Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0303597 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stochastic series expansion method for quantum Ising models with arbitrary interactions

Anders W. Sandvik

Published Nov 11, 2003• 166 citations
arXivHTMLPDF

Abstract

A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary short- or long-range interactions is presented. The algorithm is based on sampling the diagonal matrix elements of the power-series expansion of the density matrix (stochastic series expansion), and avoids the interaction summations necessary in conventional methods. In the case of long-range interactions, the scaling of the computation time with the system size N is therefore reduced from N2 to N ln(N). The method is tested on a one-dimensional ferromagnet in a transverse field, with interactions decaying as 1/r(2).

DOI
10.1103/physreve.68.056701

Related papers

Quantum-critical properties of the long-range transverse-field Ising model from quantum Monte Carlo simulations

Jan Alexander Koziol, Anja Langheld, Sebastian C. Kapfer +1 more

The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.

arXiv:2103.09469•Physical review. B./Physical review. B•Jun 23, 2021•47 citations

Quantum Monte Carlo study of long-range transverse-field Ising models on the triangular lattice

Stephan Humeniuk

Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated $M<1$ in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.

arXiv:1602.02979•Physical review. B./Physical review. B•Mar 14, 2016•35 citations

Permutation matrix representation quantum Monte Carlo

Lalit Gupta, Tameem Albash, Itay Hen

Abstract We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.

arXiv:1908.03740•Journal of Statistical Mechanics Theory and Experiment•Jul 1, 2020•15 citations

Clock factorized quantum Monte Carlo method for long-range interacting systems

Zhijie Fan, Chao Zhang, Youjin Deng

Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order \mathcal{O}(N) 𝒪 ( N ) , where N N is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from \mathcal{O}(N) 𝒪 ( N ) to \mathcal{O}(1) 𝒪 ( 1 ) . In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range z z - z z interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.

arXiv:2305.14082•SciPost Physics Core•Apr 8, 2025•3 citations

Quantum cluster algorithm for frustrated Ising models in a transverse field

Sounak Biswas, Geet Rakala, Kedar Damle

Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.

arXiv:1512.00931•Physical review. B./Physical review. B•Jun 1, 2016•14 citations

Deterministic Loop Stochastic Series Expansion Algorithm for Quantum Spin Models in Magnetic Fields

Liuyun Dao, Yan-Cheng Wang, Hui Shao

The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.

arXiv:2604.04635•Apr 6, 2026•0 citations
A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary short- or long-range interactions is presented. The algorithm is based on sampling the diagonal matrix elements of the power-series expansion of the density matrix (stochastic series expansion), and avoids the interaction summations necessary in conventional methods. In the case of long-range interactions, the scaling of the computation time with the system size N is therefore reduced from N2 to N ln(N). The method is tested on a one-dimensional ferromagnet in a transverse field, with interactions decaying as 1/r(2).
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
Abstract We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.
Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order \mathcal{O}(N) 𝒪 ( N ) , where N N is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from \mathcal{O}(N) 𝒪 ( N ) to \mathcal{O}(1) 𝒪 ( 1 ) . In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range z z - z z interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.
Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.
The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.