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Universal quantum computation in topological quantum neural networks and amplituhedron representation

Chris Fields, James F. Glazebrook, Antonino Marcianò, Emanuele Zappala

TL;DR

A formal correspondences between TQNNs and amplituhedra are exhibited to support the existence of amplituhedra for representing generic quantum processes and to point to applications areas enabled by these results.

Abstract

We study the relationship between computation and scattering both operationally (hence phenomenologically) and formally. We show how topological quantum neural networks (TQNNs) enable universal quantum computation, using the Reshetikhin-Turaev and Turaev-Viro models to show how TQNNs implement quantum error-correcting codes. We then exhibit a formal correspondences between TQNNs and amplituhedra to support the existence of amplituhedra for representing generic quantum processes. This construction shows how amplituhedra are geometric representations of underlying topological structures. We conclude by pointing to applications areas enabled by these results.

Universal quantum computation in topological quantum neural networks and amplituhedron representation

TL;DR

A formal correspondences between TQNNs and amplituhedra are exhibited to support the existence of amplituhedra for representing generic quantum processes and to point to applications areas enabled by these results.

Abstract

We study the relationship between computation and scattering both operationally (hence phenomenologically) and formally. We show how topological quantum neural networks (TQNNs) enable universal quantum computation, using the Reshetikhin-Turaev and Turaev-Viro models to show how TQNNs implement quantum error-correcting codes. We then exhibit a formal correspondences between TQNNs and amplituhedra to support the existence of amplituhedra for representing generic quantum processes. This construction shows how amplituhedra are geometric representations of underlying topological structures. We conclude by pointing to applications areas enabled by these results.

Paper Structure

This paper contains 26 sections, 2 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Computation and scattering in an operational setting
  3. LOCC protocols with external clocks
  4. Execution traces and computations
  5. Defining $Q_C$ as a QRF
  6. Turing-completeness of QRFs
  7. Implementing computation with scattering models
  8. UQC with TQFTs: spin-networks and TQNNs
  9. Spin-networks as a universal data structure
  10. Phase gating in computational processes and phase of scattering
  11. Universality of topological quantum computers
  12. Relation between Reshetikhin-Turaev model and Turaev-Viro models
  13. The chain mail invariant
  14. The chain mail and TV-invariants
  15. Quantum computation and the Turaev-Viro model
  16. ...and 11 more sections

Key Result

Theorem 1

A TQNN provides quantum processes for UQC.

Figures (3)

  • Figure 1: Process to obtain a handle decomposition of a $3$-manifold where a tetrahedron and its dual graph are the spines of the two complementary handlebodies
  • Figure 2: The spine dual to a tetrahedron is fattened to a handlebody
  • Figure 3: Handlebody component with $\delta$ curves and $\Omega$ elements represented

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof