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:2405.00012

A quantum neural network framework for scalable quantum circuit approximation of unitary matrices

Rohit Sarma Sarkar, Bibhas Adhikari

Published Feb 7, 2024• 2 citations
arXivHTMLPDF

Abstract

In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order $2^n$. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an $(n+1)$-qubit unitary can be constructed from the circuit of $n$-qubit system by adding a few CNOT gates and single-qubit gates.

DOI
10.48550/arxiv.2405.00012

Related papers

A Scalable Quantum Neural Network for Approximate Unitary Synthesis

G. Belli, Marco Mordacci, Michele Amoretti

In this work, scalable quantum neural networks are introduced to approximate unitary evolutions through the Standard Recursive Block Basis (SRBB) and, subsequently, redesigned with a reduced number of CNOTs. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain a continuous parameterization of unitary operators. First, we implemented the recursive construction of the SRBB to provide a starting tool for quickly verifying the mathematical properties needed to guarantee the original scalability scheme, already known to the literature only from a theoretical point of view. Unexpectedly, 2-qubit systems emerge as a special case outside this scheme. Furthermore, we present a method to reduce the number of CNOT gates, thus deriving a new implementable scaling scheme, which requires only one single layer and whose performance has been tested with a variety of different unitary matrices via the PennyLane library.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•3 citations

A Novel Single-Layer Quantum Neural Network for Approximate SRBB-Based Unitary Synthesis

Giacomo Belli, Marco Mordacci, Michele Amoretti

In this work, a novel quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and is subsequently redesigned with the number of CNOTs asymptotically reduced by an exponential contribution. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case of the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOT gates in the scalable variational quantum circuit, thus deriving a new implementable scaling scheme that requires only one layer of approximation. The single layer CNOT-reduced quantum neural network is implemented, and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.

arXiv:2412.03083•Quantum•Mar 20, 2026•1 citation

Probabilistic Simulation of Quantum Circuits with Neural Network Approximations

Jincheng Zhang

The simulation of quantum circuits presents a significant bottleneck in the development and understanding of quantum algorithms. Traditional methods relying on the full wave function description quickly become computationally intractable as the number of qubits and circuit depth increases. This work proposes a novel approach utilizing neural networks to approximate the probabilities of measurement outcomes within a quantum circuit. The core idea is to train a neural network to learn the complex dynamics of quantum evolution, thereby enabling efficient simulation with a significantly reduced parameter space. We introduce a framework that maps the quantum circuit to a neural network architecture, allowing for approximate but tractable simulation. The effectiveness of this approach is demonstrated through theoretical analysis and a conceptual implementation strategy, highlighting its potential to unlock the exploration of complex quantum algorithms and accelerate the development of quantum computing. The primary objective is to provide a computationally feasible alternative to traditional simulation techniques, bridging the gap between theoretical quantum algorithm design and practical implementation. The key innovation lies in leveraging the representational power of neural networks to capture the essential correlations inherent in quantum systems.

•Sep 2, 2026•0 citations

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

Emulating quantum computation with artificial neural networks

Christian Pehle, Karlheinz Meier, Markus K. Oberthaler +1 more

We demonstrate, that artificial neural networks (ANN) can be trained to emulate single or multiple basic quantum operations. In order to realize a quantum state, we implement a novel "quantumness gate" that maps an arbitrary matrix to the real representation of a positive hermitean normalized density matrix. We train the CNOT gate, the Hadamard gate and a rotation in Hilbert space as basic building blocks for processing the quantum density matrices of two entangled qubits. During the training process the neural networks learn to represent the complex structure, the hermiticity, the normalization and the positivity of the output matrix. The requirement of successful training allows us to find a critical bottleneck dimension which reflects the relevant quantum information. Chains of individually trained neural quantum gates can be constructed to realize any unitary transformation. For scaling to larger quantum systems, we propose to use correlations of stochastic macroscopic two-level observables or classical bits. This novel concept provides a path for a classical implementation of computationally relevant quantum information processing on classical neural networks, in particular on neuromorphic computing machines featuring stochastic operations.

arXiv:1810.10335•Oct 24, 2018•6 citations

A co-design framework of neural networks and quantum circuits towards quantum advantage

Weiwen Jiang, Jinjun Xiong, Yiyu Shi

Abstract Despite the pursuit of quantum advantages in various applications, the power of quantum computers in executing neural network has mostly remained unknown, primarily due to a missing tool that effectively designs a neural network suitable for quantum circuit. Here, we present a neural network and quantum circuit co-design framework, namely QuantumFlow, to address the issue. In QuantumFlow, we represent data as unitary matrices to exploit quantum power by encoding n = 2k inputs into k qubits and representing data as random variables to seamlessly connect layers without measurement. Coupled with a novel algorithm, the cost complexity of the unitary matrices-based neural computation can be reduced from O(n) in classical computing to O(polylog(n)) in quantum computing. Results show that on MNIST dataset, QuantumFlow can achieve an accuracy of 94.09% with a cost reduction of 10.85 × against the classical computer. All these results demonstrate the potential for QuantumFlow to achieve the quantum advantage.

