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:1711.10570 · The Journal of Chemical Physics

Gradient-based stochastic estimation of the density matrix

Zhentao Wang, Gia-Wei Chern, Cristian D. Batista, Kipton M. Barros

Published Mar 5, 2018• 40 citations
arXivHTMLPDF

Abstract

Fast estimation of the single-particle density matrix is key to many applications in quantum chemistry and condensed matter physics. The best numerical methods leverage the fact that the density matrix elements f(H)ij decay rapidly with distance rij between orbitals. This decay is usually exponential. However, for the special case of metals at zero temperature, algebraic decay of the density matrix appears and poses a significant numerical challenge. We introduce a gradient-based probing method to estimate all local density matrix elements at a computational cost that scales linearly with system size. For zero-temperature metals, the stochastic error scales like S−(d+2)/2d, where d is the dimension and S is a prefactor to the computational cost. The convergence becomes exponential if the system is at finite temperature or is insulating.

DOI
10.1063/1.5017741

Related papers

Locality of the Density Matrix in Metals, Semiconductors, and Insulators

Sohrab Ismail‐Beigi, T. A. Arias

We present an analytical study of the spatial decay rate $\ensuremath{\gamma}$ of the one-particle density matrix $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r},{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}})\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\ensuremath{\gamma}|\stackrel{\ensuremath{\rightarrow}}{r}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}}|)$ for systems described by single-particle orbitals in periodic potentials in arbitrary dimensions. This decay reflects electronic locality in condensed matter systems and is also crucial for $O(N)$ density functional methods. We find that $\ensuremath{\gamma}$ behaves contrary to the conventional wisdom that generically $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}\ensuremath{\Delta}$ in insulators and $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}T$ in metals, where $\ensuremath{\Delta}$ is the direct band gap and $T$ is the temperature. Rather, in semiconductors $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}$, and in metals at low temperature $\ensuremath{\gamma}\ensuremath{\propto}T$.

Physical Review Letters•Mar 8, 1999•217 citations

Implicit purification for temperature-dependent density matrices

Anders M. N. Niklasson

An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.

arXiv:cond-mat/0311469•Physical review. B, Condensed matter•Dec 30, 2003•39 citations

Markovian Matrix Product Density Operators : Efficient computation of global entropy

Isaac H. Kim

We introduce the Markovian matrix product density operator, which is a special subclass of the matrix product density operator. We show that the von Neumann entropy of such ansatz can be computed efficiently on a classical computer. This is possible because one can efficiently certify that the global state forms an approximate quantum Markov chain by verifying a set of inequalities. Each of these inequalities can be verified in time that scales polynomially with the bond dimension and the local Hilbert space dimension. The total number of inequalities scale linearly with the system size. We use this fact to study the complexity of computing the minimum free energy of local Hamiltonians at finite temperature. To this end, we introduce the free energy problem as a generalization of the local Hamiltonian problem, and study its complexity for a class of Hamiltonians that describe quantum spin chains. The corresponding free energy problem at finite temperature is in NP if the Gibbs state of such Hamiltonian forms an approximate quantum Markov chain with an error that decays exponentially with the width of the conditioning subsystem.

arXiv:1709.07828•Sep 22, 2017•8 citations

Spatial decay of the single-particle density matrix in tight-binding metals: Analytic results in two dimensions

Sergei N. Taraskin, P. A. Fry, Xiaodong Zhang +2 more

Analytical results for the asymptotic spatial decay of the density matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ in the tight-binding model of the two-dimensional metal are presented. In various dimensions D, it is found analytically and numerically that the density matrix decays with distance according to the power law $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremath{\propto}|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}{|}^{\ensuremath{-}(D+1)/2}.$

arXiv:cond-mat/0207443•Physical review. B, Condensed matter•Dec 4, 2002•22 citations

Estimation of the Reduced Density Matrix and Entanglement Entropies Using Autoregressive Networks

P. Białas, Piotr Korcyl, Tomasz Stebel +1 more

Matrix elements of the density matrix of small quantum systems have recently become experimentally accessible within the cold atoms simulations. Numerical estimation of these quantities is known to be computationally challenging. Tracing out a part of the studied system and hence accessing the reduced density matrices offers similar difficulties, but it is more interesting, as the reduced density matrix is the basis of entanglement entropy measures. In this work, we present a method of estimating the individual elements of the reduced density matrices that uses autoregressive neural networks and the path integral quantization approach. We use a hierarchy of neural networks capable of estimating conditional probabilities of consecutive spins to evaluate elements of reduced density matrices directly. Using the Ising chain as an example, we calculate the continuum limit of the ground state’s von Neumann and Rényi bipartite entanglement entropies of an interval built of up to 5 spins. We demonstrate that our architecture is able to estimate all the needed matrix elements with just a single training for a fixed temperature discretization and lattice volume. Our method can be applied to other types of spin chains, possibly with defects, as well as to estimating entanglement entropies of thermal states at non-zero temperature. Hence, it may offer an attractive computational approach as a counterpart to the experimental effort. Abstract Published by the Jagiellonian University 2025 authors

arXiv:2506.04170•Acta Physica Polonica B•Dec 4, 2025•0 citations

lattice-symmetries: A package for working with quantum many-body bases

Tom Westerhout

Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory.It is precise, unbiased, and general enough to be applicable to a huge variety of problems in condensed matter physics.Mathematically, ED is a linear algebra problem involving a matrix called the Hamiltonian.For a system of spin-1/2 particles, the size of this matrix scales exponentially (as O(2 N )) with the number of particles N .Very fast scaling of memory requirements with system size is the main computational challenge of the method.There are a few techniques allowing one to lower the amount of storage used by the Hamiltonian.For example, one can store only the non-zero elements of the Hamiltonian.This is beneficial when the Hamiltonian is sparse, which is usually the case in condensed matter physics.One can even take it one step further and avoid storing the matrix altogether by instead computing matrix elements on the fly.

