Density Matrix Purification Research Articles

Error-mitigated quantum computing of Heisenberg spin chain dynamics

We simulate the time-dependent dynamics of a three-site spin chain described by the Heisenberg XXX Hamiltonian. The quantum circuit representing the time-dependent wave function is constructed using the Suzuki-Trotter approximation, and is executed on the quantum computer ibm_kawasaki. At each time step, the density matrix of the three-qubit state is reconstructed by state tomography. By applying four different mitigation methods, Clifford data regression, Pauli twirling, density matrix purification, and density matrix orthogonalization, we demonstrate that accurate time-dependent populations and density matrices can be calculated on noisy superconducting-qubit type quantum computers.

Optimizing electronic structure simulations on a trapped-ion quantum computer using problem decomposition

Quantum computers have the potential to advance material design and drug discovery by performing costly electronic structure calculations. A critical aspect of this application requires optimizing the limited resources of the quantum hardware. Here, we experimentally demonstrate an end-to-end pipeline that focuses on minimizing quantum resources while maintaining accuracy. Using density matrix embedding theory as a problem decomposition technique, and an ion-trap quantum computer, we simulate a ring of 10 hydrogen atoms without freezing any electrons. The originally 20-qubit system is decomposed into 10 two-qubit problems, making it amenable to currently available hardware. Combining this decomposition with a qubit coupled cluster circuit ansatz, circuit optimization, and density matrix purification, we accurately reproduce the potential energy curve in agreement with the full configuration interaction energy in the minimal basis set. Our experimental results are an early demonstration of the potential for problem decomposition to accurately simulate large molecules on quantum hardware.

Sparse approximate matrix-matrix multiplication for density matrix purification with error control

We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening of submatrix products in the block-sparse matrix-matrix multiplies to reduce the computational cost. An error bound and a parameter sweep are combined to select a threshold value for the screening, such that the error can be controlled. We evaluate the performance of the method in comparison to purification without acceleration and without submatrix product screening.

Parallel Implementation of Large-Scale Linear Scaling Density Functional Theory Calculations With Numerical Atomic Orbitals in HONPAS.

Linear-scaling density functional theory (DFT) is an efficient method to describe the electronic structures of molecules, semiconductors, and insulators to avoid the high cubic-scaling cost in conventional DFT calculations. Here, we present a parallel implementation of linear-scaling density matrix trace correcting (TC) purification algorithm to solve the Kohn–Sham (KS) equations with the numerical atomic orbitals in the HONPAS package. Such a linear-scaling density matrix purification algorithm is based on the Kohn's nearsightedness principle, resulting in a sparse Hamiltonian matrix with localized basis sets in the DFT calculations. Therefore, sparse matrix multiplication is the most time-consuming step in the density matrix purification algorithm for linear-scaling DFT calculations. We propose to use the MPI_Allgather function for parallel programming to deal with the sparse matrix multiplication within the compressed sparse row (CSR) format, which can scale up to hundreds of processing cores on modern heterogeneous supercomputers. We demonstrate the computational accuracy and efficiency of this parallel density matrix purification algorithm by performing large-scale DFT calculations on boron nitrogen nanotubes containing tens of thousands of atoms.

Corrigendum: A perspective on the density matrix purification for linear scaling electronic structure calculations

Article First Published online: 7 December 2015 DOI: 10.1002/qua.25305 [Article in Int. J. Quantum Chem. 2016, 116, 563-568, DOI: 10.1002/qua.25048] When the article cited above was originally published online, the funding source was not acknowledged. The correct funding statement can be seen below. We acknowledge the support through the National Research Foundation of Korea from the Korean Government (NRF-2015R1A2A1A15055539). The authors apologize for the mistake. This error does not change the scientific conclusions of the article in any way.

A perspective on the density matrix purification for linear scaling electronic structure calculations

Density matrix purification is a linear scaling algorithm to solve the self‐consistent field (SCF) electronic structure problem. A brief review is made of the two usages of the purification method in SCF calculations, namely in direct energy minimization and for iterative calculation of density matrix from Hamiltonian. Density matrix purification can be viewed as a minimization algorithm of the particularly defined objective function such as , and hence, this objective function plays a key role in the developments of purification polynomials. Indeed, in this perspective, we show that a carefully chosen objective function can lead to significantly improved algorithms and numerical results. © 2015 Wiley Periodicals, Inc.

