Download Introduction to Quantum logic (2)

Introduction to Quantum logic (2)
2002 .10 .10
Yong-woo Choi
Classical Logic Circuits

Circuit behavior is governed implicitly
by classical physics

Signal states are simple bit vectors,
e.g. X = 01010111

Operations are defined by Boolean Algebra
Quantum Logic Circuits
Circuit behavior is governed explicitly
by quantum mechanics
 Signal states are vectors interpreted as a
superposition of binary “qubit” vectors with
complex-number coefficients


Operations are defined by linear algebra over
Hilbert Space and can be represented by unitary
matrices with complex elements
Signal state (one qubit)
0
1
 0 0  1 1
Corresponsd to
1
 
 0
Corresponsd to
 0
 
1
Corresponsd to
 1
 0   0 
 0    1     
 0
 1   1 
2
2
 0  1  1
More than one qubit

If we concatenate two qubits

0
0  1 1   0 0  1 1 
 0  1  1
2
2
 0  1  1
2
2
We have a 2-qubit system with 4 basis states
0 0  00
1 0  10
0 1  01
1 1  11
And we can also describe the state as
 0  0 00  0 1 01  1 0 10  11 11
Or by the vector
 0  0   0 1  1  0  1 1  1
2
2
2
2
2
2
2
2
   0    0  0 
 0    



 0   0     1     0 1 
      
   1  0 





 1   1    0   
 1      1 1 
  1 
Tensor product
Quantum Operations

Any linear operation that takes states
 0 0  1 1
satisfying
 0  1  1
2
2
and maps them to states
 0 0  1 1
satisfying
must be UNITARY
t
UU I
t
U U
1
 0  1  1
2
2
Find the quantum gate(operation)
t
UU I
t
 U  U 1
From upper statement
We now know the necessities
1. A matrix must has inverse , that is reversible.
2. Inverse matrix is the same as U t.
So, reversible matrix is good candidate for quantum gate
Quantum Gate

One-Input gate: Wire
c0 0  c1 1

c0 0  c1 1
WIRE
Input
By matrix
 1 0  c0   c0 

    
 0 1  c1   c1 
Wire
Output
Quantum Gate

One-Input gate: NOT
c0 0  c1 1

c1 0  c0 1
NOT
Input
By matrix
 0 1  c0   c1 

    
 1 0  c1   c0 
NOT Gate
Output
Quantum Gate

Two-Input Gate: Controlled NOT (Feynman gate)
x
y
CNOT
x
x
x
x y
y
x y
concatenate

By matrix
1 0 0

0 1 0
0 0 0

0 0 1

y
0

0
1

0 
 00 
 00 
 
 
 01 
 01 
 10  =  11 
 
 
 11 
 10 
 
 
x 0  x x
x 1  x x
Quantum Gate

3-Input gate: Controlled CNOT (Toffoli gate)
x
y
C 2 NOT
z

By matrix 1

0
0

0
0

0

0
0

x
x
x
y
y
y
xy  z
z
xy  z
0 0 0 0 0 0 0

1 0 0 0 0 0 0
0 1 0 0 0 0 0

0 0 1 0 0 0 0
0 0 0 1 0 0 0 
0 0 0 0 1 0 0

0 0 0 0 0 0 1
0 0 0 0 0 1 0 
0

0
0

0
1

1

1
1

0
0
1
1
0
0
1
1
0

1
0

1
0 
1

0
1 
=
0

0
0

0
1

1

1
1

0
0
1
1
0
0
1
1
0

1
0

1
0 
1

1
0 
Quantum circuit
G1
G2
Tensor
Product
Matrix
multiplication
G1  G2
G3
G4
G5
G6
G7
G8
G3  G4  G5 G6  G7  G8
(G 6  G 7  G8)  (G3  G 4  G5)  (G1  G 2)