Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is demonstrated to efficiently solve eigenvalue problems for graph Laplacians that appear in spectral clustering. For static graph partitioning, 10-20 iterations of LOBPCG without preconditioning result in ~10x error reduction, enough to achieve 100% correctness for all Challenge datasets with known truth partitions, e.g., for graphs with 5K/.1M (50K/1M) Vertices/Edges in 2 (7) seconds, compared to over 5,000 (30,000) seconds needed by the baseline Python code. Our Python code 100% correctly determines 98 (160) clusters from the Challenge static graphs with 0.5M (2M) vertices in 270 (1,700) seconds using 10GB (50GB) of memory. Our single-precision MATLAB code calculates the same clusters at half time and memory. For streaming graph partitioning, LOBPCG is initiated with approximate eigenvectors of the graph Laplacian already computed for the previous graph, in many cases reducing 2-3 times the number of required LOBPCG iterations, compared to the static case. Our spectral clustering is generic, i.e. assuming nothing specific of the block model or streaming, used to generate the graphs for the Challenge, in contrast to the base code. Nevertheless, in 10-stage streaming comparison with the base code for the 5K graph, the quality of our clusters is similar or better starting at stage 4 (7) for emerging edging (snowballing) streaming, while the computations are over 100-1000 faster.Comment: 6 pages. To appear in Proceedings of the 2017 IEEE High Performance Extreme Computing Conference. Student Innovation Award Streaming Graph Challenge: Stochastic Block Partition, see http://graphchallenge.mit.edu/champion

Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc

We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) publicly released recently. BLOPEX is available as a stand-alone serial library, as an external package to PETSc (``Portable, Extensible Toolkit for Scientific Computation'', a general purpose suite of tools for the scalable solution of partial differential equations and related problems developed by Argonne National Laboratory), and is also built into {\it hypre} (``High Performance Preconditioners'', scalable linear solvers package developed by Lawrence Livermore National Laboratory). The present BLOPEX release includes only one solver--the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. {\it hypre} provides users with advanced high-quality parallel preconditioners for linear systems, in particular, with domain decomposition and multigrid preconditioners. With BLOPEX, the same preconditioners can now be efficiently used for symmetric eigenvalue problems. PETSc facilitates the integration of independently developed application modules with strict attention to component interoperability, and makes BLOPEX extremely easy to compile and use with preconditioners that are available via PETSc. We present the LOBPCG algorithm in BLOPEX for {\it hypre} and PETSc. We demonstrate numerically the scalability of BLOPEX by testing it on a number of distributed and shared memory parallel systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron workstation, and IBM BlueGene/L supercomputer, using PETSc domain decomposition and {\it hypre} multigrid preconditioning. We test BLOPEX on a model problem, the standard 7-point finite-difference approximation of the 3-D Laplacian, with the problem size in the range $10^5-10^8$.Comment: Submitted to SIAM Journal on Scientific Computin

Numerical Analyses of Ground States in Strongly Correlated Electron Systems

早大学位記番号:新7521早稲田大

Strategies for solving the Fermi-Hubbard model on near-term quantum computers

The Fermi-Hubbard model is of fundamental importance in condensed-matter physics, yet is extremely challenging to solve numerically. Finding the ground state of the Hubbard model using variational methods has been predicted to be one of the first applications of near-term quantum computers. Here we carry out a detailed analysis and optimisation of the complexity of variational quantum algorithms for finding the ground state of the Hubbard model, including costs associated with mapping to a real-world hardware platform. The depth complexities we find are substantially lower than previous work. We performed extensive numerical experiments for systems with up to 12 sites. The results suggest that the variational ans\"atze we used -- an efficient variant of the Hamiltonian Variational ansatz and a novel generalisation thereof -- will be able to find the ground state of the Hubbard model with high fidelity in relatively low quantum circuit depth. Our experiments include the effect of realistic measurements and depolarising noise. If our numerical results on small lattice sizes are representative of the somewhat larger lattices accessible to near-term quantum hardware, they suggest that optimising over quantum circuits with a gate depth less than a thousand could be sufficient to solve instances of the Hubbard model beyond the capacity of classical exact diagonalisation.Comment: 14+11 pages, 19 figures, 5 tables; v3: publication versio

