13.3 Parallelism in quantum computing 289qubit| 0 〉↔| 1 〉, i.e.a| 0 〉+b| 1 〉→a| 1 〉+b| 0 〉. Similarly, pulses at angular
frequencyω 2 manipulate the state of the other spin (Qubit 2). These
changes in the state of each qubit independently are not sufficient for
quantum computing. The qubits must interact with each other so that
one qubit ‘controls’ the other qubit and influences its behaviour. A
small interaction between the spin causes the shift of the energy levels
indicated in going from Fig. 13.4(a) to (b)—the energy required to flip
Qubit 2 now depends on the state of Qubit 1 (and vice versa). The energy
levels in Fig. 13.4(b) give four separate transition frequencies between
the four states of the two qubits; a pulse of radio-frequency radiation
that drives one of these transitions can achieve the CNOT operation—it
changes the state of Qubit 2 only if Qubit 1 has state| 1 〉.^66 The coupling between trapped ions
arises from the mutual electrostatic re-
pulsion of the ions, but this interaction
between qubits doesnotgive a level
structure like that in Fig. 13.4.
Although it seems simple for the spin-1/2 system, the implementa-
tion of a quantum logic gate represents a major step towards building
a quantum computer. Rigorous proofs based on mathematical proper-
ties of unitary operators in quantum mechanics show that any unitary
operation can be constructed from a few basic operators—it is sufficient
to have just one gate which gives control of one qubit by another, such
as the CNOT, in addition to the ability to manipulate the individual
qubits in an arbitrary way.^7 These operators form a so-called universal
(^7) To rotate the nuclear spins to an ar-
bitrary point on the Bloch sphere the
NMR experiments use radio-frequency
pulses. Experiments in ion traps use
set which generates all other unitary transformations of the qubits. Raman pulses (Section 9.8).
13.3 Parallelism in quantum computing
A classical computer acts on binary numbers stored in the input regis-
ter, or registers, to output another number also represented as bits with
values 0 and 1. A quantum computer acts on the whole superposition of
all the input information in the qubits of its input register, e.g. a string
of ions in a linear Paul trap. This quantum register can be prepared
in a superposition of all possible inputs at the same time, so that the
quantum computing procedure transforms the entire register into a su-
perposition of all possible outputs. For example, withN= 3 qubits a
general initial state containing all possible inputs has the wavefunction
Ψ=A| 000 〉+B| 001 〉+C| 010 〉+D| 011 〉
+E| 100 〉+F| 101 〉+G| 110 〉+H| 111 〉.(13.12)
Quantum computing corresponds to carrying out a transformation, rep-
resented by the quantum mechanical operatorUˆ, to give the wavefunc-
tion Ψ′=UˆΨ that is a superposition of the outputs corresponding to
each input
Ψ=AUˆ| 000 〉+BUˆ| 001 〉+CUˆ| 010 〉+.... (13.13)Useful quantum algorithms combine different operations, such as
UˆCNOT, to give an overall transformationUˆ=Uˆm···Uˆ 2 Uˆ 1 ,andthese
operations have the same advantage of parallelism as for the individ-
ual operations. It appears that, instead of laboriously calculating the
output for each of the binary numbers 000 to 111, corresponding to 0