Performance Analysis of Parallel Eigensolvers of Two Libraries on BlueGene/P

Many applications in computational science and engineering require the computation of eigenvalues and vectors of dense symmetric or Hermitian matrices. For example, in DFT （density functional theory） calculations on modern supercomputers 10% to 30% of the eigenvalues and eigenvectors of huge dense matrices have to be calculated. Therefore, performance and parallel scaling of the used eigensolvers is of upmost interest. In this article different routines of the linear algebra packages ScaLAPACK and Elemental for parallel solution of the symmetric eigenvalue problem are compared concerning their performance on the BlueGene/P supercomputer. Parameters for performance optimization are adjusted for the different data distribution methods used in the two libraries. It is found that for all test cases the new library Elemental which uses a two-dimensional element by element distribution of the matrices to the processors shows better performance than the old ScaLAPACK library which uses a block-cyclic distribution.

Formulation of the Social Workers’ Problem in Quadratic Unconstrained Binary Optimization Form and Solve It on a Quantum Computer

The problem of social workers visiting their patients at home is a class of combinatorial optimization problems and belongs to the class of problems known as NP-Hard. These problems require heuristic techniques to provide an efficient solution in the best of cases. In this article, in addition to providing a detailed resolution of the social workers’ problem using the Quadratic Unconstrained Binary Optimization Problems (QUBO) formulation, an approach to mapping the inequality constraints in the QUBO form is given. Finally, we map it in the Hamiltonian of the Ising model to solve it with the Quantum Exact Solver and Variational Quantum Eigensolvers (VQE). The quantum feasibility of the algorithm will be tested on IBMQ computers.

KSSOLV-GPU:an Efficient GPU-Enabled MATLAB Toolbox for Solving the Kohn-Sham Equations within Density Functional Theory in Plane-Wave Basis Set

KSSOLV(Kohn-Sham Solver)is a MATLAB(Matrix Laboratory)toolbox for solving the Kohn-Sham density functional theory(KS-DFT)with the plane-wave basis set.In the KS-DFT calculations,the most expensive part is commonly the diagonalization of Kohn-Sham Hamiltonian in the self-consistent field(SCF)scheme.To enable a personal computer to perform medium-sized KS-DFT calculations that contain hundreds of atoms,we present a hybrid CPU-GPU implementation to accelerate the iterative diagonalization algorithms implemented in KSSOLV by using the MATLAB built-in Parallel Computing Toolbox.We compare the performance of KSSOLV-GPU on three types of GPU,including RTX3090,V100,and A100,with conventional CPU implementation of KSSOLV respectively and numerical results demonstrate that hybrid CPU-GPU implementation can achieve a speedup of about 10 times compared with sequential CPU calculations for bulk silicon systems containing up to 128 atoms.

Variational algorithms for linear algebra

Quantum algorithms have been developed for efficiently solving linear algebra tasks.However,they generally require deep circuits and hence universal fault-tolerant quantum computers.In this work,we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices.We show that the solutions of linear systems of equations and matrix–vector multiplications can be translated as the ground states of the constructed Hamiltonians.Based on the variational quantum algorithms,we introduce Hamiltonian morphing together with an adaptive ans?tz for efficiently finding the ground state,and show the solution verification.Our algorithms are especially suitable for linear algebra problems with sparse matrices,and have wide applications in machine learning and optimisation problems.The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation.We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations.We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.

Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned.Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic.Surprisingly,we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices.We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations.Our approach combines message passing and shared memory,achieving near-perfect scalability and high tolerance for network latency.We are thus able to find solutions for much larger matrices than previously possible,with the potential for extending this work to systems with greater levels of parallelism.The contributions of this work are in three areas:determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations;the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster,and the numerical results themselves.

Quantum computing for the Lipkin model with unitary coupled cluster and structure learning ansatz

We report a benchmark calculation for the Lipkin model in nuclear physics with a variational quantum eigensolver in quantum computing.Special attention is paid to the unitary coupled cluster(UCC)ansatz and structure learning(SL)ansatz for the trial wave function.Calculations with both the UCC and SL ansatz can reproduce the ground-state energy well;however,it is found that the calculation with the SL ansatz performs better than thatwith the UCC ansatz,and the SL ansatz has even fewer quantum gates than the UCC ansatz.

Eigenvalue Solver for Fluid and Kinetic Plasma Models in Arbitrary Magnetic Topology

ArbiTER(Arbitrary Topology Equation Reader)is a new code for solving linear eigenvalue problems arising from a broad range of physics and geometry models.The primary application area envisioned is boundary plasma physics in magnetic confinement devices;however ArbiTER should be applicable to other science and engineering fields as well.The code permits a variable numbers of dimensions,making possible application to both fluid and kinetic models.The use of specialized equation and topology parsers permits a high degree of flexibility in specifying the physics and geometry.