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arXiv:1802.05428

Quantum Circuit Design for Training Perceptron Models

Yu Zheng, Sicong Lu, Re-Bing Wu

Published Feb 15, 2018• 4 citations
arXivHTMLPDF

Abstract

Perceptron model is a fundamental linear classifier in machine learning and also the building block of artificial neural networks. Recently, Wiebe et al. (arXiv:1602.04799) proposed that the training of a perceptron can be quadratically speeded using Grover search with a quantum computer, which has potentially important big-data applications. In this paper, we design a quantum circuit for implementing this algorithm. The Grover oracle, the central part of the circuit, is realized by Quantum-Fourier-Transform based arithmetics that specifies whether an input weight vector can correctly classify all training data samples. We also analyze the required number of qubits and universal gates for the algorithm, as well as the success probability using uniform sampling, showing that it has higher possibility than spherical Gaussian distribution $N(0,1)$. The feasibility of the circuit is demonstrated by a testing example using the IBM-Q cloud quantum computer, where 16 qubits are used to classify four data samples.

DOI
10.48550/arxiv.1802.05428

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Perceptron model is a fundamental linear classifier in machine learning and also the building block of artificial neural networks. Recently, Wiebe et al. (arXiv:1602.04799) proposed that the training of a perceptron can be quadratically speeded using Grover search with a quantum computer, which has potentially important big-data applications. In this paper, we design a quantum circuit for implementing this algorithm. The Grover oracle, the central part of the circuit, is realized by Quantum-Fourier-Transform based arithmetics that specifies whether an input weight vector can correctly classify all training data samples. We also analyze the required number of qubits and universal gates for the algorithm, as well as the success probability using uniform sampling, showing that it has higher possibility than spherical Gaussian distribution N(0,1)N(0,1)N(0,1). The feasibility of the circuit is demonstrated by a testing example using the IBM-Q cloud quantum computer, where 16 qubits are used to classify four data samples.
This paper highlights a practical research of the possibility of forming quantum circuits for training neural networks. The demonstrated quantum circuits were based on the principles of Grover's Search Algorithm. The perceptron was chosen as the architecture for the example neural network. The multilayer perceptron is a popular neural network architecture due to its scalability and applicability for solving a wide range of problems. Large scale neural networks can require significant computing resources for training purposes, therefore Grover’s quantum search algorithm could be used as a way to speed up this process, thanks to its ability of simultaneously processing multiple input values of a given function.
Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size NNN and margin $\ensuremath{\gamma}$ (how well the given data are separated), the query complexity of the best-known quantum training algorithm scales as either $(\sqrt{N}/{\ensuremath{\gamma}}^{2})log(1/{\ensuremath{\gamma}}^{2})$ or $N/\sqrt{\ensuremath{\gamma}}$, which is achieved by a hybrid of classical and quantum search. In this paper, we improve the version space quantum training method for perceptrons such that the query complexity of our algorithm scales as $\sqrt{N/\ensuremath{\gamma}}$. This is achieved by constructing an oracle for the perceptrons using quantum counting of the number of data elements that are correctly classified. Once such an oracle is constructed, bounded-error quantum search can be used to search over the hyperplane instances. The optimality of our algorithm is proven by reducing the evaluation of a two-level and-or tree (for which the query complexity lower bound is known) to a multicriterion search. Our quantum training algorithm can be generalized to train more complex machine learning models such as neural networks, which are built on a large number of perceptrons.
This work contains the analysis of results received after running synthesized quantum circuits for training perceptron neural networks. The training is performed by creating a Grover’s algorithm with a custom oracle function. The concept of synthesizing quantum circuits was showcased in the process of generating training circuits for three perceptron topologies, which were designed to test the accuracy of the synthesis process. The test circuits serve to prove that the proposed synthesis approach could be scaled to utilize more complex quantum computing systems and to solve more practical tasks. IBM’s 100-qubit cloud quantum simulator was used as the debugging environment. Quantum circuits for described algorithms are generated by the "Naginata" quantum synthesizer, its source code is published and further documented on GitHub along with the code for the provided example algorithms. The article describes the processes behind the algorithm for synthesizing quantum circuits that perform the training process of single-layer perceptrons by finding their weights by filtering all possible input values through a predefined accuracy criterion. Since quantum computing is still in its early development phase, quantum circuits are created mainly by manual placement of logic elements. Implementing quantum algorithms, especially more use-case specific ones, directly on the quantum circuit level could lead to the circuit easily becoming too complex for human comprehension. Quantum Circuit Synthesizer "Naginata" was created to simplify the