Dimension reduction and redundancy removal through successive Schmidt decompositions

Quantum computers are believed to have the ability to process huge data sizes which can be seen in machine learning applications. In these applications, the data in general is classical. Therefore, to process them on a quantum computer, there is a need for efficient methods which can be used to map classical data on quantum states in a concise manner. On the other hand, to verify the results of quantum computers and study quantum algorithms, we need to be able to approximate quantum operations into forms that are easier to simulate on classical computers with some errors. Motivated by these needs, in this paper we study the approximation of matrices and vectors by using their tensor products obtained through successive Schmidt decompositions. We show that data with distributions such as uniform, Poisson, exponential, or similar to these distributions can be approximated by using only a few terms which can be easily mapped onto quantum circuits. The examples include random data with different distributions, the Gram matrices of iris flower, handwritten digits, 20newsgroup, and labeled faces in the wild. And similarly, some quantum operations such as quantum Fourier transform and variational quantum circuits with a small depth also may be approximated with a few terms that are easier to simulate on classical computers. Furthermore, we show how the method can be used to simplify quantum Hamiltonians: In particular, we show the application to randomly generated transverse field Ising model Hamiltonians. The reduced Hamiltonians can be mapped into quantum circuits easily and therefore can be simulated more efficiently.

A Relative Church-Turing-Deutsch Thesis from Special Relativity and Undecidability

Beginning with Turing's seminal work in 1950, artificial intelligence proposes that consciousness can be simulated by a Turing machine. This implies a potential theory of everything where the universe is a simulation on a computer, which begs the question of whether we can prove we exist in a simulation. In this work, we construct a relative model of computation where a computable \textit{local} machine is simulated by a \textit{global}, classical Turing machine. We show that the problem of the local machine computing \textbf{simulation properties} of its global simulator is undecidable in the same sense as the Halting problem. Then, we show that computing the time, space, or error accumulated by the global simulator are simulation properties and therefore are undecidable. These simulation properties give rise to special relativistic effects in the relative model which we use to construct a relative Church-Turing-Deutsch thesis where a global, classical Turing machine computes quantum mechanics for a local machine with the same constant-time local computational complexity as experienced in our universe.

Training a quantum annealing based restricted Boltzmann machine on cybersecurity data

We present a real-world application that uses a quantum computer. Specifically, we trained a Restricted Boltzmann Machine (RBM) using quantum annealing (QA) to develop an intrusion detection system. RBMs were trained on the ISCX data, which is a benchmark dataset for cybersecurity. For comparison, RBMs were also trained using contrastive divergence (CD) which is a classical method. D-Wave's 2000Q quantum annealer has been used to implement QA. Our analysis of the ISCX data shows that the dataset is imbalanced and we present two different schemes to balance the training dataset before feeding it to a classifier. The first scheme is based on the oversampling of attack instances. The imbalanced training dataset was divided into five sub-datasets that were trained separately. A majority voting was performed to get the final result. Our results show the majority vote increased the classification accuracy up from 90.24% to 95.68% in the case of CD. For the case of QA, the classification accuracy increased from 74.14% to 80.04%. In the second scheme, an RBM was used to generate synthetic data to balance the training dataset. The RBMs trained on synthetic data generated from a CD-trained RBM performed comparably to the RBMs trained on synthetic data generated from a QA-trained RBM. Balanced training data was used to evaluate several classifiers. Among the classifiers investigated, K-Nearest Neighbor (KNN) and Neural Network (NN) performed better than other classifiers. They both showed an accuracy of 93%. Our results show a proof of concept that a QA-based RBM can be trained on a binary dataset, with 64-bit records. The illustrative example suggests the possibility to migrate many practical classification problems to QA-based techniques.

Training and Classification using a Restricted Boltzmann Machine on the D-Wave 2000Q

Restricted Boltzmann Machine (RBM) is an energy based, undirected graphical model. It is commonly used for unsupervised and supervised machine learning. Typically, RBM is trained using contrastive divergence (CD). However, training with CD is slow and does not estimate exact gradient of log-likelihood cost function. In this work, the model expectation of gradient learning for RBM has been calculated using a quantum annealer (D-Wave 2000Q), which is much faster than Markov chain Monte Carlo (MCMC) used in CD. Training and classification results are compared with CD. The classification accuracy results indicate similar performance of both methods. Image reconstruction as well as log-likelihood calculations are used to compare the performance of quantum and classical algorithms for RBM training. It is shown that the samples obtained from quantum annealer can be used to train a RBM on a 64-bit `bars and stripes' data set with classification performance similar to a RBM trained with CD. Though training based on CD showed improved learning performance, training using a quantum annealer eliminates computationally expensive MCMC steps of CD.

A Generalized Circuit for the Hamiltonian Dynamics Through the Truncated Series

In this paper, we present a method for the Hamiltonian simulation in the context of eigenvalue estimation problems which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, $H(t)$, through the truncated Taylor series method described by Berry et al. \cite{berry2015simulating}. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable $t$ in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires $O(Ln)$ number of quantum gates for the simulation of a molecular Hamiltonian. Here, $n$ is the number of states of a spin orbital, and $L$ is the number of terms in the molecular Hamiltonian and generally bounded by $O(n^4)$. We also discuss how to use the circuit in adaptive processes and eigenvalue related problems along with a slight modified version of the iterative phase estimation algorithm. In addition, a simple divide and conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by $O(poly(n))$ for a matrix with $poly(n)-$sparsity.

Cuckoo Search Inspired Hybridization of the Nelder-Mead Simplex Algorithm Applied to Optimization of Photovoltaic Cells

A new hybridization of the Cuckoo Search (CS) is developed and applied to optimize multi-cell solar systems; namely multi-junction and split spectrum cells. The new approach consists of combining the CS with the Nelder-Mead method. More precisely, instead of using single solutions as nests for the CS, we use the concept of a simplex which is used in the Nelder-Mead algorithm. This makes it possible to use the flip operation introduces in the Nelder-Mead algorithm instead of the Levy flight which is a standard part of the CS. In this way, the hybridized algorithm becomes more robust and less sensitive to parameter tuning which exists in CS. The goal of our work was to optimize the performance of multi-cell solar systems. Although the underlying problem consists of the minimization of a function of a relatively small number of parameters, the difficulty comes from the fact that the evaluation of the function is complex and only a small number of evaluations is possible. In our test, we show that the new method has a better performance when compared to similar but more compex hybridizations of Nelder-Mead algorithm using genetic algorithms or particle swarm optimization on standard benchmark functions. Finally, we show that the new method outperforms some standard meta-heuristics for the problem of interest.

Group Leaders Optimization Algorithm

We present a new global optimization algorithm in which the influence of the leaders in social groups is used as an inspiration for the evolutionary technique which is designed into a group architecture. To demonstrate the efficiency of the method, a standard suite of single and multidimensional optimization functions along with the energies and the geometric structures of Lennard-Jones clusters are given as well as the application of the algorithm on quantum circuit design problems. We show that as an improvement over previous methods, the algorithm scales as N^2.5 for the Lennard-Jones clusters of N-particles. In addition, an efficient circuit design is shown for two qubit Grover search algorithm which is a quantum algorithm providing quadratic speed-up over the classical counterpart.