On Freedman’s Lattice Models for Topological
and Zhenghan Wang
Received March 7, 2003; accepted March 18, 2003
Freedman proposes a family of Hamiltonians which deﬁne quantum loop gas models
on any celluated compact surface. We study the simplest nontrivial cases:
celluations of the torus. Our numerical data support Freedman’s conjecture about
the ground states of the Hamiltonians.
KEY WORDS: topological phase; lattice model.
PACS: 71.10.w, 71.35.y
The program of topological quantum computation is to realize fault tolerant
quantum computation using topological phases of quantum systems.
central open question is whether or not there exist such physical systems
which are capable of performing universal quantum computation. In Ref. 1,
a family of Hamiltonians H
is proposed as candidates for the Chern–
Simons phases, which are known to support universal quantum computa-
tion for each level l 3; l 6¼ 4:
Freedman conjectures that the perturbed
ground states of H
are given by the Drinfeld double of the SOð3Þ-Witten–
Chern–Simons topological quantum ﬁeld theories (TQFTs). The approach
in Ref. 1 is an algebraic study of the effect of perturbation based on a
rigidity result of the picture TQFTs.
The picture TQFTs are the Drinfeld
double of the SOð3Þ-Witten–Chern–Simons TQFTs. The idea is to treat
local relations in picture TQFTs as perturbations. In this paper we will
investigate numerically the perturbed ground states of H
Department of Math., University of California, Berkeley. E-mail: JBrink@Math.Berkeley.Edu
Department of Math., Indiana University. E-mail: firstname.lastname@example.org
To whom corresspondence should be addressed.
Quantum Information Processing, Vol. 2, Nos. 1–2, April 2003 (# 2003)
1570-0755/03/0400–0081/0 # 2003 Plenum Publishing Corporation