home | literature reviews

"Quantum computation and the localization of modular functors", Michael Freedman, 2000

Reviewed July 26, 2023

Citation: Freedman, Michael H. "Quantum computation and the localization of modular functors." Foundations of Computational Mathematics 1 (2001): 183-204.

Web: https://arxiv.org/abs/quant-ph/0003128

Tags: Foundational, TQFT, Universal-scheme, Fibonacci, Quantum-groups

This is the paper which first introduced a local Hamiltonian realizing the Fibonacci anyons. By this I mean the following. The quantum double codes are naturally realized as ground spaces of certain Hamiltonians. To perform a quantum computation, you adiabatically change the Hamiltonian (i.e. change which stabilizers enforce the presence of anyons) so that the anyons braid around each other. This is done with the theory of ribbon operators.

Fibonacci anyons naturally come from TQFTs/quantum groups, and do not a-priori possess such a Hamiltonian and such a braiding. In this paper Freedman constructs a Hamiltonian whose ground space hosts Fibonacci anyons. That is, when the quasiparticles are braided the actions corresponds to the Jones representation at level = 5. The braiding is performed by adiabatically evolving the Hamiltonian in a specified fashion.

The key fact is that the Hamiltonian is local - it is the sum of a "small" number of terms, each affect a constant number of qubits. In this case, less than 30.

To quote Freedman: The idea of anyonic computation is taken from [K2] and in a more speculative form [Fr]. The new ingredient is the implementation of a computationally complete modular functor by a local Hamiltonian.

This paper is the earlier to use Temperley-Lieb theory in TQC, with Freedman's so-called "picture principle". However, the fact that Temperley-Lieb algebras are connected to TQFTs was well established before this, so this is not a particularly revolutionary milestone.

This paper is really fun because Mike goes all-out with the philosophical musings. The first sentence of the paper exemplifies it beautifully: "Reality has the habit of intruding on the prodigies of purest thought and encumbering them with unpleasant embellishments. So it is astonishing when the chthonian hammer of the engineer resonates precisely to the gossamer fluttering of theory. Such a moment may soon be at hand in the practice and theory of quantum computation."

A good follow-up paper with much more insight into Temperley-Lieb algebras and Chern-Simons theory is

> Freedman, Michael H. "A Magnetic Model with a Possible Chern-Simons Phase: (with an Appendix by F. Goodman and H. Wenzl)." Communications in mathematical physics 234 (2003): 129-183.