Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
×
then define Hm for m > 0 by:
H H
H = 1  m −1 m −1  ,
m √2
H H
 m 1 m 1
− − −
where the 1/√2 is a normalization that is sometimes omitted. Thus, other than this
normalization factor, the Hadamard matrices are made up entirely of 1 and -1.
Equivalently, we can define the Hadamard matrix by its (k, n)-th entry by writing
k = k 2m 1 +k 2m 2 + +k 2+k ,
m 1 − m 2 − 1 0
− − ···
and
n = n 2m 1 +n 2m 2 + +n 2+n ,
m 1 − m 2 − 1 0
− − ···
where the kj and nj are the binary digits (0 or 1) of n and k, respectively. In this case,
we have:
(H m) = 2m1 /2( −1)P jk jn j.
k,n
This is exactly the multidimensional 2 2 2 2 DFT, normalized to be unitary, if
× ×···× ×
the inputs and outputs are regarded as multidimensional arrays indexed by the n and k ,
j j
respectively. Some examples of the Hadamard matrices follow.
H = +1
0
1 1
 
H = 1
1 √2
1 1
 
−
(This H1 is precisely the size-2 DFT. It can also be regarded as the Fourier transform on
the two-element additive group of Z/(2).)
1 1 1 1
 
1 1 1 1
H = 1  − − 
2 2  
1 1 1 1 
 − − 
 
1 1 1 1 
− −
12
1 1 1 1 1 1 1 1
 
1 1 1 1 1 1 1 1
 
− − − −
 
1 1 1 1 1 1 1 1 
 − − − − 
 
1 1 1 1 1 1 1 1 
H = 1  − − − − .
3 23/2  
1 1 1 1 1 1 1 1 
 − − − − 
 
1 1 1 1 1 1 1 1 
 − − − − 
 
1 1 1 1 1 1 1 1 
 − − − − 
 
1 1 1 1 1 1 1 1 
− − − −
(H ) = 1 ( 1)ij
n i,j 2n/2 ·
−
where i j is the bitwise dot product of the binary representations of the numbers i and
·
j. For example, H = ( 1)32 = ( 1)(1,1)(1,0) = ( 1)1+0 = ( 1)1 = 1 , agreeing with the
32 · ·
− − − − −
above (ignoring the overall constant). Note that the first row, first column of the matrix is
denoted by H . The rows of the Hadamard matrices are the Walsh functions.
00
In quantum information processing the Hadamard transformation, more often called
Hadamard gate, is a one-qubit rotation, mapping the qubit-basis states 0 and 1 to two
| i | i
superposition states with equal weight of the computational basis states 0 and 1 . Usually
| i | i
the phases are chosen so that we have
|0 i+ |1 i 0 + |0 i−|1 i 1
√2 h | √2 h |
in Dirac notation. This corresponds to the transformation matrix
1 1
 
H = 1
1 √2
1 1
 
−
in the 0 , 1 basis.