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

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.

The Physical Meaning of the Holographic Principle

“…[6, p. 826] Our purpose here is to respond to this remark of Bousso's by developing a clear, consistent, and at bottom, a very simple picture of the physical meaning of the HP. Building on previous results [12][13][14][15][16][17], we show that the HP can be generalized to describe the maximum classical information flow implemented by any physical interaction between mutually separable, i.e. unentangled, finite physical systems.…”

“…Quantum theory is traditionally considered to have a "measurement problem"; indeed since Schrödinger first introduced his cat [40][41][42][43], an enormous literature has devoted itself to the question of how observers can obtain classical information from a quantum world (see [44] for a thorough review and [45] for a recent compendium of philosophical positions). We will, in this section, employ quantum theory and the GHP to construct a fully-general theory of measurement; this construction was developed in [12][13][14][15][16][17], to which readers are referred for further details. We show in [18] that this theory reduces, in its classical limit, to the well-established theory of active inference derived from the classical free-energy principle [19][20][21][22][23].…”

“…that reversibly maps between "raw" data representable as n-bit strings and meaningful observational outcomes representable as m-bit strings, with the forward mapping implementing measurement and the reverse mapping implementing preparation. As developed in previous work [14,15] and proven in the general case in [17], any such computation can be given a category-theoretic representation as a Cone-Cocone Diagram (CCCD):…”

“…A Barwise-Seligman classifier implements a satisfaction relation ⊩ A between "tokens" and "types" in some language. Letting the tokens be bit strings in {0, 1} n and the types be bit strings in {0, 1} m , we can consider ⊩ A to be given by an n × m real matrix P i j , where each element p i j represents the probability that the i th token belongs to the j th type [15,17]; when all p i j ∈ {0, 1}, binary classifiers as originally defined in [49] are recovered. Letting A and B be classifiers with tokens in Tok(A) and Tok(B) and types in Typ(A) and Typ(B), respectively, an infomorphism is a pair of maps − → f and ← − f such that the following diagram commutes:…”

“…CCCDs are naturally interpreted as automorphisms {0, 1} k → {0, 1} k implemented by passage through a constraint network having the classifier C ′ as its apex; they can be interpreted as implementing variational auto-encoders (VAEs) or arbitrary Bayesian networks as discussed in [18,55]. More generally, they can be taken to represent most types of functional (directed) graph networks along with their underlying quiver representations [56] as applied in [17].…”