Celebratio Mathematica

article M. H. Freedman, A. Kitaev, and Z. Wang: “Simulation of topological field theories by quantum computers,” Comm. Math. Phys. 227 : 3 (2002), pp. 587–603. MR 1910832 Zbl 1014.81006 ArXiv quant-ph/0001071

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, \( H \equiv 0 \). These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:

TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.
TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

Zhenghan Wang	Related
Alexei Kitaev	Related

article M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang: “Topological quantum computation,” Bull. Amer. Math. Soc. (N.S.) 40 : 1 (2003), pp. 31–38. MR 1943131 Zbl 1019.81008 ArXiv quant-ph/0101025

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten–Chern–Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and \( 2D \)-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like \( e^{-\alpha l} \), where \( l \) is a length scale, and \( \alpha \) is some positive constant. In contrast, the “presumptive” qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about \( 10^{-4} \)) before computation can be stabilized.

Zhenghan Wang	Related
Michael J. Larsen	Related
Alexei Kitaev	Related

article M. Freedman, A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, and Z. Wang: “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett. 98 (2007), pp. 160409. ArXiv cond-mat/0612341

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (“identity”) channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent \( z=1 \), and described by a two-dimensional (\( 2D \)) conformal field theory with central charge \( c=7/10 \). An exact mapping of the anyonic chain onto the \( 2D \) tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley–Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Zhenghan Wang	Related
Adrian Esteban Feiguin	Related
A. Ludwig	Related
Simon Trebst	Related
Matthias Troyer	Related
Alexei Kitaev	Related

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Michael H. Freedman

Quantum computing

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