We extend the recently developed quantum quasi Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multidimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitude. A core concept of QQMC is the a priori construction of a ``model function'' that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $\ensuremath{\sim}1/{N}^{0.86}$ in the best cases and comparable to Monte Carlo scaling $\ensuremath{\sim}1/{N}^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the fork tensor product state (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pad\'e approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.We extend the recently developed quantum quasi Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multidimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitude. A core concept of QQMC is the a priori construction of a ``model function'' that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $\ensuremath{\sim}1/{N}^{0.86}$ in the best cases and comparable to Monte Carlo scaling $\ensuremath{\sim}1/{N}^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the fork tensor product state (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pad\'e approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.

arXiv:2012.02803Physical review. B./Physical review. BApr 1, 202128 citations

Cluster Monte Carlo: Extending the range

Henk W. J. Blöte, Jouke R. Heringa, Erik Luijten

Computer Physics CommunicationsAug 1, 200222 citations

Statistical and systematic errors in Monte Carlo sampling

Alan M. Ferrenberg, David Paul Landau, Kurt Binder

Journal of Statistical PhysicsJun 1, 199197 citations

High-dimensional quasi-Monte Carlo via combinatorial discrepancy

Jiaheng Chen, Haotian Jiang, Nathan Kirk

Quasi-Monte Carlo (QMC) methods are known to achieve faster convergence rates than Monte Carlo (MC), but their effectiveness in high dimensions often relies on additional structure, such as low effective dimension or carefully chosen coordinate weights. Moreover, in many applications one has access only to random samples rather than deterministic QMC constructions. In this work, we extend the recent method of N. Bansal and H. Jiang and construct high-dimensional QMC point sets from random samples via combinatorial discrepancy. We establish error bounds for these constructions in weighted function spaces, including settings with low effective dimension in both the superposition and truncation senses. We also implement the resulting Subgaussian Transference algorithm and present numerical experiments to assess empirically the performance of these constructions.Quasi-Monte Carlo (QMC) methods are known to achieve faster convergence rates than Monte Carlo (MC), but their effectiveness in high dimensions often relies on additional structure, such as low effective dimension or carefully chosen coordinate weights. Moreover, in many applications one has access only to random samples rather than deterministic QMC constructions. In this work, we extend the recent method of N. Bansal and H. Jiang and construct high-dimensional QMC point sets from random samples via combinatorial discrepancy. We establish error bounds for these constructions in weighted function spaces, including settings with low effective dimension in both the superposition and truncation senses. We also implement the resulting Subgaussian Transference algorithm and present numerical experiments to assess empirically the performance of these constructions.

arXiv:2508.18426Journal of ComplexityMay 5, 20260 citations

On the Power of Restricted Monte Carlo Algorithms

Stefan Heinrich

MATRIX book seriesJan 1, 20203 citations

Diffusion in the Continuous-Imaginary-Time Quantum World-Line Monte Carlo Simulations with Extended Ensembles

Kenji Harada, Yuto Kuge

The dynamics of samples in the continuous-imaginary-time quantum world-line Monte Carlo simulations with extended ensembles are investigated. In the case of a conventional flat ensemble on the one-dimensional quantum S=1 bi-quadratic model, the asymmetric behavior of Monte Carlo samples appears in the diffusion process in the space of the number of vertices. We prove that a local diffusivity is asymptotically proportional to the number of vertices, and we demonstrate the asymmetric behavior in the flat ensemble case. On the basis of the asymptotic form, we propose the weight of an optimal ensemble as $1/\sqrt{n}$ , where $n$ denotes the number of vertices in a sample. It is shown that the asymmetric behavior completely vanishes in the case of the proposed ensemble on the one-dimensional quantum S=1 bi-quadratic model.The dynamics of samples in the continuous-imaginary-time quantum world-line Monte Carlo simulations with extended ensembles are investigated. In the case of a conventional flat ensemble on the one-dimensional quantum S=1 bi-quadratic model, the asymmetric behavior of Monte Carlo samples appears in the diffusion process in the space of the number of vertices. We prove that a local diffusivity is asymptotically proportional to the number of vertices, and we demonstrate the asymmetric behavior in the flat ensemble case. On the basis of the asymptotic form, we propose the weight of an optimal ensemble as $1/\sqrt{n}$, where $n$ denotes the number of vertices in a sample. It is shown that the asymmetric behavior completely vanishes in the case of the proposed ensemble on the one-dimensional quantum S=1 bi-quadratic model.

arXiv:0710.3973Journal of the Physical Society of JapanJan 10, 20081 citation

N

is $\ensuremath{\sim}1/{N}^{0.86}$ in the best cases and comparable to Monte Carlo scaling $\ensuremath{\sim}1/{N}^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the fork tensor product state (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pad\'e approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.

1/\sqrt{n}

, where

n

denotes the number of vertices in a sample. It is shown that the asymmetric behavior completely vanishes in the case of the proposed ensemble on the one-dimensional quantum S=1 bi-quadratic model.