Parallel scalability of Hartree-Fock calculations.

Quantum chemistry is increasingly performed using large cluster computers consisting of multiple interconnected nodes. For a fixed molecular problem, the efficiency of a calculation usually decreases as more nodes are used, due to the cost of communication between the nodes. This paper empirically investigates the parallel scalability of Hartree-Fock calculations. The construction of the Fock matrix and the density matrix calculation are analyzed separately. For the former, we use a parallelization of Fock matrix construction based on a static partitioning of work followed by a work stealing phase. For the latter, we use density matrix purification from the linear scaling methods literature, but without using sparsity. When using large numbers of nodes for moderately sized problems, density matrix computations are network-bandwidth bound, making purification methods potentially faster than eigendecomposition methods.

HONPAS: A linear scaling open‐source solution for large system simulations

HONPAS is an ab initio electronic structure program for linear scaling or O(N) first‐principles calculations of large and complex systems using standard norm‐conserving pseudopotentials, numerical atomic orbitals (NAOs) basis sets, and periodic boundary conditions. HONPAS is developed in the framework of the SIESTA methodology and focuses on the development and implementation of efficient O(N) algorithms for ab initio electronic structure calculations. The Heyd–Scuseria–Ernzerhof (HSE) screened hybrid density functional has been implemented using a NAO2GTO scheme to evaluate the electron repulsion integrals (ERIs) with NAOs. ERI screening techniques allow the HSE functional calculations to be very efficient and scale linearly. The density matrix purification algorithms have been implemented, and the PSUTC2 and SUTC2 methods have been developed to deal with spin unrestricted systems with or without predetermined spin multiplicity, respectively. After the self‐consistent field (SCF) process, additional O(N) post‐SCF calculations for frontier molecular orbitals and maximally localized Wannier functions are also developed and implemented. Finally, an O(N) method based on the density matrix perturbation theory has been proposed and implemented to treat electric field in solids. This article provides an overall introduction to capabilities of HONPAS and implementation details of different O(N) algorithms. © 2014 Wiley Periodicals, Inc.

On the inherited “purity” of certain extrapolated density matrices

The method of Löwdin transform – inverse Löwdin transform is extended to approximate density matrix extrapolation in cases where the reference density matrix in not “pure” that is, if it fails the test of idempotency. It is shown that a matrix expression provides a scalar measure of the overall level of impurity and for large, sparse matrices of approximate macromolecular fragment-based quantum chemistry methods the one-step Diophantine density matrix purification method is recommended if a charge-preserving improvement of the extrapolated density matrix is needed.

Energy conserving, linear scaling Born-Oppenheimer molecular dynamics

Born-Oppenheimer molecular dynamics simulations with long-term conservation of the total energy and a computational cost that scales linearly with system size have been obtained simultaneously. Linear scaling with a low pre-factor is achieved using density matrix purification with sparse matrix algebra and a numerical threshold on matrix elements. The extended Lagrangian Born-Oppenheimer molecular dynamics formalism [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] yields microcanonical trajectories with the approximate forces obtained from the linear scaling method that exhibit no systematic drift over hundreds of picoseconds and which are indistinguishable from trajectories computed using exact forces.

Comment on “On the optimal symmetric purification scheme of the one-particle density matrix” [Chem. Phys. Lett. 511 (2011) 159–160

Recently, Kim and Jung reported on the efficiency of certain density matrix purification polynomials for calculation of the one-particle density matrix. In this comment, it is shown that a different conclusion is reached if a more efficient method is used for polynomial evaluation. The impact of the choice of purification polynomial on sparsity and memory usage is also discussed. We comment on how the use of nonmonotonic polynomials can make the convergence even faster.

On the optimal symmetric purification scheme of the one-particle density matrix

Density matrix purification is an efficient way of avoiding the expensive cubic scaling diagonalization in self-consistent field calculations. Although there are a number of different algorithms suggested to reduce the number of matrix multiplications for purification, there is no rigorous mathematical proof which scheme is optimal. In this Letter, we show analytically that the repeated application of the fifth-order Holas polynomial throughout all iterations is an optimal scheme that reduces the error symmetrically for both occupied and virtual occupations, and either the use of lower/higher-order polynomials throughout or mixed use of polynomials of different degree at different iteration results in higher cost.