•Jan 1, 2021•54 citations
In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order 2n2^n2n. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an (n+1)(n+1)(n+1)-qubit unitary can be constructed from the circuit of nnn-qubit system by adding a few CNOT gates and single-qubit gates.
In this work, scalable quantum neural networks are introduced to approximate unitary evolutions through the Standard Recursive Block Basis (SRBB) and, subsequently, redesigned with a reduced number of CNOTs. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain a continuous parameterization of unitary operators. First, we implemented the recursive construction of the SRBB to provide a starting tool for quickly verifying the mathematical properties needed to guarantee the original scalability scheme, already known to the literature only from a theoretical point of view. Unexpectedly, 2-qubit systems emerge as a special case outside this scheme. Furthermore, we present a method to reduce the number of CNOT gates, thus deriving a new implementable scaling scheme, which requires only one single layer and whose performance has been tested with a variety of different unitary matrices via the PennyLane library.
In this work, a novel quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and is subsequently redesigned with the number of CNOTs asymptotically reduced by an exponential contribution. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case of the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOT gates in the scalable variational quantum circuit, thus deriving a new implementable scaling scheme that requires only one layer of approximation. The single layer CNOT-reduced quantum neural network is implemented, and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.
The simulation of quantum circuits presents a significant bottleneck in the development and understanding of quantum algorithms. Traditional methods relying on the full wave function description quickly become computationally intractable as the number of qubits and circuit depth increases. This work proposes a novel approach utilizing neural networks to approximate the probabilities of measurement outcomes within a quantum circuit. The core idea is to train a neural network to learn the complex dynamics of quantum evolution, thereby enabling efficient simulation with a significantly reduced parameter space. We introduce a framework that maps the quantum circuit to a neural network architecture, allowing for approximate but tractable simulation. The effectiveness of this approach is demonstrated through theoretical analysis and a conceptual implementation strategy, highlighting its potential to unlock the exploration of complex quantum algorithms and accelerate the development of quantum computing. The primary objective is to provide a computationally feasible alternative to traditional simulation techniques, bridging the gap between theoretical quantum algorithm design and practical implementation. The key innovation lies in leveraging the representational power of neural networks to capture the essential correlations inherent in quantum systems.
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.
We demonstrate, that artificial neural networks (ANN) can be trained to emulate single or multiple basic quantum operations. In order to realize a quantum state, we implement a novel "quantumness gate" that maps an arbitrary matrix to the real representation of a positive hermitean normalized density matrix. We train the CNOT gate, the Hadamard gate and a rotation in Hilbert space as basic building blocks for processing the quantum density matrices of two entangled qubits. During the training process the neural networks learn to represent the complex structure, the hermiticity, the normalization and the positivity of the output matrix. The requirement of successful training allows us to find a critical bottleneck dimension which reflects the relevant quantum information. Chains of individually trained neural quantum gates can be constructed to realize any unitary transformation. For scaling to larger quantum systems, we propose to use correlations of stochastic macroscopic two-level observables or classical bits. This novel concept provides a path for a classical implementation of computationally relevant quantum information processing on classical neural networks, in particular on neuromorphic computing machines featuring stochastic operations.
Abstract Despite the pursuit of quantum advantages in various applications, the power of quantum computers in executing neural network has mostly remained unknown, primarily due to a missing tool that effectively designs a neural network suitable for quantum circuit. Here, we present a neural network and quantum circuit co-design framework, namely QuantumFlow, to address the issue. In QuantumFlow, we represent data as unitary matrices to exploit quantum power by encoding n = 2k inputs into k qubits and representing data as random variables to seamlessly connect layers without measurement. Coupled with a novel algorithm, the cost complexity of the unitary matrices-based neural computation can be reduced from O(n) in classical computing to O(polylog(n)) in quantum computing. Results show that on MNIST dataset, QuantumFlow can achieve an accuracy of 94.09% with a cost reduction of 10.85 × against the classical computer. All these results demonstrate the potential for QuantumFlow to achieve the quantum advantage.