Introduction 7

capacitance. This interaction can be controlled. A quantum dot is a mi-

crostructure which can contain few electrons or even a single electron. The

spin of this electron can be used as a qubit. The diﬃculty is that one needs

to control each granule or quantum dot individually with high precision.

This seems harder than in the case of free atoms, because all atoms of the

same type are identical while parameters of fabricated structures fluctuate.

This approach may eventually succeed, but a new technology is required for

its realization.

4. Anyons. Anyons are quasi-particles (excitations) in certain two-dimen-

sional quantum systems, e.g. in a two-dimensional electron liquid in mag-

netic field. What makes them special is their topological properties, which

are stable to moderate variation of system parameters. One of the authors

(A. K.) considers this approach especially interesting (in view of it being

his own invention, cf. [35]), so that we will describe it in more detail. (At

a more abstract level, the connection between quantum computation and

topology was discussed by M. Freedman [26].)

The fundamental diﬃculty in constructing a quantum computer is the

necessity for realizing unitary transformations with precision δ δ0, where

δ0 is between 10−2 and 10−6. To achieve this it is necessary, as a rule, to

control the parameters of the system with still greater precision. However,

we can imagine a situation where high precision is achieved automatically,

i.e., where error correction occurs on the physical level. An example is given

by two-dimensional systems with anyonic excitations.

All particles in three-dimensional space are either bosons or fermions.

The wave function of bosons does not change if the particles are permuted.

The wave function of fermions is multiplied by −1 under a transposition

of two particles. In any case, the system is unchanged when each of the

particles is returned to its prior position. In two-dimensional systems, more

complex behavior is possible. Note, however, that the discussion is not about

fundamental particles, such as an electron, but about excitations (“defects”)

in a two-dimensional electron liquid. Such excitations can move, transform

to each other, etc., just like “genuine”

particles.5

However, excitations in

the two-dimensional electron liquid display some unusual properties. An

excitation can have a fractional charge (for example, 1/3 of the charge of

an electron). If one excitation makes a full turn around another, the state

of the surrounding electron liquid changes in a precisely defined manner

that depends on the types of the excitations and on the topology of the

path, but not on the specific trajectory. In the simplest case, the wave

function gets multiplied by a number (which is equal to

e2πi/3

for anyons in

5Fundamental particles can also be considered as excitations in the vacuum which is, actually,

a nontrivial quantum system. The difference is that the vacuum is unique, whereas the electron

liquid and other “quantum media” can be designed to meet our needs.