Topological Quantum Computers
In this chapter non-abelian anyons are used for quantum computing, aiming at
universality. The approximation of Jones evaluations is seen to be just a special
case of the approximation of general quantum invariants of links.
Each non-abelian anyon type leads to an anyonic quantum computing model.
Information is encoded in the collective states of many anyons at well-separated
positions. The lack of continuous evolution due to H ≡ 0 naturally protects the
encoded information, which is processed by braiding the anyons along prescribed
paths. The computational outcome is encoded in the amplitude of this process,
which is accessed by bringing anyons together and fusing them. The amplitude of
creating an array of anyons, braiding, and fusing them back to the ground state is
given by the quantum invariant of certain colored links. Hence anyonic quantum
computers approximate quantum invariants of links, and are very robust against
local errors. More elaborate schemes are based on topological change, e.g., mea-
surement during computation or hybridization with nontopological gates. Such
adaptive schemes are more powerful than braiding anyons alone.
7.1. Anyonic quantum computers
Every non-abelian anyon type gives rise to an anyonic model of quantum com-
puting [FKLW]. Quantum gates are realized by the afforded representations of
the braid groups. Topological quantum compiling is to realize, by braiding, uni-
tary transformations desired for algorithms such as Shor’s factoring. Of particular
interest are gates.
Abstractly, a quantum computing model consists of
(1) A sequence of Hilbert spaces Vn whose dimensions are exponential in n.
For each n, a state |ψ0 to initialize the computation.
(2) A collection of unitary matrices in U(Vn) which can be eﬃciently compiled
(3) A readout scheme based on measurement of quantum states to give the
It is not intrinsic for Vn to have a tensor decomposition, though desirable for cer-
tain architectures such as QCM. In TQC, tensor decomposition is unnecessary and
inconvenient. Leakage error in TQC arises when a tensor decomposition is forced
in order to simulate QCM, because dim Vn is rarely a power of a fixed integer for all
n. It would be interesting to find algorithms native to TQC beyond approximation
of quantum invariants.