Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2206.03053 · Physical Review A

Amplitude-based implementation of the unit step function on a quantum computer

Jonas Koppe, Mark-Oliver Wolf

Published Feb 7, 2023• 3 citations
arXivHTMLPDF

Abstract

Modeling nonlinear activation functions on quantum computers is vital for quantum neurons employed in fully quantum neural networks, however, remains a challenging task. We introduce an amplitude-based implementation for approximating nonlinearity in the form of the unit step function on a quantum computer. Our approach expands upon repeat-until-success protocols, suggesting a modification that requires a single measurement only. We describe two distinct circuit types which receive their input either directly from a classical computer or as a quantum state when embedded in a more advanced quantum algorithm. All quantum circuits are theoretically evaluated using numerical simulation and executed on Noisy Intermediate-Scale Quantum hardware. We demonstrate that reliable data with high precision can be obtained from our quantum circuits involving up to eight qubits and up to 25 CX-gate applications, enabled by state-of-the-art hardware-optimization techniques and measurement error mitigation.

DOI
10.1103/physreva.107.022606

Related papers

Nonlinear quantum neuron: A fundamental building block for quantum neural networks

Shilu Yan, Hongsheng Qi, Wei Cui

Quantum computing enables quantum neural networks (QNNs) to have great potential to surpass artificial neural networks. The powerful generalization of neural networks is attributed to nonlinear activation functions. Although various models related to QNNs have been developed, they are facing the challenge of merging the nonlinear, dissipative dynamics of neural computing into the linear, unitary quantum system. In this paper, we establish different quantum circuits to approximate nonlinear functions and then propose a generalizable framework to realize any nonlinear quantum neuron. We present two quantum neuron examples based on the proposed framework. The quantum resources required to construct a single quantum neuron are polynomial in function of the input size. Finally, both IBM Quantum Experience results and numerical simulations illustrate the effectiveness of the proposed framework.

arXiv:2011.03429•Physical Review A•Nov 25, 2020•46 citations

Realization of a quantum neural network using repeat-until-success circuits in a superconducting quantum processor

M. S. Moreira, Gian Giacomo Guerreschi, Wouter Vlothuizen +14 more

Abstract Artificial neural networks are becoming an integral part of digital solutions to complex problems. However, employing neural networks on quantum processors faces challenges related to the implementation of non-linear functions using quantum circuits. In this paper, we use repeat-until-success circuits enabled by real-time control-flow feedback to realize quantum neurons with non-linear activation functions. These neurons constitute elementary building blocks that can be arranged in a variety of layouts to carry out deep learning tasks quantum coherently. As an example, we construct a minimal feedforward quantum neural network capable of learning all 2-to-1-bit Boolean functions by optimization of network activation parameters within the supervised-learning paradigm. This model is shown to perform non-linear classification and effectively learns from multiple copies of a single training state consisting of the maximal superposition of all inputs.

arXiv:2212.10742•npj Quantum Information•Nov 21, 2023•16 citations

Towards a Real Quantum Neuron

Wei Hu

Google’s AlphaGo represents the impressive performance of deep learning and the backbone of deep learning is the workhorse of highly versatile neural networks. Each network is made up of layers of interconnected neurons and the nonlinear activation function inside each neuron is one of the key factors that account for the unprecedented achievement of deep learning. Learning how to create quantum neural networks has been a long time pursuit since 1990’s from many researchers, unfortunately without much success. The main challenge is to know how to design a nonlinear activation function inside the quantum neuron, because the laws in quantum mechanics require the operations on quantum neurons be unitary and linear. A recent discovery uses a special quantum circuit technique called repeat-until-success to make a nonlinear activation function inside a quantum neuron, which is the hard part of creating this neuron. However, the activation function used in that work is based on the periodic tangent function. Because of this periodicity, the input to this function has to be restricted to the range of [0, π/2), which is a serious constraint for its applications in real world problems. The function’s periodicity also makes its neurons not suited for being trained with gradient descent as its derivatives oscillate. The purpose of our study is to propose a new nonlinear activation function that is not periodic so it can take any real numbers and its neurons can be trained with efficient gradient descent. Our quantum neuron offers the full benefit as a quantum entity to support superposition, entanglement, interference, while also enjoys the full benefit as a classical entity to take any real numbers as its input and can be trained with gradient descent. The performance of the quantum neurons with our new activation function is analyzed on IBM’s 5Q quantum computer and IBM’s quantum simulator.

