Obtaining a linear combination of the principal components of a matrix on quantum computers

Daskin, Ammar

doi:10.1007/s11128-016-1388-7

Cited by 14 publications

(30 citation statements)

References 41 publications

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“…In the following, we shall first explain two well-known quantum algorithms and then describe how they are used in Ref. [35] to obtain the linear combination of the eigenvectors.…”

Section: B Quantum Algorithms Used In the Modelmentioning

confidence: 99%

“…In this paper, we present a quantum implementation model for the artificial neural networks by employing the algorithm in Ref. [35]. In particular, we show how to construct QQ T x on quantum computers in linear time.…”

Section: Introductionmentioning

confidence: 99%

“…In the following section, we give the necessary description of Widrow-Hoff learning rule and QPCA described in Ref. [35]. In Sec.III, we shall show how to apply QPCA to the neural networks given by the Widrow-Hoff learning rule and discuss the possible implementation issues such as the circuit implementation of W , the preparation of the input x as a quantum circuit, and determining the number of iterations in the algorithm.…”

Section: Introductionmentioning

confidence: 99%

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A Quantum Implementation Model for Artificial Neural Networks

Daskin

2018

Quanta

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The learning process for multi layered neural networks with many nodes makes heavy demands on computational resources. In some neural network models, the learning formulas, such as the Widrow-Hoff formula, do not change the eigenvectors of the weight matrix while flatting the eigenvalues. In infinity, this iterative formulas result in terms formed by the principal components of the weight matrix: i.e., the eigenvectors corresponding to the non-zero eigenvalues.In quantum computing, the phase estimation algorithm is known to provide speed-ups over the conventional algorithms for the eigenvalue-related problems. Combining the quantum amplitude amplification with the phase estimation algorithm, a quantum implementation model for artificial neural networks using the Widrow-Hoff learning rule is presented. The complexity of the model is found to be linear in the size of the weight matrix. This provides a quadratic improvement over the classical algorithms.

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“…In the following, we shall first explain two well-known quantum algorithms and then describe how they are used in Ref. [35] to obtain the linear combination of the eigenvectors.…”

Section: B Quantum Algorithms Used In the Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Quantum Implementation Model for Artificial Neural Networks

Daskin

2018

Quanta

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“…Then |χ i and ri can be obtained by sampling the eigenvalue estimate register of σ and this procedure will be highly efficient when ρ is approximately low-rank [30]. For a more general Hermitian matrix H with eigenvalues λ i and corresponding eigenvectors |ϕ i , Daskin [31] subsequently proposed another PCA-based quantum algorithm that uses amplitude amplification [32] to obtain a linear combination of eigenvectors |ϕ i , together with λ i , a≤λi≤b α i |λ i |ϕ i , where the eigenvalues λ i lie in a range [a, b] and α i are the coefficients dependent on |ϕ i . Later, Cong and Duan [11] suggested a quantum LDA algorithm for DR, which is similar to Llyod's algorithm [30] and also yields a quantum state as σ, but the state ρ involved encodes some scatter matrices of a dataset.…”

Section: Introductionmentioning

confidence: 99%

Quantum data compression by principal component analysis

Yu,

Gao,

Lin

et al. 2018

Preprint

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Data compression can be achieved by reducing the dimensionality of high-dimensional but approximately low-rank datasets, which may in fact be described by the variation of a much smaller number of parameters. It often serves as a preprocessing step to surmount the curse of dimensionality and to gain efficiency, and thus it plays an important role in machine learning and data mining. In this paper, we present a quantum algorithm that compresses an exponentially large high-dimensional but approximately low-rank dataset in quantum parallel, by dimensionality reduction (DR) based on principal component analysis (PCA), the most popular classical DR algorithm. We show that the proposed algorithm achieves exponential speedup over the classical PCA algorithm when the original dataset are projected onto a polylogarithmically low-dimensional space. The compressed dataset can then be further processed to implement other tasks of interest, with significantly less quantum resources. As examples, we apply this algorithm to reduce data dimensionality for two important quantum machine learning algorithms, quantum support vector machine and quantum linear regression for prediction. This work demonstrates that quantum machine learning can be released from the curse of dimensionality to solve problems of practical importance.

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“…Quantum phase estimation is an efficient algorithm to solve eigen-value related problems on quantum computers. In earlier two works [12], [13], we have showed respectively how to use quantum phase estimation algorithm to do principal component analysis of a classical data and how this approach can be employed for neural networks using Widrow-Hoff Learning rule. In this paper, we employ the same algorithm with slight modifications for clustering.…”

Section: Introductionmentioning

confidence: 99%

Quantum Spectral Clustering through a Biased Phase Estimation Algorithm

Daskin¹

2017

Preprint

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In this brief paper, we go through the theoretical steps of the spectral clustering on quantum computers by employing the phase estimation and the amplitude amplification algorithms. We discuss circuit designs for each step and show how to obtain the clustering solution from the output state. In addition, we introduce a biased version of the phase estimation algorithm which significantly speeds up the amplitude amplification process. The complexity of the whole process is analyzed: it is shown that when the circuit representation of a data matrix of order N is produced through an ancilla based circuit in which the matrix is written as a sum of L number of Householder matrices; the computational complexity is bounded by O(2 m LN ) number of quantum gates. Here, m represents the number of qubits (e.g., 6) involved in the phase register of the phase estimation algorithm.

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Obtaining a linear combination of the principal components of a matrix on quantum computers

Cited by 14 publications

References 41 publications

A Quantum Implementation Model for Artificial Neural Networks

A Quantum Implementation Model for Artificial Neural Networks

Quantum data compression by principal component analysis

Quantum Spectral Clustering through a Biased Phase Estimation Algorithm

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