8 Introduction

the two-dimensional electron liquid in a magnetic field at the filling factor

1/3). Excitations with such properties are called Abelian anyons. Another

example of Abelian anyons is described (in a mathematical language) in

Section 15.11.

More interesting are non-Abelian anyons, which have not yet been ob-

served experimentally. (Theory predicts their existence in a two-dimensional

electron liquid in a magnetic field at the filling factor 5/2.) In the presence

of non-Abelian anyons, the state of the surrounding electron liquid is de-

generate, the multiplicity of the degeneracy depending on the number of

anyons. In other words, there exist not one, but many states, which can

display arbitrary quantum superpositions. It is utterly impossible to act on

such a superposition without moving the anyons, so the system is ideally

protected from perturbations. If one anyon is moved around another, the

superposition undergoes a certain unitary transformation. This transforma-

tion is absolutely precise. (An error can occur only if the anyon “gets out of

hand” as a result of quantum tunneling.)

At first glance, the design using anyons seems least realistic. Firstly,

Abelian anyons will not do for quantum computation, and non-Abelian ones

are still awaiting experimental discovery. But in order to realize a quantum

computer, it is necessary to control (i.e., detect and drag by a specified

path) each excitation in the system, which will probably be a fraction of

a micron apart from each other. This is an exceedingly complex technical

problem. However, taking into account the high demands for precision, it

may not be at all easier to realize any of the other approaches we have men-

tioned. Beyond that, the idea of topological quantum computation, lying at

the foundation of the anyonic approach, might be expedited by other means.

For example, the quantum degree of freedom protected from perturbation,

might shoot up at the end of a “quantum wire” (a one-dimensional conduc-

tor with an odd number of propagating electronic modes, placed in contact

with a three-dimensional superconductor).

Thus, the idea of a quantum computer looks so very attractive, and

so very unreal. It is likely that the design of an ordinary computer was

perceived in just that way at the time of Charles Babbage, whose invention

was realized only a hundred years later. We may hope that in our time the

science and the industry will develop faster, so that we will not have to wait

that long. Perhaps a couple of fresh ideas plus a few years for working out

a new technology will do.