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

Variations on PIGS: Non-standard approaches for imaginary-time path integrals

Dmitri Iouchtchenko

Published Aug 21, 2015• 0 citations
OpenAlex

Abstract

The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. \n \nThe Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector.

Related papers

A path integral ground state replica trick approach for the computation of entanglement entropy of dipolar linear rotors

Tapas Kumar Sahoo, Dmitri Iouchtchenko, C. M. Herdman +1 more

We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole-dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).

The Journal of Chemical Physics•May 14, 2020•15 citations

A path integral ground state Monte Carlo algorithm for entanglement of lattice bosons

Emanuel Casiano-Diaz, C. M. Herdman, Adrian Del Maestro

A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256 L = 256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.

arXiv:2207.11301•SciPost Physics•Mar 29, 2023•6 citations

Numerical evaluation of Rényi entropy with path integrals

Miha Srdinšek

Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.

•Sep 26, 2023•0 citations

Estimating Ground State Entanglement Entropy Using Path Integral Molecular Dynamics

Dmitri Iouchtchenko, Pierre–Nicholas Roy

We provide a perspective on the calculation of ground state entanglement properties of many-body bosonic systems with continuous degrees of freedom using path integral molecular dynamics sampling. To date, such ground state calculations have been performed using quantum Monte Carlo sampling and we show here that the Langevin Equation path integral ground state (LePIGS) dynamics sampling approach can be practical. A key challenge is the computation of reduced density matrices required for the estimation of the so-called Rényi entanglement entropy. We provide proof of concept results using a novel path integral molecular dynamics implementation of the permutation operator. This implementation allows us to efficiently estimate particle entanglement entropy in arbitrary continuum systems, among other quantities.

ACS symposium series•Jan 1, 2016•4 citations

Renyi Entanglement Entropy of Molecules: Interaction Effects and Signatures of Bonding

Norm M. Tubman, Jeremy McMinis

Recent proposals of measuring bipartite Renyi entropy experimentally involve techniques that hold exactly for non-interacting quantum particles. Here we consider the difference between such measurements and the actual Renyi entropy for ground state fermion ab initio molecular systems. To calculate various entanglement measures we extended several different techniques for use with variational Monte Carlo, Hartree-Fock and quantum chemistry methods. Our results show that in systems with strong electron correlations the Renyi entropy may not be accurately determined with the proposed measurements. In addition, we find significant physical insight can be gained by calculating entanglement properties in molecular systems. We see that the Renyi entropy and the entanglement Hamiltonian encode information about the character of the covalent bonds in a molecular system and that such information may lead to better descriptions of bonded systems that have been traditionally hard to describe with standard techniques. In particular our results for the C$_{2}$ dimer suggests all eight valence electrons play a significant role in covalent bonding.

arXiv:1204.4731•Apr 20, 2012•9 citations

Exact ground state Monte Carlo method for Bosons without importance sampling

M. Rossi, Marco Nava, L. Reatto +1 more

Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

arXiv:0907.4430•The Journal of Chemical Physics•Oct 19, 2009•62 citations
The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. \n \nThe Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector.
We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole-dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).
A ground state path integral quantum Monte Carlo algorithm is introduced that allows for the study of entanglement in lattice bosons at zero temperature. The Rényi entanglement entropy between spatial subregions is explored across the phase diagram of the one dimensional Bose-Hubbard model for systems consisting of up to L=256 L = 256 sites at unit-filling without any restrictions on site occupancy, far beyond the reach of exact diagonalization. The favorable scaling of the algorithm is demonstrated through a further measurement of the Rényi entanglement entropy at the two dimensional superfluid-insulator critical point for large system sizes, confirming the existence of the expected entanglement boundary law in the ground state. The Rényi estimator is extended to measure the symmetry resolved entanglement that is operationally accessible as a resource for experimentally relevant lattice gases with fixed total particle number.
Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.
We provide a perspective on the calculation of ground state entanglement properties of many-body bosonic systems with continuous degrees of freedom using path integral molecular dynamics sampling. To date, such ground state calculations have been performed using quantum Monte Carlo sampling and we show here that the Langevin Equation path integral ground state (LePIGS) dynamics sampling approach can be practical. A key challenge is the computation of reduced density matrices required for the estimation of the so-called Rényi entanglement entropy. We provide proof of concept results using a novel path integral molecular dynamics implementation of the permutation operator. This implementation allows us to efficiently estimate particle entanglement entropy in arbitrary continuum systems, among other quantities.
Recent proposals of measuring bipartite Renyi entropy experimentally involve techniques that hold exactly for non-interacting quantum particles. Here we consider the difference between such measurements and the actual Renyi entropy for ground state fermion ab initio molecular systems. To calculate various entanglement measures we extended several different techniques for use with variational Monte Carlo, Hartree-Fock and quantum chemistry methods. Our results show that in systems with strong electron correlations the Renyi entropy may not be accurately determined with the proposed measurements. In addition, we find significant physical insight can be gained by calculating entanglement properties in molecular systems. We see that the Renyi entropy and the entanglement Hamiltonian encode information about the character of the covalent bonds in a molecular system and that such information may lead to better descriptions of bonded systems that have been traditionally hard to describe with standard techniques. In particular our results for the C2_{2}2​ dimer suggests all eight valence electrons play a significant role in covalent bonding.
Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.