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.
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