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quantum

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0 1 0 0
00= , 01= , 10= , 11= . (3.9)
0 0 1 0
0 0 0 1
       
Thecorrespondingtwo-qubitstates 00 , 01 , 10 and 11 ,haveexactlythe
| i | i | i | i
samevectorrepresentation. Following[26]Icallthesevectorstheclassical basis.
When the numbersofbits orqubits arelargeitis convenientto use a short-
hand notation using the base-10representationof the bit strings interpreted as
binarynumbers. For example, the bit string 0101 willbe denoted 54 andsimi-
larlyfor 0101 whichisdenotedby 5 . Thesuffixisneededinordertoremove
4
| i | i
any ambiguity as to the numbers of bits or qubits that the number represents.
One-bit classical computations
The bit-flip program, flip, can now be represented by the matrix
0 1
FLIP = ,
(cid:18)1 0(cid:19)
which, of course, corresponds to the logical operation NOT. In the context of
quantum computation, this matrix is also called X for reason that will become
clear subsequently.
Furthermore, there are three more distinct programs, namely for a one-bit
state, namely
0 0 0 1 0 0
id: → , set: → , reset: → .
1 1 1 1 1 0
→ → →
All these can be represented by 2 2-matrices. Not that the two last compu-
×
tations, set and reset, are not reversible, whereas the first two, not and id are
reversible.
Reversibility in this context means that the input can be deduced from the
output.
Two-bit classical computations
On the 2-bit states (3.9), certain 4 4-matrices, represents computations. To
×
take just one example, consider the operation of exchanging the values of the
two bits.
00 00
→
01 10
swap: → . (3.10)
10 01
→
11 11
→
A matrix effecting this transformation is
62
1 0 0 0
0 0 1 0
SWAP = . (3.11)
0 1 0 0
0 0 0 1
 
Multi-bit classical computations
In this view of computation, n -bit strings are represented by 2n dimensional
vectors and computations are represented by 2n 2n square matrices. The
×
number of input and output bits are the same.
Some restrictions on the allowed matrices can be derived. The input bit
string is represented by a column vector with just one 1 and the rest entries 0.
The output bit string must also be represented in the same way. This puts a
severe restriction on the possible matrices. Consider applying a certain matrix
C to an n -bit state
0
c c c .
11 12 1n .
···  . 
c c c
 21 22 2n
. .. . .. ··· . .. δ j,
. . .
  c. n. 1 c. n. 2 c n.. n     0. . .   
···  
 
where the n-bit state is represented by a column vector with just on entry δ
j
different from 0 and equal to 1. Carrying out the matrix multiplication yields
c
1j
c
 2j 
. ,
.
.
 
c 
 nj
i.e. it pulls out the j-th column from the matrix C. If this column vector is to
represent an n-bit state, only one of the coefficients c can be equal to 1.
ij
This shows that the computation matrix C is a matrix of zeros, except for
preciselyoneentryineachcolumnwhichisequalto1. Asanaside,notethatfor
n-bit computations, represented by 2n 2n -matrices, there are (2n)2n =2n2n
·
×
different possible matrices.
In the classical case working with this model of computation is very uneco-