Quantum Perceptron Revisited: Computational-Statistical Tradeoffs

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.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.

arXiv:2106.02496Jun 4, 20211 citation

On quantum perceptron learning via quantum search

Xiaoyu Sun, Mathieu Roget, Giuseppe Di Molfetta +1 more

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.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.

arXiv:2503.17308Quantum Machine IntelligenceJun 19, 20260 citations

Quadratic quantum speedup for perceptron training

Pengcheng Liao, Barry C. Sanders, Tim Byrnes

Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size $N$ 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.Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size $N$ 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.

arXiv:2109.04695Physical Review ADec 9, 20246 citations

Quantum Perceptron Models

Nathan Wiebe, Ashish Kapoor, Krysta M. Svore

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 $N$ , namely $O(\sqrt{N})$ . The second algorithm illustrates how the classical mistake bound of $O(\frac{1}{γ^2})$ can be further improved to $O(\frac{1}{\sqrtγ})$ through quantum means, where $γ$ denotes the margin. Such improvements are achieved through the application of quantum amplitude amplification to the version space interpretation of the perceptron model.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 $N$, namely $O(\sqrt{N})$. The second algorithm illustrates how the classical mistake bound of $O(\frac{1}{γ^2})$ can be further improved to $O(\frac{1}{\sqrtγ})$ through quantum means, where $γ$ denotes the margin. Such improvements are achieved through the application of quantum amplitude amplification to the version space interpretation of the perceptron model.

arXiv:1602.04799neural information processing systemsFeb 15, 2016103 citations

A Grover-search based quantum learning scheme for classification

Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu +1 more

Abstract The hybrid quantum–classical learning scheme provides a prominent way to achieve quantum advantages on near-term quantum devices. A concrete example toward this goal is the quantum neural network (QNN), which has been developed to accomplish various supervised learning tasks such as classification and regression. However, there are two central issues that remain obscure when QNN is exploited to accomplish classification tasks. First, a quantum classifier that can well balance the computational cost such as the number of measurements and the learning performance is unexplored. Second, it is unclear whether quantum classifiers can be applied to solve certain problems that outperform their classical counterparts. Here we devise a Grover-search based quantum learning scheme (GBLS) to address the above two issues. Notably, most existing QNN-based quantum classifiers can be seamlessly embedded into the proposed scheme. The key insight behind our proposal is reformulating the classification tasks as the search problem. Numerical simulations exhibit that GBLS can achieve comparable performance with other quantum classifiers under various noise settings, while the required number of measurements is dramatically reduced. We further demonstrate a potential quantum advantage of GBLS over classical classifiers in the measure of query complexity. Our work provides guidance to develop advanced quantum classifiers on near-term quantum devices and opens up an avenue to explore potential quantum advantages in various classification tasks.Abstract The hybrid quantum–classical learning scheme provides a prominent way to achieve quantum advantages on near-term quantum devices. A concrete example toward this goal is the quantum neural network (QNN), which has been developed to accomplish various supervised learning tasks such as classification and regression. However, there are two central issues that remain obscure when QNN is exploited to accomplish classification tasks. First, a quantum classifier that can well balance the computational cost such as the number of measurements and the learning performance is unexplored. Second, it is unclear whether quantum classifiers can be applied to solve certain problems that outperform their classical counterparts. Here we devise a Grover-search based quantum learning scheme (GBLS) to address the above two issues. Notably, most existing QNN-based quantum classifiers can be seamlessly embedded into the proposed scheme. The key insight behind our proposal is reformulating the classification tasks as the search problem. Numerical simulations exhibit that GBLS can achieve comparable performance with other quantum classifiers under various noise settings, while the required number of measurements is dramatically reduced. We further demonstrate a potential quantum advantage of GBLS over classical classifiers in the measure of query complexity. Our work provides guidance to develop advanced quantum classifiers on near-term quantum devices and opens up an avenue to explore potential quantum advantages in various classification tasks.

arXiv:1809.06056New Journal of PhysicsJan 22, 202138 citations

Simulation of a Variational Quantum Perceptron using Grover's Algorithm

Nouhaila Innan, Mohamed Bennai

The quantum perceptron, the variational circuit, and the Grover algorithm have been proposed as promising components for quantum machine learning. This paper presents a new quantum perceptron that combines the quantum variational circuit and the Grover algorithm. However, this does not guarantee that this quantum variational perceptron with Grover's algorithm (QVPG) will have any advantage over its quantum variational (QVP) and classical counterparts. Here, we examine the performance of QVP and QVP-G by computing their loss function and analyzing their accuracy on the classification task, then comparing these two quantum models to the classical perceptron (CP). The results show that our two quantum models are more efficient than CP, and our novel suggested model QVP-G outperforms the QVP, demonstrating that the Grover can be applied to the classification task and even makes the model more accurate, besides the unstructured search problems.The quantum perceptron, the variational circuit, and the Grover algorithm have been proposed as promising components for quantum machine learning. This paper presents a new quantum perceptron that combines the quantum variational circuit and the Grover algorithm. However, this does not guarantee that this quantum variational perceptron with Grover's algorithm (QVPG) will have any advantage over its quantum variational (QVP) and classical counterparts. Here, we examine the performance of QVP and QVP-G by computing their loss function and analyzing their accuracy on the classification task, then comparing these two quantum models to the classical perceptron (CP). The results show that our two quantum models are more efficient than CP, and our novel suggested model QVP-G outperforms the QVP, demonstrating that the Grover can be applied to the classification task and even makes the model more accurate, besides the unstructured search problems.

arXiv:2305.11040May 18, 20230 citations

Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size

N

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.

N

, namely

O(\sqrt{N})

. The second algorithm illustrates how the classical mistake bound of

O(\frac{1}{γ^2})

can be further improved to

O(\frac{1}{\sqrtγ})

through quantum means, where

γ

denotes the margin. Such improvements are achieved through the application of quantum amplitude amplification to the version space interpretation of the perceptron model.