QMCPy: A Python Framework for (Quasi-)Monte Carlo Algorithms

Summary

Monte Carlo (MC) methods estimate high-dimensional integrals by computing sample averages at independent and identically distributed (IID) random points. Quasi-Monte Carlo (QMC) methods replace IID samples with low-discrepancy (LD) sequences which more uniformly cover the integration domain, leading to faster convergence and reduced computational requirements. The figure below visualizes IID and LD sequences.

QMCPy (https://qmcsoftware.github.io/QMCSoftware) (Choi et al. 2026) is our Python package for high-dimensional numerical integration using MC and QMC methods, collectively “(Q)MC.” Its object-oriented design enables researchers to easily implement novel (Q)MC algorithms. The framework offers user-friendly APIs, diverse (Q)MC algorithms, adaptive error estimation techniques, and integration with scientific libraries following reproducible research practices (Choi 2014; Choi et al. 2022). Compared to previous versions, QMCPy v2.2 (which is easily installed with pip install -U qmcpy) includes

• improved documentation,
• strengthened tests and demos, and
• expanded support for randomized LD sequences.

An IID sequence with gaps and clusters alongside LD sequences with more uniform coverage. Each sequence contains purple stars (initial 32 points), green triangles (next 32), and blue circles (subsequent 64).

Statement of Need

(Q)MC methods are essential for computational finance (Lemieux 2004; X. Wang and Sloan 2005; Giles and Waterhouse 2009; Zhang et al. 2021), uncertainty quantification (Seelinger et al. 2023; Parno et al. 2014; Parno, Davis, and Seelinger 2021; Marzouk et al. 2016; Kaarnioja et al. 2021), machine learning (Dick and Feischl 2021; Chen et al. 2018), and physics (Albert and Barabási 2002; Landau and Binder 2014; Bernhard et al. 2015). While (Q)MC methods are well established (Dick and Pillichshammer 2010; Dick, Kuo, and Sloan 2013), practical implementation demands numerical and algorithmic expertise. QMCPy follows MATLAB’s Guaranteed Automatic Integration Library (GAIL) (Choi et al. 2021; Tong et al. 2022) in consolidating a broad range of cutting-edge (Q)MC algorithms into a unified framework (Choi et al. 2022, 2024; Sorokin and Rathinavel 2022; Sorokin 2025; Hickernell, Kirk, and Sorokin 2026). QMCPy features

• intuitive APIs for (Q)MC components,
• flexible integrations with NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), and PyTorch (Paszke et al. 2019),
• robust and adaptive sampling with theoretically grounded error estimation, and
• extensible components enabling researchers to implement and test new algorithms.

While popular modules like scipy.stats.qmc (Roy et al. 2023) and torch.quasirandom (Paszke et al. 2019) provide basic (Q)MC sequences such as Sobol’ and Halton, QMCPy provides (Q)MC researchers and practitioners an end-to-end framework with additional capabilities to enable state-of-the-art (Q)MC techniques. Advanced features unique to QMCPy include

• customizable LD sequences with diverse randomization techniques,
• efficient generators of LD sequences with multiple independent randomizations,
• automatic variable transformations for (Q)MC compatibility, and
• rigorous adaptive error estimation algorithms.

Components

(Q)MC methods approximate the multivariate integral $$\begin{equation}\label{eq:mu-general} \mu := \mathbb{E}[g(\mathbf{T})] = \int_{\mathcal{T}} g(\mathbf{t}) \, \lambda(\mathbf{t}) \, d\mathbf{t}, \qquad \mathbf{T} \sim \lambda, \end{equation}$$ where $g$ is the integrand and $\lambda$ is the probability density of a random variable $\mathbf{T}$ whose distribution we call the true measure. To accommodate LD samples (approximately uniform on $[0,1]^d$), a transformation $\boldsymbol{\psi}$ is performed to rewrite $\mu$ as $$\begin{equation}\label{eq:mu-uniform} \mu = \mathbb{E}[f(\mathbf{X})] = \int_{[0,1]^d} f(\mathbf{x}) \, d\mathbf{x}, \qquad \mathbf{X} \sim \mathcal{U}[0,1]^d. \end{equation}$$ If $\mathbf{T} \sim \boldsymbol{\psi}(\mathbf{X})$, then $f = g \circ \boldsymbol{\psi}$.

(Q)MC methods estimate the population mean $\mu$ in this transformed form via the sample mean $$\begin{equation}\label{eq:mu-hat} \widehat{\mu} := \frac{1}{n} \sum_{i=1}^{n} f(\mathbf{X}_i). \end{equation}$$ MC methods use IID $\mathbf{X}_1,\dots,\mathbf{X}_n$ and have error $|\widehat{\mu}-\mu|$ like $\mathcal{O}(n^{-1/2})$ (Niederreiter 1978). QMC methods choose dependent LD nodes that fill $[0,1]^d$ more evenly, i.e., the discrepancy between the discrete distribution of $\mathbf{X}_1,\dots,\mathbf{X}_n$ and the uniform distribution is small. QMC methods can achieve errors like $\mathcal{O}(n^{-1+\delta})$ where $\delta>0$ is arbitrarily small (Hickernell and Wang 2002; X. Wang 2003). A key feature of QMCPy is stopping criteria that automatically determine $n$ so $|\mu - \widehat{\mu}| \le \varepsilon$ for a user-specified tolerance $\varepsilon>0$, deterministically or with high probability.

