Quantum Neural Networks and Topological Quantum Field Theories

Marcianò, Antonino; Deen, Chen,; Fabrocini, Filippo; Fields, Chris; Greco, Enrico; Gresnigt, Niels G.; Jinklub, Krid; Lulli, Matteo; Terzidis, Kostas; Zappala, Emanuele

doi:10.1016/j.neunet.2022.05.028

Cited by 16 publications

(23 citation statements)

References 25 publications

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“…We have shown in [17] that given the GHP, sequential measurements of any sector S of a holographic screen B induce a TQFT on S . We also show how this TQFT can be realized as a quantum topological neural network, a generalized representation of a standard deep-learning system [63]. Here we briefly summarize the main result and mention some of its consequences, referring readers to [17] for details.…”

Section: Sequential Measurements Induce Tqftsmentioning

confidence: 90%

“…Quantum artificial neural networks generalize classical artificial neural networks, which are Turing equivalent. Conventional quantum neural networks can be further generalized to topological quantum neural networks, which as structured as spin networks, are tensor network representations of TQFTs, and hence fully compliant with the GHP as discussed in §3.4 above [17,63]. It is in view of such tensor networks, and the development of several sections here in relation to the boundary B (as in e.g §4.…”

Section: Multiple Realizability and Virtual Machinesmentioning

confidence: 99%

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The Physical Meaning of the Holographic Principle

Fields

Glazebrook

Marcianò

2022

Quanta

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We show in this pedagogical review that far from being an apparent law of physics that stands by itself, the holographic principle is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Utilizing the notion of holographic screens, which are information encoding boundaries between physical subsystems, we demonstrate that the physical interaction is an information exchange during which information is strictly conserved. Then we use generalized holographic principle in order to flesh out a fully-general quantum theory of measurement in which the measurement produces finite-resolution, classical outcomes. Further, we show that the measurements are given meaning by quantum reference frames and sequential measurements induce topological quantum field theories. Finally, we discuss principles equivalent to the holographic principle, including Markov blankets and the free-energy principle in biology, multiple realizability and virtual machines in computer science, and active inference and interface theories in cognitive science. This appearance in multiple disciplines suggests that the holographic principle is not just a fundamental principle of physics, but of all of science.Quanta 2022; 11: 72–96.

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Section: Sequential Measurements Induce Tqftsmentioning

confidence: 90%

Section: Multiple Realizability and Virtual Machinesmentioning

confidence: 99%

The Physical Meaning of the Holographic Principle

Fields

Glazebrook

Marcianò

2022

Quanta

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“…A framework has been constructed in [17] that associates spin‐network states that are colored with irreducible representations of some Lie group G to training and test sample states in (quantum) machine learning. A typical case‐study is represented by training and test samples that are images composed of pixels.…”

Section: Mapping Sequential Measurements To Tqnnsmentioning

confidence: 99%

“…The structure itself of the model shares similarity with partition functions emerging in TQFTs characterizing quantum gravity. The spin‐network simulator, as introduced in [127], is a combinatorial TQFT model for computation, which can be extended to the TQNNs, as proposed in [17]. A q ‐deformation of the spin‐network simulator has been investigated as well.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

“…[ 16 ] ), we have that sequential observations induce TQFTs, in particular, sequential observations of S induce TQFTs with copies of the Hilbert space

\mathcal {H}_S

as boundaries. In §6 we establish a link between sequential measurements and topological quantum neural networks (TQNNs), a framework that encodes deep neural networks (DNNs) in their semi‐classical limit, moving from the relation between TQFT and TQNN constructed in [17]. We then show how topological data processed by the functorial evolution of TQNNs, encoded by topological quantum neural 2‐complexes (TQN2Cs), can be classified resorting to the Atiyah‐Singer theorems, when spin manifolds are taken into account.…”

Section: Introductionmentioning

confidence: 99%

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

Fields

Glazebrook

Marcianò

2022

Fortschritte der Physik

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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.

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Communication Protocols and QECC From the Perspective of TQFT, Part II: QECCs as Spacetimes

Fields,

Glazebrook,

Marcianò

2024

Fortschritte der Physik

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Topological quantum field theories (TQFTs) provide a general, minimal‐assumption language for describing quantum‐state preparation and measurement. They therefore provide a general language in which to express multi‐agent communication protocols, e.g., local operations, classical communication (LOCC) protocols. In the accompanying Part I, we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error‐correcting codes (QECCs) on the agent‐environment boundary. Such QECCs can be regarded as implementing or inducing the emergence of spacetimes on such boundaries. Here connection between inter‐agent communication and spacetime is investigated, by exploiting different realizations of TQFT. The authors delved into TQFTs that support on their boundaries spin‐networks as computational systems: these are known as topological quantum neural networks (TQNNs). TQNNs, which have a natural representation as tensor networks, implement QECC. The HaPPY code is recognized to be a paradigmatic example. How generic QECCs, as bulk‐boundary codes, induce effective spacetimes is then shown. The effective spatial and temporal separations that take place in QECC enables LOCC protocols between spatially separated observers. The implementation of QECCs in BF and Chern‐Simons theories are then considered, and QECC‐induced spacetimes are shown to provide the classical redundancy required for LOCC. Finally, the topological M‐theory is considered as an implementation of QECC in higher spacetime dimensions.

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Quantum Neural Networks and Topological Quantum Field Theories

Cited by 16 publications

References 25 publications

The Physical Meaning of the Holographic Principle

The Physical Meaning of the Holographic Principle

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

Communication Protocols and QECC From the Perspective of TQFT, Part II: QECCs as Spacetimes

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