Spin-resolved microscopy of strongly correlated fermionic many-body states

Mit ultrakalte Gasen in optischen Gittern lassen sich stark wechselwirkende Quantenvielteilchensysteme auf der Ebene einzelner Spins untersuchen. Diese Doktorarbeit fasst den Aufbau und die ersten Ergebnisse eines Quantengasmikroskops mit fermionischen Li-6 Atomen zusammen. Wir konnten erstmals antiferromagnetische Spinkorrelationen in Hubbard-Systemen beobachten und in eindimensionalen Systeme die Trennung von Ladungs- und Spinfreiheitsgraden mit Korrelationen nachweisen, die im thermischen Gleichgewicht gemessen wurden. Die Grundlage für diese Experimente ist das Quantengasmikroskop, welches während der Doktorarbeit geplant und aufgebaut wurde. Die Bilder, die man damit nehmen kann, sind Momentaufnahmen eines Quantenvielteilchensystems, auf denen man alle Atome einzeln auf ihren jeweiligen Gitterplätzen erkennen kann. Wir produzieren ein ultrakaltes Quantengas mit Standardverfahren wie Laserkühlung, optischen Fallen und Verdunstungskühlung und laden es in eine einzelne Ebene eines dreidimensionalen optischen Gitters. Vor dem Abbilden wird jedes Atom entsprechend seines Spin mit einem Stern-Gerlach-Magnetfeld um einen halben Gitterplatz nach links oder rechts verschoben. Zur Abbildung der Atome messen wir die Fluoreszenz eines Raman-Seitenbandkühlprozesses, welcher in einem zusätzlichen sehr tiefen optischen Gitter abläuft, und können so 97% der Atome erfolgreich detektieren. Ein Kapitel dieser Arbeit widmet sich den Details dieses Prozesses und kann die verbliebenen Verluste durch eine Nichtgleichgewichtsverteilung der lokalen Anregungen erklären. Die Messung der Dichteverteilung der stark wechelwirkenden Atome im inhomogenen Gitter erlaubt es die Zustandsgleichung des Fermi-Hubbard-Models zu bestimmen. Dabei beobachten wir die starke Unterdrückung der Kompressibilität in der Mott-Isolator-Phase. Mit unseren hochaufgelösten Bildern können wir auch die Dichtekorrelationen des Systems messen und so das Fluktuations-Dissipations-Theorem bestätigen, welches die Kompressibilität in Beziehung zu der Summe aller Dichtefluktuationen setzt. Für eine Entropie pro Teilchen von weniger als log(2) kB zeigt der Mott-Isolator antiferromagnetische Spinkorrelationen aufgrund der Austauschwechselwirkung. In eindimensionalen Spinketten konnten wir diese magnetische Ordnung bis zu einer Distanz von vier Gitterplätzen direkt messen. Die Stärke der beobachteten Korrelationen stimmt sehr gut überein mit Quanten-Monte-Carlo-Rechnungen bei einer Temperatur von einem Achtel der Bandbreite, welches einer Entropie von 0.4 kB pro Atom entspricht. Für Spinketten mit weniger als einem Atom pro Gitterplatz sehen wir eine charakteristische Verschiebung der Spinkorrelationen zu größeren Wellenlängen, die der einer Luttinger Flüssigkeit entspricht. Besonders interessante physikalische Phänomene treten auf, wenn man den Spinfreiheitsgrad mit der Bewegung der Atome koppelt. In eindimensionalen Systemen tritt hier die Spin-Ladungs-Trennung auf, die einem Loch eine freie Bewegung durch eine Spinkette ermöglicht. Allerdings scheint diese Delokalisierung zu einer Reduktionen der magnetischen Ordnung zu führen, da die Position der Teilchen nun fluktuiert. Normale Zweipunktkorrelatoren messen so kleinere Werte. Allerdings konnten wir zeigen, dass die Spins um einzelne Löcher herum primär antiparallel ausgerichtet sind und so nachweisen, dass die Spinordnung fast unabhängig vom Grad der Dotierung ist, wenn man die Lochposition mitberücksichtigt. Diese Messungen lassen sich als erster direkten Nachweis von Spin-Ladungs-Trennung durch Gleichgewichtskorrelationen interpretieren. Diese Arbeit umfasst die ersten Messungen von Spin-Loch-Korrelationen mit ultrakalten Atomen und sie stellt damit einen wichtigen Schritt auf dem Weg zu Quantensimulationen dotierter Antiferromagneten dar, die einen Beitrag zum Verständnis von Hochtemperatursupraleitern leisten könnten.Ultracold fermionic atoms in optical lattices allows to simulate the behavior of electrons in strongly correlated materials. In this thesis, we demonstrate the preparation and site- and spin-resolved imaging of Hubbard systems with fermionic Li-6 atoms. We realize and measure strong antiferromagnetic spin correlations and study their amplitude for various temperatures, interactions and dopings. In one-dimensional systems we observe spin-charge separation signatures by measuring equilibrium correlations for spin and density. The basis for these measurements is a quantum gas microscope for fermionic Li-6 atoms, which was built during this PhD thesis. It allows to take snapshots of the quantum many-body system with single-atom and single-site resolution. Using standard techniques of laser cooling, optical trapping, and evaporative cooling, ultracold Fermi gases are prepared and loaded into a single plane of a three-dimensional optical lattice of tunable geometry. The spin of each atom is converted to a spatial information via a local Stern-Gerlach splitting. The imaging is performed by collection of fluorescence light from Raman sideband cooling in an additional, deep optical lattice. A detailed analysis of this cooling process, which explains our imaging fidelity of 97% from the non-thermal distribution of excitations is presented. A study of the density distribution of the strongly interacting atoms in the lattice allows to derive the equatioän of state of the Fermi-Hubbard model, which shows a strongly reduced compressibility in the Mott insulating regime. From the high-resolution images we can, in addition, extract all density correlations. This allows us to experimentally confirm the fluctuation-dissipation theorem linking the compressibility to the sum of all density fluctuations. At entropies below log(2) kB per particle, antiferromagnetic correlations arise from exchange interactions in a Mott insulator. We directly observe magnetic correlations up to four sites in one-dimensional spin chains. The measured antiferromagnetic spin correlations agree well with quantum Monte-Carlo calculations at temperatures of 1/8 of the band width, which corresponds to an entropy per particle of only 0.4 kB. At fillings below one atom per site, we observe characteristic oscillations of the spin correlations vs density as predicted by Luttinger liquid theory. Interesting physics arises when one couples the spin degree of freedom with the motion of the quantum particles. In one dimension, the phenomenon of spin-charge separation allows the holes to propagate through a spin chain without energy cost. Their motion, however, hides the magnetic order from local observables. Thanks to our simultaneous imaging of spins and holes, we can directly study the spin alignment around individual holes. We reveal spin correlations which are almost fully independent of the degree of hole doping with string spin-density correlation functions. These measurement are the first experimental observation of spin-charge separation in equilibrium correlation measurements. This work demonstrates the experimental study of doped quantum magnetism with individual spin resolution and paves the way for quantum simulations of doped two-dimensional antiferromagnets relevant to high temperature superconductivity