QMCPy contains four main abstract classes:

1. Discrete Distributions generate IID or randomized LD sequences (Sorokin 2025) including

   • Lattices with random shifts (Cranley and Patterson 1976; Hickernell, Lemieux, and Owen 2005; Richtmyer 1951; Coveyou and MacPherson 1967; Y. Wang and Hickernell 2002).
   • Digital Sequences (including Sobol’ and Faure constructions) with digital shifts (DS), linear matrix scrambling (LMS), or nested uniform scrambling (NUS, also called Owen scrambling) (Sobol’ 1967; Dick and Pillichshammer 2005; Matoušek 1998; Owen 1995, 2003; Dick 2011; Niederreiter 1987, 1992; Dick and Pillichshammer 2010). Higher-order digital sequences are available to enable QMC convergence like $\mathcal{O}(n^{-\alpha+\delta})$ when $f$ has $\alpha$ degrees of smoothness (Dick 2011).
   • Halton Sequences with digital permutations, DS, LMS, or NUS (Halton 1960; X. Wang and Hickernell 2000; Matoušek 1998; Owen and Pan 2024; Morokoff and Caflisch 1994).

   Internally, QMCPy’s LD generators call our C package QMCToolsCL (Sorokin 2026). We also integrate with the LDData repository (Sorokin, Hickernell, and L’Ecuyer 2025) which collects lattice generating vectors and digital sequence generating matrices from Kuo’s websites (Cools, Kuo, and Nuyens 2006; Nuyens and Cools 2006; Kuo 2007; Joe and Kuo 2003, 2010; Joe and Kuo 2008), the Magic Point Shop (Kuo and Nuyens 2016), and LatNet Builder (L’Ecuyer et al. 2022).

2. True Measures come with default transformations $\boldsymbol{\psi}$ satisfying $\boldsymbol{\psi}(\mathbf{X}) \sim \mathbf{T}$. For example, if $\mathbf{T} \sim \mathcal{N}(\mathbf{m},\Sigma= \mathbf{A}\mathbf{A}^T)$ is a $d$-dimensional Gaussian, then $\boldsymbol{\psi}(\mathbf{X}) = \mathbf{A} \Phi^{-1}(\mathbf{X}) + \mathbf{m}$ where $\Phi^{-1}$ is the inverse Gaussian distribution function applied elementwise. We support the broad range of measures included in scipy.stats (Virtanen et al. 2020).

3. Integrands $g$, given a transformation $\boldsymbol{\psi}$, automatically set $f = g \circ \boldsymbol{\psi}$ so that $\mu = \mathbb{E}[g(\mathbf{T})] = \mathbb{E}[f(\mathbf{X})]$.

4. Stopping Criteria (SC) adaptively increase the sample size $n$ until (Q)MC estimates satisfy user-defined error tolerances (Hickernell et al. 2018; Tong et al. 2022; Owen 2024). SC include guaranteed MC algorithms (Hickernell et al. 2013) and QMC algorithms based on:

   • multiple randomizations of LD sequences (L’Ecuyer et al. 2023),
   • quickly tracking the decay of Fourier coefficients (Hickernell and Jiménez Rugama 2016; Jiménez Rugama and Hickernell 2016; Hickernell, Jiménez Rugama, and Li 2018; Ding, Hickernell, and Jiménez Rugama 2020), or
   • fast Bayesian cubature (Rathinavel 2019; Rathinavel and Hickernell 2019, 2022).

   QMCPy is also capable of simultaneously approximating functions of multiple integrands (Sorokin and Rathinavel 2022), and we are actively expanding support for multilevel (Q)MC algorithms following Julia’s MultilevelEstimators.jl (Robbe 2024).

   The figure below compares (Q)MC SC for Asian option pricing with 100 independent trials per error tolerance $\varepsilon$. The left and middle plots show median lines and shaded regions for 10%–90% quantiles. While MC SC require $n = \mathcal{O}(1/\varepsilon^2)$ samples (and time), QMC SC require only $n = \mathcal{O}(1/\varepsilon)$. (Q)MC SC consistently meet tolerances, with the right plot showing distributions of errors for a single error tolerance.

(Q)MC SC for Asian option pricing.

Acknowledgements

The authors acknowledge support from the U.S. National Science Foundation grant DMS-2316011 and the Department of Energy Office of Science Graduate Student Research Program. We thank the international (Q)MC research community as well as JOSS reviewers and editors for invaluable and timely feedback and support.

This article has been co-authored by employees of National Technology and Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employees co-own right, title and interest in and to the article and are responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (https://www.energy.gov/downloads/doe-public-access-plan).

AS is partially supported by DARPA The Right Space HR0011-25-9-0031.

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