arXiv:2104.04011•The Journal of Open Source Software•Aug 26, 2021•17 citations
Fast estimation of the single-particle density matrix is key to many applications in quantum chemistry and condensed matter physics. The best numerical methods leverage the fact that the density matrix elements f(H)ij decay rapidly with distance rij between orbitals. This decay is usually exponential. However, for the special case of metals at zero temperature, algebraic decay of the density matrix appears and poses a significant numerical challenge. We introduce a gradient-based probing method to estimate all local density matrix elements at a computational cost that scales linearly with system size. For zero-temperature metals, the stochastic error scales like S−(d+2)/2d, where d is the dimension and S is a prefactor to the computational cost. The convergence becomes exponential if the system is at finite temperature or is insulating.
We present an analytical study of the spatial decay rate $\ensuremath{\gamma}$ of the one-particle density matrix $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r},{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}})\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\ensuremath{\gamma}|\stackrel{\ensuremath{\rightarrow}}{r}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}}|)$ for systems described by single-particle orbitals in periodic potentials in arbitrary dimensions. This decay reflects electronic locality in condensed matter systems and is also crucial for O(N)O(N)O(N) density functional methods. We find that $\ensuremath{\gamma}$ behaves contrary to the conventional wisdom that generically $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}\ensuremath{\Delta}$ in insulators and $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}T$ in metals, where $\ensuremath{\Delta}$ is the direct band gap and TTT is the temperature. Rather, in semiconductors $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}$, and in metals at low temperature $\ensuremath{\gamma}\ensuremath{\propto}T$.
An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.
We introduce the Markovian matrix product density operator, which is a special subclass of the matrix product density operator. We show that the von Neumann entropy of such ansatz can be computed efficiently on a classical computer. This is possible because one can efficiently certify that the global state forms an approximate quantum Markov chain by verifying a set of inequalities. Each of these inequalities can be verified in time that scales polynomially with the bond dimension and the local Hilbert space dimension. The total number of inequalities scale linearly with the system size. We use this fact to study the complexity of computing the minimum free energy of local Hamiltonians at finite temperature. To this end, we introduce the free energy problem as a generalization of the local Hamiltonian problem, and study its complexity for a class of Hamiltonians that describe quantum spin chains. The corresponding free energy problem at finite temperature is in NP if the Gibbs state of such Hamiltonian forms an approximate quantum Markov chain with an error that decays exponentially with the width of the conditioning subsystem.
Analytical results for the asymptotic spatial decay of the density matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ in the tight-binding model of the two-dimensional metal are presented. In various dimensions D, it is found analytically and numerically that the density matrix decays with distance according to the power law $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremath{\propto}|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}{|}^{\ensuremath{-}(D+1)/2}.$
Matrix elements of the density matrix of small quantum systems have recently become experimentally accessible within the cold atoms simulations. Numerical estimation of these quantities is known to be computationally challenging. Tracing out a part of the studied system and hence accessing the reduced density matrices offers similar difficulties, but it is more interesting, as the reduced density matrix is the basis of entanglement entropy measures. In this work, we present a method of estimating the individual elements of the reduced density matrices that uses autoregressive neural networks and the path integral quantization approach. We use a hierarchy of neural networks capable of estimating conditional probabilities of consecutive spins to evaluate elements of reduced density matrices directly. Using the Ising chain as an example, we calculate the continuum limit of the ground state’s von Neumann and Rényi bipartite entanglement entropies of an interval built of up to 5 spins. We demonstrate that our architecture is able to estimate all the needed matrix elements with just a single training for a fixed temperature discretization and lattice volume. Our method can be applied to other types of spin chains, possibly with defects, as well as to estimating entanglement entropies of thermal states at non-zero temperature. Hence, it may offer an attractive computational approach as a counterpart to the experimental effort. Abstract Published by the Jagiellonian University 2025 authors
Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory.It is precise, unbiased, and general enough to be applicable to a huge variety of problems in condensed matter physics.Mathematically, ED is a linear algebra problem involving a matrix called the Hamiltonian.For a system of spin-1/2 particles, the size of this matrix scales exponentially (as O(2 N )) with the number of particles N .Very fast scaling of memory requirements with system size is the main computational challenge of the method.There are a few techniques allowing one to lower the amount of storage used by the Hamiltonian.For example, one can store only the non-zero elements of the Hamiltonian.This is beneficial when the Hamiltonian is sparse, which is usually the case in condensed matter physics.One can even take it one step further and avoid storing the matrix altogether by instead computing matrix elements on the fly.