Natural Science•Jan 1, 2018•28 citations

Implementing Any Nonlinear Quantum Neuron

Fernando M. de Paula Neto, Teresa B. Ludermir, Wilson Rosa de Oliveira +1 more

The ability of artificial neural networks (ANNs) to adapt to input data and perform generalizations is intimately connected to the use of nonlinear activation and propagation functions. Quantum versions of ANN have been proposed to take advantage of the possible supremacy of quantum over classical computing. To date, all proposals faced the difficulty of implementing nonlinear activation functions since quantum operators are linear. This brief presents an architecture to simulate the computation of an arbitrary nonlinear function as a quantum circuit. This computation is performed on the phase of an adequately designed quantum state, and quantum phase estimation recovers the result, given a fixed precision, in a circuit with linear complexity in function of ANN input size.

IEEE Transactions on Neural Networks and Learning Systems•Sep 25, 2019•31 citations

Quantum Neuron: an elementary building block for machine learning on quantum computers

Yudong Cao, Gian Giacomo Guerreschi, Alán Aspuru‐Guzik

Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.

arXiv:1711.11240•Nov 30, 2017•113 citations

Optimization of quantum neuron based on repeat-until-success circuit

Fanghua Gao, Xianglong Yu, Zhimin Wang

Although the Repeat-Until-Success (RUS) neuron is functionally analogous to the activation function in classical neural networks, its practical application still faces limitations such as high operational complexity and the requirement for binary input states. To address these issues, this paper optimizes the RUS neuron. By merging specific quantum gate sequences into CNOT gates, we propose an optimized RUS neuron, which significantly reduces quantum resource overhead while maintaining functional integrity. By conducting an in-depth analysis of their dynamic behavior under both binary and superposition input states, we reveal the characteristic mechanisms of their activation-like functionality. To validate the practicality of the optimized RUS neuron, we design two typical application scenarios: constructing a quantum perceptron model to solve the XOR problem and developing a Hopfield neural network for image restoration. Simulation experiments demonstrate that the proposed quantum perceptron achieves 100% classification accuracy on the XOR problem, while the Hopfield network successfully reconstructs occluded images, fully confirming the practical feasibility and technical superiority of RUS neurons in quantum neural networks.

