Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A modular functor which is universal for quantum computation

Michael Freedman, Michael Larsen, Zhenghan Wang

TL;DR

The paper demonstrates that the SU(2) Witten–Chern–Simons topological modular functor at a fifth root of unity provides a universal, fault-tolerant model for quantum computation by embedding qubits into TMF state spaces and implementing gates via braiding. It shows that braids densely realize poly-local unitaries through the Jones representation, and establishes a density theorem for relevant sectors to guarantee universality with polynomial overhead, including both fault-tolerant (CS5) and exact unitary (ECS5) variants. By connecting the Jones representations with the modular functor CS_r and providing explicit r = 5 examples, the work bridges topological quantum field theory with scalable quantum computation. The results imply that a topological quantum computer based on these TMFs can efficiently simulate any quantum circuit, with fault-tolerant mechanisms anchored in label measurements and ancilla management.

Abstract

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation, have topological implications which will be considered elsewhere.

A modular functor which is universal for quantum computation

TL;DR

The paper demonstrates that the SU(2) Witten–Chern–Simons topological modular functor at a fifth root of unity provides a universal, fault-tolerant model for quantum computation by embedding qubits into TMF state spaces and implementing gates via braiding. It shows that braids densely realize poly-local unitaries through the Jones representation, and establishes a density theorem for relevant sectors to guarantee universality with polynomial overhead, including both fault-tolerant (CS5) and exact unitary (ECS5) variants. By connecting the Jones representations with the modular functor CS_r and providing explicit r = 5 examples, the work bridges topological quantum field theory with scalable quantum computation. The results imply that a topological quantum computer based on these TMFs can efficiently simulate any quantum circuit, with fault-tolerant mechanisms anchored in label measurements and ancilla management.

Abstract

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation, have topological implications which will be considered elsewhere.

Paper Structure

This paper contains 4 sections, 7 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. A universal quantum computer
  3. Jones' representation of the braid groups
  4. A Density theorem

Key Result

Theorem 2.1

There is a constant $C>0$ so that for all unitary $g: {{\mathbb C}}_{i}^{2}\otimes {{\mathbb C}}_{j}^{2} \rightarrow {{\mathbb C}}_{i}^{2}\otimes {{\mathbb C}}_{j}^{2}$, there is a braid $b_{l}$ of length $\leq l$ in the generators $\sigma_{i}$ and their inverses $\sigma_{i}^{-1}, 1\leq i\leq n-1$, for some unit complex numbers (phases) $\omega_{i},i=0,2$ whenever $\epsilon$ satisfies

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Lemma 4.2