Simulation of topological field theories by quantum computers
Michael H. Freedman, Alexei Kitaev, Zhenghan Wang
TL;DR
The paper addresses whether topological quantum field theories (TQFTs) can define computation beyond the standard quantum model. It shows that unitary TQFTs (via unitary topological modular functors, or UTMFs) can be efficiently simulated on a quantum computer by embedding their state spaces into poly-local qupit systems and implementing mapping-class-group actions with local gates; this remains within the BQP class even though H=0 in TQFTs. The key contributions are (i) a rigorous simulation framework using pants decompositions, Dehn twists, and F/S moves with a quantitative gate-depth bound of $c \, n \, \text{log}\big(b_1(\Sigma)\big)$ and an extension bound $11 \, \ell(h)$, and (ii) an extension to 3D TQFTs via bordisms, showing maps $b_*$ can be simulated by partial quantum circuits with length $c'(V) \, \text{complexity}(b)$. The work provides a topology-inspired lens on quantum computation and a concrete negative answer to universal speedups from TQFTs, while suggesting avenues for new quantum algorithms rooted in topological structure.
Abstract
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.