International Conference on Algorithms, Network, and Communication Technology (ICANCT)•Feb 17, 2026•0 citations
Modeling nonlinear activation functions on quantum computers is vital for quantum neurons employed in fully quantum neural networks, however, remains a challenging task. We introduce an amplitude-based implementation for approximating nonlinearity in the form of the unit step function on a quantum computer. Our approach expands upon repeat-until-success protocols, suggesting a modification that requires a single measurement only. We describe two distinct circuit types which receive their input either directly from a classical computer or as a quantum state when embedded in a more advanced quantum algorithm. All quantum circuits are theoretically evaluated using numerical simulation and executed on Noisy Intermediate-Scale Quantum hardware. We demonstrate that reliable data with high precision can be obtained from our quantum circuits involving up to eight qubits and up to 25 CX-gate applications, enabled by state-of-the-art hardware-optimization techniques and measurement error mitigation.
Quantum computing enables quantum neural networks (QNNs) to have great potential to surpass artificial neural networks. The powerful generalization of neural networks is attributed to nonlinear activation functions. Although various models related to QNNs have been developed, they are facing the challenge of merging the nonlinear, dissipative dynamics of neural computing into the linear, unitary quantum system. In this paper, we establish different quantum circuits to approximate nonlinear functions and then propose a generalizable framework to realize any nonlinear quantum neuron. We present two quantum neuron examples based on the proposed framework. The quantum resources required to construct a single quantum neuron are polynomial in function of the input size. Finally, both IBM Quantum Experience results and numerical simulations illustrate the effectiveness of the proposed framework.
Abstract Artificial neural networks are becoming an integral part of digital solutions to complex problems. However, employing neural networks on quantum processors faces challenges related to the implementation of non-linear functions using quantum circuits. In this paper, we use repeat-until-success circuits enabled by real-time control-flow feedback to realize quantum neurons with non-linear activation functions. These neurons constitute elementary building blocks that can be arranged in a variety of layouts to carry out deep learning tasks quantum coherently. As an example, we construct a minimal feedforward quantum neural network capable of learning all 2-to-1-bit Boolean functions by optimization of network activation parameters within the supervised-learning paradigm. This model is shown to perform non-linear classification and effectively learns from multiple copies of a single training state consisting of the maximal superposition of all inputs.
Google’s AlphaGo represents the impressive performance of deep learning and the backbone of deep learning is the workhorse of highly versatile neural networks. Each network is made up of layers of interconnected neurons and the nonlinear activation function inside each neuron is one of the key factors that account for the unprecedented achievement of deep learning. Learning how to create quantum neural networks has been a long time pursuit since 1990’s from many researchers, unfortunately without much success. The main challenge is to know how to design a nonlinear activation function inside the quantum neuron, because the laws in quantum mechanics require the operations on quantum neurons be unitary and linear. A recent discovery uses a special quantum circuit technique called repeat-until-success to make a nonlinear activation function inside a quantum neuron, which is the hard part of creating this neuron. However, the activation function used in that work is based on the periodic tangent function. Because of this periodicity, the input to this function has to be restricted to the range of [0, π/2), which is a serious constraint for its applications in real world problems. The function’s periodicity also makes its neurons not suited for being trained with gradient descent as its derivatives oscillate. The purpose of our study is to propose a new nonlinear activation function that is not periodic so it can take any real numbers and its neurons can be trained with efficient gradient descent. Our quantum neuron offers the full benefit as a quantum entity to support superposition, entanglement, interference, while also enjoys the full benefit as a classical entity to take any real numbers as its input and can be trained with gradient descent. The performance of the quantum neurons with our new activation function is analyzed on IBM’s 5Q quantum computer and IBM’s quantum simulator.
The ability of artificial neural networks (ANNs) to adapt to input data and perform generalizations is intimately connected to the use of nonlinear activation and propagation functions. Quantum versions of ANN have been proposed to take advantage of the possible supremacy of quantum over classical computing. To date, all proposals faced the difficulty of implementing nonlinear activation functions since quantum operators are linear. This brief presents an architecture to simulate the computation of an arbitrary nonlinear function as a quantum circuit. This computation is performed on the phase of an adequately designed quantum state, and quantum phase estimation recovers the result, given a fixed precision, in a circuit with linear complexity in function of ANN input size.
Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.
Although the Repeat-Until-Success (RUS) neuron is functionally analogous to the activation function in classical neural networks, its practical application still faces limitations such as high operational complexity and the requirement for binary input states. To address these issues, this paper optimizes the RUS neuron. By merging specific quantum gate sequences into CNOT gates, we propose an optimized RUS neuron, which significantly reduces quantum resource overhead while maintaining functional integrity. By conducting an in-depth analysis of their dynamic behavior under both binary and superposition input states, we reveal the characteristic mechanisms of their activation-like functionality. To validate the practicality of the optimized RUS neuron, we design two typical application scenarios: constructing a quantum perceptron model to solve the XOR problem and developing a Hopfield neural network for image restoration. Simulation experiments demonstrate that the proposed quantum perceptron achieves 100% classification accuracy on the XOR problem, while the Hopfield network successfully reconstructs occluded images, fully confirming the practical feasibility and technical superiority of RUS neurons in quantum neural networks.