Sequential measurements, TQFTs, and TQNNs
- URL: http://arxiv.org/abs/2205.13184v1
- Date: Thu, 26 May 2022 06:37:57 GMT
- Title: Sequential measurements, TQFTs, and TQNNs
- Authors: Chris Fields, James F. Glazebrook and Antonino Marciano
- Abstract summary: We introduce novel methods for implementing generic quantum information within a scale-free architecture.
We show how observational outcomes are taken to be finite bit strings induced by measurement operators derived from a holographic screen bounding the system.
We extend the analysis so develop topological quantum neural networks (TQNNs), which enable machine learning with functorial evolution of quantum neural 2-complex quiveres.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 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.
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