CHAPTER 7

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

classically.

(3) A readout scheme based on measurement of quantum states to give the

answer.

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.

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http://dx.doi.org/10.1090/cbms/112/07