development and debugging process of quantum algorithms, by adding better clarity to their development process. In our case, better clarity for the development process is achieved by composing functions for commonly used operations performed in the implemented quantum algorithm. The programmer could now implement the quantum algorithm as a set of functions, instead of manually creating a circuit from single logic elements. After this, the synthesizer would handle the task of creating the data for placing logic elements on the circuit. This enables an opportunity of implementing quantum algorithms with higher-level commands. In the scope of this work, parametrically generated generic blocks for frequently used operations such as: the adder, multiplier and digital comparator were created and utilized to form the training circuits. The test results, proved that with the help of the proposed quantum synthesizer, these compositions could be used efficiently as building blocks for implementing quantum algorithms. And by visually comparing sizes of both code and circuit representations of the synthesized circuits, to the code examples used to synthesize these circuits, it is determined that the proposed approach for implementing quantum circuits greatly simplifies the processes of development and debugging a quantum algorithm.
Abstract This research is motivated by the incomplete use of quantum in existing learning algorithms, so that the proposed learning algorithm is not optimal. Research (Fahri & Neven, 2018) shows that the proposed method of architectural form still uses classical architecture but inputs, weights and targets already use a quantum approach. Based on the results of previous studies, it shows that quantum computing is better than classical computation. Many researchers use quantum computing in the proposed learning algorithm. The model proposed is a quantum circuit architecture with the quantum perceptron method consisting of a quantum bit gate that uses a quantum computational approach as the architecture of the quantum perceptron learning algorithm. Then the authors conduct training and testing of the proposed quantum circuit architecture to test the quantum circuit model that the author proposes. The result of this research is a quantum circuit model with the quantum perceptron method which can be used to solve the learning optimization problem by using a quantum circuit architecture with 5 measurement measurements to show error training and testing = 0, with 9 measurements showing an error training of 1.13%, error testing 2.06%.
Quantum machine learning algorithms could provide significant speed-ups over their classical counterparts; however, whether they could also achieve good generalization remains unclear. Recently, two quantum perceptron models which give a quadratic improvement over the classical perceptron algorithm using Grover's search have been proposed by Wiebe et al. arXiv:1602.04799 . While the first model reduces the complexity with respect to the size of the training set, the second one improves the bound on the number of mistakes made by the perceptron. In this paper, we introduce a hybrid quantum-classical perceptron algorithm with lower complexity and better generalization ability than the classical perceptron. We show a quadratic improvement over the classical perceptron in both the number of samples and the margin of the data. We derive a bound on the expected error of the hypothesis returned by our algorithm, which compares favorably to the one obtained with the classical online perceptron. We use numerical experiments to illustrate the trade-off between computational complexity and statistical accuracy in quantum perceptron learning and discuss some of the key practical issues surrounding the implementation of quantum perceptron models into near-term quantum devices, whose practical implementation represents a serious challenge due to inherent noise. However, the potential benefits make correcting this worthwhile.
This paper showcases a method of parametric synthesis of quantum circuits for training perceptron neural networks. Synapse weights are found using Grover's algorithm with a modified oracle function. The results of running these parametrically synthesized circuits for training perceptrons of three different topologies are described. The circuits were run on a 100-qubit IBM quantum simulator. The synthesis of quantum circuits is carried out using quantum synthesizer "Naginata", which was developed in the scope of this work, the source code of which is published and further documented on GitHub. The article describes the quantum circuit synthesis algorithm for training single-layer perceptrons. At the moment, quantum circuits are created mainly by manually placing logic elements on lines that symbolize quantum bits. The purpose of creating Quantum Circuit Synthesizer "Naginata" was due to the fact that even with a slight increase in the number of operations in a quantum algorithm, leads to the significant increase in size of the corresponding quantum circuit. This causes serious difficulties both in creating and debugging these quantum circuits. The purpose of our quantum synthesizer is enabling users an opportunity to implement quantum algorithms using higher-level commands. This is achieved by creating generic blocks for frequently used operations such as: the adder, multiplier, digital comparator (comparison operator), etc. Thus, the user could implement a quantum algorithm by using these generic blocks, and the quantum synthesizer would create a suitable circuit for this algorithm, in a format that is supported by the chosen quantum computation environment. This approach greatly simplifies the processes of development and debugging a quantum algorithm.