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

Fortschritte der Physik

Sequential Measurements, Topological Quantum Field Theories, and Topological Quantum Neural Networks

Chris Fields, James F. Glazebrook, Antonino Marcianò

Published Sep 2, 2022• 24 citations
Publisher

Abstract

Abstract We introduce novel methods for implementing generic quantum information within a scale‐free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically‐regulated information flows through distributed systems of finite sets of binary‐valued Barwise‐Seligman classifiers. Specifically, we construct a functor from the category of cone‐cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis too develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2‐complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah‐Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin‐networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.

DOI
10.1002/prop.202200104

Related papers

Sequential measurements, TQFTs, and TQNNs

Chris Fields, James F. Glazebrook, Antonino Marcianò

We introduce novel methods for implementing generic quantum information within a scale-free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis so develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.

arXiv:2205.13184•May 26, 2022•0 citations

Exact Evaluation of Hexagonal Spin‐Networks for Topological Quantum Neural Networks

Matteo Lulli, Antonino Marcianò, Emanuele Zappala

Abstract The physical scalar product between spin‐networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNNs). These are a class of quantum neural networks supported on graphs and related to topological quantum field theory (TQFT), which have been previously introduced by the authors, recovering deep neural networks (DNNs) as their semiclassical limit. However, the effective evaluation of the scalar product remains an obstacle for the applicability of the theory. Inspired by decimation techniques for the computation of the partition function in statistical mechanics, an analytical technique is introduced for the exact evaluation of hexagonal spin‐networks of arbitrary size, and describe the corresponding algorithm for the evaluation of the physical scalar product defined by Noui and Perez. The transition amplitudes on certain classes of spin‐networks with both classical and quantum recoupling are investigated, obtaining a “continuous” spectrum of the transitions for the former and a discrete one for the latter. The theoretical and computational framework is expected to impact applications in string/tensor‐networks for solid state physics, lattice gauge theories, and quantum gravity approaches.

Fortschritte der Physik•Apr 20, 2025•0 citations

Deep Neural Networks as the Semi-classical Limit of Quantum Neural Networks

Antonino Marcianò, Chen, Deen, Fabrocini*, Filippo +7 more

Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) A number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological features of the graphs structures involved.

arXiv:2007.00142•Jun 30, 2020•1 citation

Getting large-scale quantum neural networks ready for quantum hardware

Mario Boneberg, Simon Kochsiek, Igor Lesanovsky

Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many-body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise -- which is inherent to any quantum measurement -- sets practical limits for the estimation of training loss. Here, we study physics-informed large-scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many-body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.

quant-pharXiv:2604.24886•Apr 29, 2026•0 citations

TQFTs and quantum computing

Mahmud Azam, Steven Rayan

Quantum computing is captured in the formalism of the monoidal subcategory of 𝐕 𝐞 𝐜 𝐭 ℂ generated by ℂ 2 — in particular, quantum circuits are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ — while topological quantum field theories , in the sense of Atiyah, are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ . We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ , which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

arXiv:2210.03556•Bulletin des Sciences Mathématiques•May 28, 2024•2 citations

Quantum Neural Networks: Bridging Topological Structures and Variational Quantum Circuits

Sergey M. Gushanskiy, Viktor Potapov, Maxim Polenov

This paper presents the concept of quantum neural networks (QNN), a new framework that combines quantum computing with topological data analysis to process data defined on simplicial complexes. Based on the fundamentals of topological signal processing and variational quantum circuits, QNN aim to use the structure of simplicial complexes to perform efficient calculations and extract higher-order relational patterns from data. The proposed architecture uses quantum layers (QL), which map interactions between simplices into quantum circuits, providing both in-order and inter-order interactions. The paper describes two QL variants: basic and schematic, providing flexibility for applications with varying complexity. In addition, the paper explores the theoretical foundations of simplicial complexes, quantum neural networks, and the interaction between classical and quantum approaches in data encoding, state evolution and optimization. By leveraging quantum algorithms and eliminating current hardware limitations, QNN promise scalability and enhanced performance for applications in optimization, numerical simulation, and complex systems modeling. This research demonstrates the potential of QNN to advance the field of quantum machine learning, paving the way for future developments of quantum algorithms and their practical applications.

