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

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Researchers Giacomo Belli, Marco Mordacci, and Michele Amoretti developed a single-layer quantum neural network that approximates arbitrary unitary operations using the Standard Recursive Block Basis (SRBB) framework. The novel architecture exponentially reduces CNOT gate requirements compared to traditional SRBB methods, addressing a key bottleneck in quantum circuit complexity by leveraging Lie algebra properties for scalable parameterization. A specialized algorithm enables efficient implementation of 2-qubit operators as a simplified case, while maintaining high expressivity through a complete Hermitian unitary basis. Performance was validated on up to 6-qubit systems using PennyLane, with gradient-based and Nelder-Mead optimizers demonstrating robust approximation across sparse and dense unitary matrices. Real-hardware tests confirmed the approach outperforms existing decomposition methods, offering a practical path toward low-overhead unitary synthesis in near-term quantum devices.

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

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AbstractIn 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.Featured image: The SRBB-based single-layer quantum neural network.Popular summaryWhile exact gate synthesis algorithms typically involve exponential growth in the number of independent parameters, variational approaches allow for lower-depth circuit ansatzes at the cost of non-convex optimization landscapes and limited expressivity. In this context, the scalable design of variational quantum circuits (VQCs) capable of approximating arbitrary unitary evolutions with high expressivity, while reducing their complexity – specifically the CNOT gate count – remains a current challenge. This work introduces a novel quantum neural network (QNN) architecture as a means to approximate any unitary operator through the Standard Recursive Block Basis (SRBB) formalism, which allows for robust and scalable parameterizations thanks to the underlying properties of Lie algebras. The expressivity of the QNN is guaranteed by the completeness of the Hermitian unitary basis, and its existing theoretical scaling scheme 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 to exponentially reduce the number of CNOT gates compared to the original SRBB scheme is proposed and validated, thus facilitating the implementation of a new QNN architecture that requires only one layer of approximation. The single layer CNOT-reduced QNN is implemented, and its performance is assessed with a variety of different unitary matrices 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 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.► BibTeX data@article{Belli2026novelsinglelayer, doi = {10.22331/q-2026-03-20-2034}, url = {https://doi.org/10.22331/q-2026-03-20-2034}, title = {A {N}ovel {S}ingle-{L}ayer {Q}uantum {N}eural {N}etwork for {A}pproximate {SRBB}-{B}ased {U}nitary {S}ynthesis}, author = {Belli, Giacomo and Mordacci, Marco and Amoretti, Michele}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2034}, month = mar, year = {2026} }► References [1] Sahel Ashhab, Naoki Yamamoto, Fumiki Yoshihara, and Kouichi Semba. Numerical analysis of quantum circuits for state preparation and unitary operator synthesis. 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Could not fetch ADS cited-by data during last attempt 2026-03-20 07:36:10: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractIn 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.Featured image: The SRBB-based single-layer quantum neural network.Popular summaryWhile exact gate synthesis algorithms typically involve exponential growth in the number of independent parameters, variational approaches allow for lower-depth circuit ansatzes at the cost of non-convex optimization landscapes and limited expressivity. In this context, the scalable design of variational quantum circuits (VQCs) capable of approximating arbitrary unitary evolutions with high expressivity, while reducing their complexity – specifically the CNOT gate count – remains a current challenge. This work introduces a novel quantum neural network (QNN) architecture as a means to approximate any unitary operator through the Standard Recursive Block Basis (SRBB) formalism, which allows for robust and scalable parameterizations thanks to the underlying properties of Lie algebras. The expressivity of the QNN is guaranteed by the completeness of the Hermitian unitary basis, and its existing theoretical scaling scheme 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 to exponentially reduce the number of CNOT gates compared to the original SRBB scheme is proposed and validated, thus facilitating the implementation of a new QNN architecture that requires only one layer of approximation. The single layer CNOT-reduced QNN is implemented, and its performance is assessed with a variety of different unitary matrices 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 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.► BibTeX data@article{Belli2026novelsinglelayer, doi = {10.22331/q-2026-03-20-2034}, url = {https://doi.org/10.22331/q-2026-03-20-2034}, title = {A {N}ovel {S}ingle-{L}ayer {Q}uantum {N}eural {N}etwork for {A}pproximate {SRBB}-{B}ased {U}nitary {S}ynthesis}, author = {Belli, Giacomo and Mordacci, Marco and Amoretti, Michele}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2034}, month = mar, year = {2026} }► References [1] Sahel Ashhab, Naoki Yamamoto, Fumiki Yoshihara, and Kouichi Semba. Numerical analysis of quantum circuits for state preparation and unitary operator synthesis. Physical Review A, 106 (2): 022426, 2022. 10.1103/physreva.106.022426. https://doi.org/10.1103/physreva.106.022426 [2] Adriano Barenco, Charles H Bennett, Richard Cleve, David P DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical review A, 52 (5): 3457, 1995. 10.1103/physreva.52.3457. https://doi.org/10.1103/physreva.52.3457 [3] Giacomo Belli, Marco Mordacci, and Michele Amoretti. A scalable quantum neural network for approximate unitary synthesis. In 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 02, pages 49–54, 2024. 10.1109/QCE60285.2024.10251. https://doi.org/10.1109/QCE60285.2024.10251 [4] Giacomo Belli, Marco Mordacci, and Michele Amoretti. Srbb-based quantum state preparation. arXiv preprint arXiv:2503.13647, 2025a. 10.48550/arXiv.2503.13647. https://doi.org/10.48550/arXiv.2503.13647 arXiv:2503.13647 [5] Giacomo Belli, Marco Mordacci, and Michele Amoretti.

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

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