294 Quantum computing
The theoretical principles of quantum computing are well understood
but there are many practical difficulties to overcome before it becomes
a reality. All potential systems must balance the need to have interac-
tions between the qubits to give coherent control and the minimisation
of the interactions with the external environment that perturb the sys-
tem. Trapped ions have decoherence times much longer than the time
needed to execute quantum logic gates and this makes them one of the
most promising possibilities. In quantum computing many new and in-
teresting multiple-particle systems have been analysed and, even if they
cannot be realised yet, thinking about them sharpens our understand-
ing of the quantum world, just as the EPR paradox did for many years
before it could be tested experimentally.Further reading
TheContemporary Physicsarticle by Cummins and Jones (2000) gives
an introduction to the main ideas and their implementation by NMR
techniques. The books by Nielsen and Chuang (2000) and Stolze and
Suter (2004) give a very comprehensive treatment. The article on the
ion-trap quantum information processor by Steane (1997) is also useful
background for Chapter 12. Quantum computing is a fast-moving field
with new possibilities emerging all the time. The latest information can
be found on the World-Wide Web.Exercises
(13.1)Entanglement
(a) Show that the two-qubit state in eqn 13.8 is
notentangled because it can be written as a
simple product of states of the individual par-
ticles in the basis| 0 ′〉=(| 0 〉−| 1 〉)/√
2and
| 1 ′〉=(| 0 〉+| 1 〉)/√
2.
(b) Write the maximally-entangled state| 00 〉+
| 11 〉in the new basis.
(c) Is| 00 〉+| 01 〉−| 10 〉+| 11 〉an entangled state?
[Hint.Try to write it in the form of eqn 13.6
and find the coefficients.]
(d) Show that the two states given in eqns 13.9
and 13.10 are both entangled.(e) Discuss whether the three-qubit state Ψ =
| 000 〉+| 111 〉possesses entanglement.(13.2)Quantum logic gates
This question goes through a particular example
of the statement that any operation can be con-
structed from a combination of a control gate and
arbitrary rotations of the individual qubits. For
trapped ions the most straightforward logic gate
to build is a controlled ‘rotation’ of Qubit 2 when
Qubit 1 is| 1 〉, i.e.UˆCROT{A| 00 〉+B| 01 〉+C| 10 〉+D| 11 〉}
=A| 00 〉+B| 01 〉+C| 10 〉−D| 11 〉.