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chapters on quantum mechanics.
3.3 Multiple qubit states
Multiple qubit statesaremodeledontheir classicalanalogue,the bit strings. A
classical two-bit register can store any of the bit strings 00, 01, 10 or 11. The
quantumanalogue ofthese strings are 00 , 01 , 10 and 11 respectively,and
| i | i | i | i
they can be regarded as a basis for a four-dimensional vector space.
A general 2-qubit state can now be written as a linear combination of the
basis states,
ψ =α 00 +α 01 +α 10 +α 11
00 01 10 11
| i | i | i | i | i
Two facts can be noted at this stage. Firstly, as already noted for the single
qubit, whereas a classical memory register must be in a definite state corre-
sponding to the actual values of the stored bits, the quantum memory register
canbe in linearcombinationofallthe basisstates. This is referredto as super-
position of states. Secondly, there are quantum states of the memory register
that cannot be expressed as direct products of the basis states. One example
is the state 1 (00 + 11 ) which in no way can be written as a product of
√2 | i | i
single qubit states 0 and 1 . This property of quantum mechanics is called
| i | i
entanglement.
These features of quantum mechanics, superposition of states and entangle-
ment, are crucial to the theory of quantum computation.
If we denote a single bit by b, a classical n-bit string can be written as
b b ...b . The quantum analogue is b b ...b . A general state is a linear
1 2 n 1 2 n
| i
combinationofthese2n basisstates. Thesestatesarecalledcomputationalbasis
states.
3.4 Computation
Computation can be seen as a transformation of an input state to an output
state. If both input and output are represented by n -bit strings, then the
computation can be performed by applying an 2n 2n matrix to the input.
×
At first sight one might be tempted to use n n matrices, representing the
×
states by n -dimensional vectors, the components of which are taken to be the
bits of the bit strings. However, that cannot work, as can be seen even in the
simplestcaseofjustonebit. Representingtheinputbitbythe(one-dimensional)
vectoriandthe outputbyo,the computationalrelationconnectingoutputand
input is
60
i o=Ci
→
for some matrix C which in this case is just a number. But then the bit flip
0 1, 1 0 cannot be represented with one and the same number C. Thus
→ →
onebitofinformationmustactuallyberepresentedbyatwodimensionalspace.
Representations of classical bits and quantum bits
We will introduce a convenient representation for both classical och quantum
bits. Theconstructionsareactuallythesame,butdifferentnotationwillbeused
inordertohighlightthedifferencesbetweenclassicalandquantumcomputation.
The values for a classical bit will be denoted in boldface as 0 and 1 and they
will be represented as two-dimensional vectors as
1 0
0= , 1= . (3.5)
(cid:18)0(cid:19) (cid:18)1(cid:19)
Correspondingly,the quantum bits will be represented in terms of the same
vectors as
1 0
0 = , 1 = . (3.6)
| i (cid:18)0(cid:19) | i (cid:18)1(cid:19)
Noteonedifferenceininterpretationinthiscontext. Asintheprecedingsec-
tion, the quantumstates 0 and 1 are basis vectorsina complex vectorspace
| i | i
and consequently it makes sense to consider linear combinations as in equation
(3.1). For the classical bit values 0 and 1 we introduce no such structure.2
Next,bitstringsandmulti-qubitstateswillberepresentedbydirectproducts
of these two-dimensional vectors. The rules are very simple, and we will write
them out in the case of products of two and three vectors.
a b
0 0
a b a b
0 0 = 0 1, (3.7)
(cid:18)a 1(cid:19)⊗(cid:18)b 1(cid:19) a 1b 0
a b 
 1 1
a b c
0 0 0
a b c
 0 0 1
a b c
0 1 0
a b b a b c 
0 0 0 = 0 1 1. (3.8)
(cid:18)a 1(cid:19)⊗(cid:18)b 1(cid:19)⊗(cid:18)c 1(cid:19)  a 1b 0c 0 
a b c 
 1 0 1
a b c 
 1 1 0
a b c 
 1 1 1
Fromthesetwocases,theprincipleshouldbeclear. Asanexample,thefour
different two-bit strings can be represented by the vectors
2Itcouldbesomewhat artificiallyintroducedinordertorepresentprobabilisticcomputa-
tion. Still,evensotherearefundamental differences ascomparedtoquantum computation.
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1 0 0 0