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Chapman & Hall/CRC eBooks

Quantum Perceptron

Andreas Wichert

Published Nov 28, 2023• 0 citations
Publisher

Abstract

The classical perceptron describes an algorithm for supervised learning that considers only linearly separable problems in which groups can be separated by a line or hyperplane. A quantum perceptron can be used as building block of larger systems, it can process an arbitrary number of input vectors in parallel. This chapter presents a simple example of the quantum perceptron for two-dimensional input. The classical perceptron describes an algorithm for supervised learning that considers only linearly separable problems in which groups can be separated by a line or hyperplane. A quantum Perceptron can be used as building block of larger systems, it can process an arbitrary number of input vectors in parallel.

DOI
10.1201/9781003374404-21

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The classical perceptron describes an algorithm for supervised learning that considers only linearly separable problems in which groups can be separated by a line or hyperplane. A quantum perceptron can be used as building block of larger systems, it can process an arbitrary number of input vectors in parallel. This chapter presents a simple example of the quantum perceptron for two-dimensional input. The classical perceptron describes an algorithm for supervised learning that considers only linearly separable problems in which groups can be separated by a line or hyperplane. A quantum Perceptron can be used as building block of larger systems, it can process an arbitrary number of input vectors in parallel.
We demonstrate how quantum computation can provide non-trivial improvements in the computational and statistical complexity of the perceptron model. We develop two quantum algorithms for perceptron learning. The first algorithm exploits quantum information processing to determine a separating hyperplane using a number of steps sublinear in the number of data points NNN, namely O(N)O(\sqrt{N})O(N​). The second algorithm illustrates how the classical mistake bound of O(1γ2)O(\frac{1}{γ^2})O(γ21​) can be further improved to O(1γ)O(\frac{1}{\sqrtγ})O(γ​1​) through quantum means, where γγγ denotes the margin. Such improvements are achieved through the application of quantum amplitude amplification to the version space interpretation of the perceptron model.
Abstract With the growing interest in quantum machine learning, the perceptron, a fundamental building block in traditional machine learning, has emerged as a valuable model for exploring the potential of quantum algorithms. In this work, we make two principal contributions. First, we revisit the quantum version space perceptron algorithm proposed by Kapoor et al. (2016), by identifying and correcting a flawed complexity assumption. We show that the query complexity of the algorithm is dimension-dependent, which has significant implications for its behaviour in high-dimensional regimes under worst-case scenarios. Second, we propose and analyse two quantum-enhanced cutting-plane algorithms for perceptron learning. Specifically, we leverage established quantum subroutines such as Grover’s search and quantum walk search , and provide detailed algorithmic constructions together with query and arithmetic complexity analyses. Our results establish improved complexity bounds under an idealised implementation framework and noise-free quantum computational models, offering insights into the trade-offs between margin dependence, dimensional dependence, and quantum resources. These findings provide a refined understanding of quantum perceptron models and their theoretical computational complexity properties.
We propose a method for learning a quantum probabilistic model of a perceptron. By considering a cross entropy between two density matrices we can learn a model that takes noisy output labels into account while learning. Although some work has been done that aims to utilize the curious properties of quantum systems to build a quantum perceptron, these proposals rely on the ad hoc introduction of a classical cost function for the optimization procedure. We demonstrate the usage of a quantum probabilistic model by considering a quantum equivalent of the classical log-likelihood, which allows for both a quantum model and training procedure. We show that this allows us to better capture noisiness in data compared to a classical perceptron. By considering entangled qubits we can learn nonlinear separation boundaries, such as xor.
The use of advanced quantum neuron models for pattern recognition applications requires fault tolerance. Therefore, it is not yet possible to test such models on a large scale in currently available quantum processors. As an alternative, we propose a quantum perceptron (QP) model that uses a reduced number of multi-qubit gates and is therefore less susceptible to quantum errors in current actual quantum computers with limited tolerance. The proposed quantum algorithm is superior to its classical counterpart, although since it does not take full advantage of quantum entanglement, it provides a lower encoding power than other quantum algorithms using multiple qubit entanglement. However, the use of separable-sate encoding allows for testing the algorithm and different training schemes at a large scale in currently available non-fault tolerant quantum computers. We demonstrate the performance of the proposed model by implementing a few qubits version of the QP in a simulated quantum computer. The proposed QP uses an N-ary encoding of the binary input data characterizing the patterns. We develop a hybrid (quantum-classical) training procedure for simulating the learning process of the QP and test their efficiency.
This chapter covers a number of disparate applications of quantum computing in the area of machine learning. We only consider situations where the dataset is classical (rather than quantum). We cover quantum algorithms for big-data problems relying upon high-dimensional linear algebra, such as Gaussian process regression and support vector machines. We discuss the prospect of achieving a quantum speedup with these algorithms, which face certain input/output caveats and must compete against quantum-inspired classical algorithms. We also cover heuristic quantum algorithms for energy-based models, which are generative machine learning models that learn to produce outputs similar to those in a training dataset. Next, we cover a quantum algorithm for the tensor principal component analysis problem, where a quartic speedup may be available, as well as quantum algorithms for topological data analysis, which aim to compute topologically invariant properties of a dataset. We conclude by covering quantum neural networks and quantum kernel methods, where the machine learning model itself is quantum in nature.