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ical system to be useful as a computer, it must be possible to exercise precise
control over the states of the system. Typically, it must be possible to prepare
the system in an input state, and then let the system evolve according to dy-
namical laws (this corresponds to the program) and subsequently to measure
an output state after the computation is completed.
To conclude, in both classical and quantum physics one speaks of the state
ofaphysicalsystem,andthe statesarecharacterizedbythe valuesofanumber
of variables. The state spaces in classical physics are continuous. This is true
also for systems like the bit. In solid state devices, voltage levels represent the
two states of the bit and there is a certain range within which these voltage
levels are allowed to vary. However, the levels must be well separated so that
no overlap of the ranges occur. This ensures the discrete digital nature of the
device.
We will return to the physics of computing system in the two last chapters.
3.2 Two-state quantum systems and the quan-
tum bit
The basic building block in most quantum computation models is the qubit.
The qubit is a quantum generalization of the classical bit. A bit can be in any
of the two well defined states 0 and 1, and a classical memory register can be
modeled by a string of bits. There is no interaction between the separate bits
in the register. Information processing, or computation, can be regardedas bit
flips performed on the register. Reading and writing single bits are the most
primitive acts of computation.
A qubit is a quantum system having two states. These states are denoted
by 0 and 1 .1 These states are the quantum versions of two states of the bit,
| i | i
0 and 1. The fundamental difference between classical and quantum physics is
that whereas a classical system must be in a definite state, a quantum system
can be in a superposition of a set of states. The bit must be either 0 or 1. But
the qubit can be in a complex linear combination of 0 and 1 , namely
| i | i
1Thenotationwillbeexplainedlateron.
58
ψ =α0 +β 1 . (3.1)
| i | i | i
Here α and β are complex numbers and ψ is used to denote the general
| i
state. Precise definitions of the quantum mechanical notations will be given in
thenexttwochapters. Sufficeitheretonotethattheproperframeworkforthis
is complex linear vector spaces, and consequently, the states 0 and 1 can be
| i | i
thought of as basis states in such a space.
The typical example of a two-state quantum system is the spin states of
spin-1 particle like the electron, but the precise physical nature of the system
2
will not concern us at the moment. We will instead develop the theory of
quantum computation based on generic two-state quantum systems. Questions
of practical implementations will be returned to in chapter 9.
There is a certainrestrictiononthe complex numbers α and β having to do
with the interpretation of quantum mechanics. Classically, one can determine
whichstate the bit is in, andone will get0 or1 accordingto the actual state of
the bit. Quantum mechanically, the situation is different.
The process of obtaining information out of the qubit is called a measure-
ment. If the qubit is in (or is known to be in) either the state 0 or the state
| i
1 , a measurement performed on it will give the result 0 or 1 respectively. If
| i
howeverthe qubit is in the generalstate ψ , the measurementwill give 0 with
| i
probability α2 and 1 withprobability β 2. There is no way,for a single qubit,
| | | |
to determine its precise state, i.e. there is no way to determine the values of
α and β. If however we have a large collection of identically prepared qubits,
repeated measurements on the qubits will yield statistical values for α2 and
| |
β 2. No single measurement can ever determine the values of α and β.
| |
However, since a measurement must yield either 0 or 1, this probabilistic
interpretation gives the restriction
α2+ β 2 =1 (3.2)
| | | |
on the numbers α and β.
A quantummeasurementwillhaveaneffect onthe state ofthe systemafter
the measurement. If a measurement is performed on the general state ψ and
| i
the result is 1, the state will be 1 after the measurement. Likewise, if the
| i
resultis 0,the state afterthe measurementwill be 0 . This is generalproperty
| i
of measurements.
There is a further difference with regard to classical physics. Classically it
does not make sense to consider measuring the bit in any other state than 0 or
1 because there are no other states. However, quantum mechanically we can
consider, for example, the special states
1 1
+ = 0 + 1 (3.3)
| i √2| i √2| i
1 1
= 0 1 . (3.4)
|−i √2| i− √2| i
59
Just as one can make a measurement on a general state ψ with respect to
| i
the states 0 and 1 , one can measure with respect to the states + and .
| i | i | i |−i
A qubit in the state 0 , say, will when measured with respect to the states +
| i | i
and , yield the result ’+’ with probability 1/2 and ’ ’ with probability 1/2.
|−i −
These non-classicalfeatures ofthe theory willbe elaboratedinthe next two