International Business Machines Corporation

Tel Aviv University

Vasileios Kalantzis

Lior Horesh

Shashanka Ubaru

Mark Steven Squillante

Tayfun Gokmen

Oguzhan Murat Onen

Wilfried Haensch

Tomasz J Nowicki

Kenneth Lee Clarkson

Haim Avron

Chai Wah Wu

Malte Johannes Rasch

Ismail Yunus Akhalwaya

Georgios Kollias

Naoki Abe

Ogunzhan Murat Onen

A quantum computer-implemented system, method, and computer program product for quantum topological domain analysis (QTDA). The QTDA method achieves an improved exponential speedup and depth complexity of O(n log(1/(δ∈))) where n is the number of data points, ∈ is the error tolerance, δ is the smallest nonzero eigenvalue of the restricted Laplacian, and achieves quantum advantage on general classical data. The QTDA system and method efficiently realizes a combinatorial Laplacian as a sum of Pauli operators; performs a quantum rejection sampling and projection approach to build the relevant simplicial complex repeatedly and restrict the superposition to the simplices of a desired order in the complex; and estimates Betti numbers using a stochastic trace/rank estimation method that does not require Quantum Phase Estimation. The quantum circuit and QTDA method exhibits computational time and depth complexities for Betti number estimation up to an error tolerance ∈.

Linear-depth quantum system for topological data analysis

Identifying influential nodes of a graph is provided. The graph is represented by an adjacency matrix and graph updates are represented using different subgraphs corresponding to the graph. Each update of the graph is calculated by approximating partial spectral factorization data for a subgraph of the graph using a previously stored adjacency matrix and information about additional nodes added to the subgraph during the update. Each subgraph is accessed only once. The most influential nodes of the graph are identified based on the partial spectral factorization data.

System to update the most influential nodes of dynamic graphs

A processor performing machine learning including spectral clustering can receive data from the sensor. Graph Laplacian of the data can be created and stored in a memory device. Spectral characteristic can be created by applying density of states and spectral gaps can be detected in an unsupervised manner in the spectral characteristic to determine r as number of clusters to cluster the data. A range space of a rational matrix of the graph Laplacian can be determined. K-means clustering can be performed on the range space of rational matrix of the graph Laplacian using r as the number of clusters, the K-means clustering returning r clusters of the received data.

Spectral clustering of high-dimensional data

Techniques are provided to implement hardware accelerated computing of eigenpairs of a matrix. For example, a system includes a processor, and a resistive processing unit coupled to the processor. The resistive processing unit includes an array of cells which include respective resistive devices, wherein at least a portion of the resistive devices are tunable to encode values of a given matrix which is storable in the array of cells. When the given matrix is stored in the array of cells, the processor is configured to determine an eigenvector of the stored matrix by executing a process which includes performing analog matrix-vector multiplication operations on the stored matrix to converge an initial vector to an estimate of the eigenvector of the stored matrix.

Hardware acceleration for computing eigenpairs of a matrix

A method is presented for computing an equilibrium distribution of Markov processes. The method includes storing weight values in an analog crossbar array of transition probability matrices, where the analog crossbar array of transition probability matrices represents a weight matrix with m rows and n columns, computing an eigenvector associated with a real eigenvalue of modulus one for each of the transition probability matrices, applying a gradient-based eigenvalue solver to converge to a dominant eigenpair, and determining a probability of changing from one state to another state in a stochastic entity based on outcomes of the gradient-based eigenvalue solver.

Markov processes using analog crossbar arrays

Techniques and a system to facilitate estimation of a quantum phase, and more specifically, to facilitate estimation of an expectation value of a quantum state, by utilizing a hybrid of quantum and classical methods are provided. In one example, a system is provided. The system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can include an encoding component and a learning component. The encoding component can encode an expectation value associated with a quantum state. The learning component can utilize stochastic inference to determine the expectation value based on an uncollapsed eigenvalue pair.

Short-depth active learning quantum amplitude estimation without eigenstate collapse

A system, method, and computer program product are disclosed. The method includes loading a first set of data as an initial matrix and determining a truncated singular value decomposition (SVD) of the initial matrix. The method also includes loading a second set of data as a new matrix, generating a first projection matrix, which approximates k leading left singular vectors of the updated matrix, and generating a second projection matrix, which approximates k leading right singular vectors of the updated matrix. Further, the method includes determining based on the initial matrix, the new matrix, the SVD of the existing matrix, and the first or second projection matrix, an approximate truncated SVD of the updated matrix.

