Dataset Information

Finding Hadamard Matrices by a Quantum Annealing Machine.

ABSTRACT: Finding a Hadamard matrix (H-matrix) among the set of all binary matrices of corresponding order is a hard problem, which potentially can be solved by quantum computing. We propose a method to formulate the Hamiltonian of finding H-matrix problem and address its implementation limitation on existing quantum annealing machine (QAM) that allows up to quadratic terms, whereas the problem naturally introduces higher order ones. For an M-order H-matrix, such a limitation increases the number of variables from M² to (M³?+?M²?-?M)/2, which makes the formulation of the Hamiltonian too exhaustive to do by hand. We use symbolic computing techniques to manage this problem. Three related cases are discussed: (1) finding N?

SUBMITTER: Suksmono AB

PROVIDER: S-EPMC6779766 | biostudies-literature | 2019 Oct

REPOSITORIES: biostudies-literature

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Finding Hadamard Matrices by a Quantum Annealing Machine.

Suksmono Andriyan Bayu AB Minato Yuichiro Y

Scientific reports 20191007 1

Finding a Hadamard matrix (H-matrix) among the set of all binary matrices of corresponding order is a hard problem, which potentially can be solved by quantum computing. We propose a method to formulate the Hamiltonian of finding H-matrix problem and address its implementation limitation on existing quantum annealing machine (QAM) that allows up to quadratic terms, whereas the problem naturally introduces higher order ones. For an M-order H-matrix, such a limitation increases the number of varia ...[more]

PMID: 31591411

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Project description:A hybrid quantum-classical method for learning Boltzmann machines (BM) for a generative and discriminative task is presented. BM are undirected graphs with a network of visible and hidden nodes where the former is used as the reading site. In contrast, the latter is used to manipulate visible states' probability. In Generative BM, the samples of visible data imitate the probability distribution of a given data set. In contrast, the visible sites of discriminative BM are treated as Input/Output (I/O) reading sites where the conditional probability of output state is optimized for a given set of input states. The cost function for learning BM is defined as a weighted sum of Kullback-Leibler (KL) divergence and Negative conditional Log-likelihood (NCLL), adjusted using a hyper-parameter. Here, the KL Divergence is the cost for generative learning, and NCLL is the cost for discriminative learning. A Stochastic Newton-Raphson optimization scheme is presented. The gradients and the Hessians are approximated using direct samples of BM obtained through quantum annealing. Quantum annealers are hardware representing the physics of the Ising model that operates on low but finite temperatures. This temperature affects the probability distribution of the BM; however, its value is unknown. Previous efforts have focused on estimating this unknown temperature through regression of theoretical Boltzmann energies of sampled states with the probability of states sampled by the actual hardware. These approaches assume that the control parameter change does not affect the system temperature; however, this is usually untrue. Instead of using energies, the probability distribution of samples is employed to estimate the optimal parameter set, ensuring that the optimal set can be obtained from a single set of samples. The KL divergence and NCLL are optimized for the system temperature, and the result is used to rescale the control parameter set. The performance of this approach, as tested against the theoretically expected distributions, shows promising results for Boltzmann training on quantum annealers.

Dataset Information

Finding Hadamard Matrices by a Quantum Annealing Machine.

Publications

Finding Hadamard Matrices by a Quantum Annealing Machine.

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