Nonmonotonic Recursive Polynomial Expansions for Linear Scaling Calculation of the Density Matrix.

As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this letter, there is room for improvements. The key is to allow for nonmonotonicity in the recursive polynomial expansion. On the basis of this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.

Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification

Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.

Density matrix purification with rigorous error control

Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification.

Charge-conserving electron density averaging for a set of nuclear configurations

The Lowdin-Inverse Lowdin Transformation (LILT) technique combined with the Diophantine Density Matrix Purification (DDMP) method is suggested for a charge-preserving density averaging approach for a range of nuclear configurations.

Linear scaling calculation of band edge states and doped semiconductors

Linear scaling methods provide total energy, but no energy levels and canonical wave functions. From the density matrix computed through the density matrix purification methods, we propose an order-N [O(N)] method for calculating both the energies and wave functions of band edge states, which are important for optical properties and chemical reactions. In addition, we also develop an O(N) algorithm to deal with doped semiconductors based on the O(N) method for band edge states calculation. We illustrate the O(N) behavior of the new method by applying it to boron nitride (BN) nanotubes and BN nanotubes with an adsorbed hydrogen atom. The band gap of various BN nanotubes are investigated systematically and the acceptor levels of BN nanotubes with an isolated adsorbed H atom are computed. Our methods are simple, robust, and especially suited for the application in self-consistent field electronic structure theory.

Magnetic interactions of Mn clusters supported on Cu

A first-principles linear scaling real-space method for investigating non-collinear magnetic behaviour of nanostructured materials has been developed. With this method, the magnetic structures of small supported transition metal clusters have been examined. The geometric constraints imposed on the clusters by the underlying surface is found to cause non-collinear behaviour for V, Cr, and Mn clusters on Cu(111). Fe clusters supported on Cu and Ni have been studied and both spin and orbital moments are found to be enhanced for the Fe atoms, which is attributed to the recuced symmetry present at the surface. Atoms in Co clusters have been found to order antiferromagnetically, and some times in a non-collinear fasion, when deposited on a W surface. Small clusters of fcc Fe embedded in Cu have been examined and a new type of ordering, not present in larger fcc Fe systems was found. Several theoretical studies of Fe and Co based nanostructures consisting of multilayers or embedded clusters have been conducted, with the aim of predicting high moment materials for use in data storage applications. In agreement with previous experiments an enhancement of the magnetic moment is found compared to the magnetic moment of bcc Fe. The enhancement has been shown to be caused by increased spin moments for Fe atoms in close proximity with Co atoms, and this enhancement depends on the number of Co neighbours. As a result of these studies, a possible method of increasing the magnetic moment of cluster based materials has been proposed. Fermi surface analysis have been performed both on bulk materials, in order to investigate mechanisms for stabilizing non-collinear magnetic states, and in layered structures where the effect of the Fermi surface on the interlayer exchange coupling has been investigated. In addition to the development of a real-space electronic structure method for non-collinear magnetism, a density matrix purification method has been implemented in the framework of linear muffin-tin orbitals.

Systematic sparse matrix error control for linear scaling electronic structure calculations

Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.

A one-step, diophantine solution to the density matrix purification problem

Approximate first-order density matrices can be ‘purified', and converted into idempotent density matrices satisfying the McWeeny condition using a new, single-step process based on a Diophantine matrix approach. The new, direct, single-step method replaces the conventional many-step McWeeny iterative method for density matrix purification. Besides the obvious advantages of a single-step, direct method over the traditional iterative method of many steps that also requires convergence criteria to be tested at each step and eventually approximately satisfied, the new approach requires no convergence tests, it is exact within the numerical accuracy of the eigenvalue routine employed, and it is also conceptually simple and has some additional advantages when compared with the traditional iterative density matrix purification method based on the McWeeny condition. One application of the technique is suggested for the generation of idempotent macromolecular density matrices of the linear scaling, adjustable density matrix assembler (ADMA) method, within the framework of the additive fuzzy density fragmentation (AFDF) approach.