Projection-based techniques for updating singular value decomposition in evolving data sets

Techniques are provided to implement hardware accelerated application of preconditioners to solve linear equations. For example, a system includes a processor, and a resistive processing unit coupled to the processor. The resistive processing unit includes an array of cells which include respective resistive devices, wherein at least a portion of the resistive devices are tunable to encode entries of a preconditioning matrix which is storable in the array of cells. When the preconditioning matrix is stored in the array of cells, the processor is configured to apply the preconditioning matrix to a plurality of residual vectors by executing a process which includes performing analog matrix-vector multiplication operations on the preconditioning matrix and respective ones of the plurality of residual vectors to generate a plurality of output vectors used in one or more subsequent operations.

Hardware acceleration with preconditioners

A computer implemented method for speeding up execution of a convex optimization operation one or more quadratic complexity operations to be performed by an analog crossbar hardware switch, and identifying one or more linear complexity operations to be performed by a CPU. At least one of the quadratic complexity operations is performed by the analog crossbar hardware, and at least one of the linear complexity operations is performed by the CPU. An iteration of an approximation of a solution to the convex optimization operation is updated by the CPU.

Accelerated Quasi-Newton methods on analog crossbar hardware

A device solves for eigenvalues of a matrix system. The device performs a domain decomposition of a matrix system into non-overlapping subdomains and a reordering of matrices of the matrix system. An interface variable projection subspace associated with interface variables of an adjacency graph of the matrix system is created. The interface variables are related to nodes of the adjacency graph which are connected with nodes located in neighboring partitions. An internal variable projection subspace is created that is associated with internal variables of the adjacency graph of the matrix system, wherein the internal variables are related to nodes of the adjacency graph which are connected only to nodes located in the same partition. A projection matrix is built based on the interface variable projection subspace and the internal variable projection subspace. The device determines eigenvalues that solve a Raleigh-Ritz eigenvalue problem utilizing the projection matrix.

Determination and use of spectral embeddings of large-scale systems by substructuring

Methods and systems for solving a linear system include setting resistances in an array of settable electrical resistances in accordance with values of an input matrix. A series of input vectors is applied to the array as voltages to generate a series of respective output vectors. Each input vector in the series of vectors is updated based on comparison of the respective output vectors to a target vector. A solution of a linear system is determined that includes the input matrix based on the updated input vectors.

Analog hardware matrix computation

A computer-implemented method is presented for performing matrix sketching by employing an analog crossbar architecture. The method includes low rank updating a first matrix for a first period of time, copying the first matrix into a dynamic correction computing device, switching to a second matrix to low rank update the second matrix for a second period of time, as the second matrix is low rank updated, feeding the first matrix with first stochastic pulses to reset the first matrix back to a first matrix symmetry point, copying the second matrix into the dynamic correction computing device, switching back to the first matrix to low rank update the first matrix for a third period of time, and as the first matrix is low rank updated, feeding the second matrix with second stochastic pulses to reset the second matrix back to a second matrix symmetry point.

Matrix sketching using analog crossbar architectures

A method of performing Principal Component Analysis is provided. The method includes receiving, by a computing device, evolving data for processing/visualization. The method further includes, by the computing device, a dimensionality for reducing of the evolving data using the PCA, wherein the PCA is performed on analog crossbar hardware. The method also includes using, by the computing device, the evolving data for visualization having the dimensionality thereof reduced by the principal component analysis for a further application.

Fast PCA of evolving data using analog crossbar array

A computer-implemented method for Eigenpair computation is provided. The method includes computing, her a hardware processor, an Eigenvector and respective Eigenvalues of the Eigenvector of a matrix by using a modified Stochastic Optimization process including performing a matrix vector product on a Resistive Processing Unit (RPU) crossbar array operatively coupled to the hardware processor and performing a scalar vector product on a digital device operatively coupled to the hardware processor and representing, for each of an Eigenpair, an initial guess for the Eigenvector and the respective Eigenvalues. The computing step includes storing the matrix in the RPU crossbar array.

Growing community of inventors

White Plains, NY, United States of America

Vasileios Kalantzis