International Russian Smart Industry Conference (SmartIndustryCon)•Mar 24, 2025•2 citations
Abstract We introduce novel methods for implementing generic quantum information within a scale‐free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically‐regulated information flows through distributed systems of finite sets of binary‐valued Barwise‐Seligman classifiers. Specifically, we construct a functor from the category of cone‐cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis too develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2‐complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah‐Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin‐networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.
We introduce novel methods for implementing generic quantum information within a scale-free architecture. For a given observable system, we show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system. In this framework, measurements of identified systems with respect to defined reference frames are represented by semantically-regulated information flows through distributed systems of finite sets of binary-valued Barwise-Seligman classifiers. Specifically, we construct a functor from the category of cone-cocone diagrams (CCCDs) over finite sets of classifiers, to the category of finite cobordisms of Hilbert spaces. We show that finite CCCDs provide a generic representation of finite quantum reference frames (QRFs). Hence the constructed functor shows how sequential finite measurements can induce TQFTs. The only requirement is that each measurement in a sequence, by itself, satisfies Bayesian coherence, hence that the probabilities it assigns satisfy the Kolmogorov axioms. We extend the analysis so develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complexes (TQN2Cs) governed by TQFTs amplitudes, and resort to the Atiyah-Singer theorems in order to classify topological data processed by TQN2Cs. We then comment about the quiver representation of CCCDs and generalized spin-networks, a basis of the Hilbert spaces of both TQNNs and TQFTs. We finally review potential implementations of this framework in solid state physics and suggest applications to quantum simulation and biological information processing.
Abstract The physical scalar product between spin‐networks has been shown to be a fundamental tool in the theory of topological quantum neural networks (TQNNs). These are a class of quantum neural networks supported on graphs and related to topological quantum field theory (TQFT), which have been previously introduced by the authors, recovering deep neural networks (DNNs) as their semiclassical limit. However, the effective evaluation of the scalar product remains an obstacle for the applicability of the theory. Inspired by decimation techniques for the computation of the partition function in statistical mechanics, an analytical technique is introduced for the exact evaluation of hexagonal spin‐networks of arbitrary size, and describe the corresponding algorithm for the evaluation of the physical scalar product defined by Noui and Perez. The transition amplitudes on certain classes of spin‐networks with both classical and quantum recoupling are investigated, obtaining a “continuous” spectrum of the transitions for the former and a discrete one for the latter. The theoretical and computational framework is expected to impact applications in string/tensor‐networks for solid state physics, lattice gauge theories, and quantum gravity approaches.
Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) A number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological features of the graphs structures involved.
Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many-body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise -- which is inherent to any quantum measurement -- sets practical limits for the estimation of training loss. Here, we study physics-informed large-scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many-body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.
Quantum computing is captured in the formalism of the monoidal subcategory of 𝐕 𝐞 𝐜 𝐭 ℂ generated by ℂ 2 — in particular, quantum circuits are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ — while topological quantum field theories , in the sense of Atiyah, are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ . We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ , which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.
This paper presents the concept of quantum neural networks (QNN), a new framework that combines quantum computing with topological data analysis to process data defined on simplicial complexes. Based on the fundamentals of topological signal processing and variational quantum circuits, QNN aim to use the structure of simplicial complexes to perform efficient calculations and extract higher-order relational patterns from data. The proposed architecture uses quantum layers (QL), which map interactions between simplices into quantum circuits, providing both in-order and inter-order interactions. The paper describes two QL variants: basic and schematic, providing flexibility for applications with varying complexity. In addition, the paper explores the theoretical foundations of simplicial complexes, quantum neural networks, and the interaction between classical and quantum approaches in data encoding, state evolution and optimization. By leveraging quantum algorithms and eliminating current hardware limitations, QNN promise scalability and enhanced performance for applications in optimization, numerical simulation, and complex systems modeling. This research demonstrates the potential of QNN to advance the field of quantum machine learning, paving the way for future developments of quantum algorithms and their practical applications.