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Quantum Computation and the localization of Modular Functors

Michael H. Freedman

TL;DR

This work connects the localization of topological modular functors to concrete quantum engineering by formulating a Picture Principle that recasts modular functors in terms of local pictures and moves. It provides a constructive pathway from combinatorial localization on disks to a physical local Hamiltonian, showing that braiding point-like defects yields the Jones representation at level $5$ and is universal for quantum computation. The paper develops a detailed combinatorial localization for CS$r$ on marked disks using an $r$-collared tree and proves Lemma 2.1 to equate smooth and combinatorial equivalence under a roomy condition, then builds a local Hamiltonian $H$ whose ground space matches the modular functor space $V$ (up to duals), with adiabatic braiding realizing the desired unitary operations. It discusses energy-gap conjectures and potential renormalization pathways to simpler lattice models, highlighting the engineering relevance of topological quantum computation and the broader Freed–Walker program for localization of modular functors.

Abstract

The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus $=0$ surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation.

Quantum Computation and the localization of Modular Functors

TL;DR

This work connects the localization of topological modular functors to concrete quantum engineering by formulating a Picture Principle that recasts modular functors in terms of local pictures and moves. It provides a constructive pathway from combinatorial localization on disks to a physical local Hamiltonian, showing that braiding point-like defects yields the Jones representation at level and is universal for quantum computation. The paper develops a detailed combinatorial localization for CS on marked disks using an -collared tree and proves Lemma 2.1 to equate smooth and combinatorial equivalence under a roomy condition, then builds a local Hamiltonian whose ground space matches the modular functor space (up to duals), with adiabatic braiding realizing the desired unitary operations. It discusses energy-gap conjectures and potential renormalization pathways to simpler lattice models, highlighting the engineering relevance of topological quantum computation and the broader Freed–Walker program for localization of modular functors.

Abstract

The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation.

Paper Structure

This paper contains 2 sections, 2 theorems, 9 equations.

Table of Contents

  1. The Picture Principle
  2. Combinatorial localization of CS5 on marked disks, and the proof of the theorem.

Key Result

Theorem 1.1

Consider a rectangle $R$ of Euclidian square lattice consisting of $15$ boxes by $30\, n$ boxes. Associate a $2-$level spin system ${\mathbb C}^2$ with each of the $e :=960n + 36$ box edges in $R$. The disjoint union of these spin systems has Hilbert space $({\mathbb C}^{2})^{\otimes e}=:X$. There i

Theorems & Definitions (2)

  • Theorem 1.1
  • Lemma 2.1