integrals, with a limited number of measurements. By using the exponentially sized Hilbert space of a Parameterized Quantum Circuit (PQC), we manipulate the Probability Density Function (PDF) over a grid in its entirety. With adequate expressivity and entanglement, we effectively capture correlations among different integrand variables, thus bypassing the separable PDF assumption, as implemented in VEGAS. By optimizing the PQC, we adapt and ultimately load a PDF that approximates the integrand's behavior. Direct sampling from it, allocates samples to the important regions, allowing a highly accurate integral estimation. As an application, we study a benchmark two-dimensional integral, and compare the precision of the best grid of VEGAS against that of the quantum-generated proposal PDF, as a function of the number of samples.
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. Theoretical calculations in particle physics rely on multidimensional integrals that become difficult to evaluate in the presence of correlations. Here, the authors show that a hybrid quantum-classical algorithm implementing Quantum Adaptive Importance Sampling can learn correlated structures efficiently and deliver precise results for nontrivial integrals.
Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.
Abstract Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the standard strategies of rejection sampling, importance sampling, and Markov-chain sampling can be adapted to this context, where the samples must obey the constraints imposed by the positivity of the statistical operator. For illustration, we generate sample points in the probability space of qubits, qutrits, and qubit pairs, both for tomographically complete and incomplete measurements. We use these samples for various purposes: establish the marginal distribution of the purity; compute the fractional volume of separable two-qubit states; and calculate the size of regions with bounded likelihood.
Monte Carlo integration using quantum computers has been widely investigated, including applications to concrete problems. It is known that quantum algorithms based on quantum amplitude estimation (QAE) can compute an integral with a smaller number of iterative calls of the quantum circuit which calculates the integrand, than classical methods call the integrand subroutine. However, the issues about the iterative operations in the integrand circuit have not been discussed so much. That is, in the high-dimensional integration, many random numbers are used for calculation of the integrand and in some cases similar calculations are repeated to obtain one sample value of the integrand. In this paper, we point out that we can reduce the number of such repeated operations by a combination of the nested QAE and the use of pseudorandom numbers (PRNs), if the integrand has the separable form with respect to contributions from distinct random numbers. The use of PRNs, which the authors originally proposed in the context of the quantum algorithm for Monte Carlo, is the key factor also in this paper, since it enables parallel computation of the separable terms in the integrand. Furthermore, we pick up one use case of this method in finance, the credit portfolio risk measurement, and estimate to what extent the complexity is reduced.
Introduction We present Quantum Adaptive Search (QAGS), a hybrid quantum-classical algorithm for global optimization of multivariate functions. The method employs an adaptive mechanism that dynamically narrows the search space based on a quantum-estimated probability distribution of the objective function. Methods A quantum state encodes information about solution quality through a complex-amplitude mapping, enabling identification of promising regions and progressive tightening of the search bounds; a classical optimizer then performs local refinement. The analysis shows contraction of the search space toward global optima with controlled computational complexity. Results In simulation on standard benchmarks (Rastrigin, Styblinski-Tang, Rosenbrock), QAGS attains solutions at or near the true minima with very small absolute errors. Against an Adaptive Grid Search on the Sphere function, QAGS achieves comparable accuracy and shows increasing efficiency with dimensionality. Discussion These results indicate that amplitude-encoded region selection combined with classical refinement effectively contracts the search space and can reduce time and space requirements, especially at higher dimensions, while practical hardware implementations of amplitude encoding remain challenging.