Download Matter Waves and Uncertainty Principle

qubits, quantum registers and gates
Anu Venugopalan
Guru Gobind Singh Indraprastha Univeristy
Delhi
_______________________________________________
INTERNATIONAL PROGRAM ON QUANTUM INFORMATION
(IPQI-2010)
Institute of Physics (IOP), Bhubaneswar
January 2010
IPQI-2010-Anu Venugopalan
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The Qubit
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‘Bit’ : fundamental concept of classical computation & info. - 0 or 1
‘Qubit’ : fundamental concept of quantum computation & info
 0  1
|  |2  |  |2  1
1
Normalization
0
- can
be thought of
mathematical objects having
some specific properties
Physical implementations - Photons,
electron, spin, nuclear spin
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The Qubit- computational basis
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  0   1
State space : C2
General state of a qubit

and

are complex coefficients that can take
any possible values and satisfy the normalization
condition:
   1
2
Computational basis
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1 
0   ;
 0
 0
1   
1 
orthonormal basis
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The Qubit- measurement in the
computational basis
_________________________________
  0   1
•In general the state of a qubit is a unit vector in a
two dimensional complex vector space.
• Unlike a bit you cannot ‘examine’ a qubit to determine
its quantum state
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The Qubit- measurement in the
computational basis
_________________________________
•A qubit
between
can
0
exist
and
in
a
1
continuum of states
until it is observed
•Measuring on the qubit:
-
0
  0   1
IPQI-2010-Anu Venugopalan
measurement
1
with prob

2
with prob

2
5
How much information does a qubit hold?
_________________________________
Geometric representation in terms of
a Bloch sphere:
 
 
i
  cos 0  e sin 1 
2
2  A point on a unit 3D sphere

  0   1



IPQI-2010-Anu Venugopalan
There are an infinite number of
points on the unit sphere, so that in
principle one could store a large
amount of information
But – from a single measurement one
obtains only a single bit of
information.
In the state of a qubit, Nature conceals a great
deal of hidden information
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multiple qubits - quantum registers
_________________________________
More than one qubit…..
The state space of a composite physical system is the
tensor product of the state spaces of the component
systems.
Example: for a two qubits, the state space is C2  C2=C4
computational basis for
C2
-
:
0 ;1
computational basis for C4 : 0  0 ; 0  1 ; 1  0 ; 1  1
alternate representation :
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00 ; 01 ; 10 ; 11
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Two qubit register – computational basis
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state of qubit 1

1
state of qubit 2

2
computational basis :
The state vector for two qubits:

1, 2
  0  1
  0  1
1
0
1
1
2
0
2
1
00 ; 01 ; 10 ; 11

1, 2

1

2
  00 00   01 01  10 10  11 11
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Two qubit register – matrix representation
_________________________________
1



0
1
1
1
state of qubit 1
   0 0  1 1   1 
 1 
2
state of qubit 2



0
2
2
2
   0 0  1 1   2 
1 

computational basis :
1 
 0
 0
0
 
 
 
 
 0
1 
 0
0
00   ; 01   ; 10   ; 11   
0
0
1
0
 
 
 
 
 0
 0
 0
1 
 
 
 
 
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Two qubit register – matrix representation
_________________________________
The state vector for two qubits:

1, 2

1, 2

1

2
  00 00   01 01  10 10  11 11
IPQI-2010-Anu Venugopalan
  01 02    00 

  
  0112    01 
 1 2 
 1 0   10 
 1 2   
 11   11 
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n-qubit register
_________________________________
Two-qubit register- state space C2 ; 4 basis states
4 terms in the superposition for the state vector

1, 2
  00 00   01 01  10 10  11 11
C8 :three-qubit register : 23=8 terms in the superposition
C16 :four-qubit register : 24=16 terms in the superposition
Cn :n-qubit register : 2n terms in the superposition
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n-qubit register
_________________________________
n-qubit register- 2n basis states
2n terms in the superposition for the state vector

1, 2 , 3.......... .. n
: a superposition specified by 2n amplitudes
e.g. for n=500 (a quantum register of 500 qubits),
the number of terms in the superposition, i.e., 2500 ,
is larger than the number of atoms in the Universe!
A few hundered atoms can store an enormous amount of
data - an exponential amount of classical info. in only a
polynomial number of qubits because of the superposition.
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Computing – gates – quantum analogs
_________________________________
Quantum Mechanics as computation
Classical computer circuits consist of logic gates. The
logic gates perform manipulations of the information,
converting it from one form to another.
Quantum analogs of logic
gates are Unitary
operators which can be represented as matrices.
Unitary operators (quantum gates ) operate on
qubits and quantum registers.
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Computing –gates – quantum analogs
_________________________________
example: single qubit quantum gates:
The NOT Gate
0 1 ˆ
  X
 x  
1 0
Xˆ 0  1 ; Xˆ 1  0
Xˆ  0   1    1   0
IPQI-2010-Anu Venugopalan
1
0
0
1








ˆ
ˆ
X      ; X     
 0  1  1   0 






Xˆ     
    
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Computing –gates – quantum analogs
_________________________________
example: single qubit quantum gates:
The phase flip Gate
1 0  ˆ
  Z
 z  
 0  1
Zˆ 0  0 ; Zˆ 1   1
1
0
0
1








Zˆ      ; Zˆ     
 0  1  1   0 
Zˆ  0   1    0   0
IPQI-2010-Anu Venugopalan







Xˆ    
    
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Computing –gates – quantum analogs
_________________________________
example: single qubit quantum gates:
The Hadamard Gate
1
1


1
ˆ


H
2 1  1
1
ˆ
0  1 
H0 
2
1
ˆ
0  1 
H1 
2
1  1 1 ˆ  0  1 1 
ˆ
H      ; H     
gate is uniquely
 0  2 1 1  2   1 This
quantum-mechanical with
no classical counterpart
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Computing –gates – quantum analogs
_________________________________
example: single qubit quantum gates:
The Hadamard Gate
1 1 1 
ˆ


H
2 1  1
1
ˆ
    0      1 
H  0   1  
2
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Computing –gates – quantum analogs
_________________________________
example: two-qubit quantum gates:
The quantum controlled NOT gate
The classical
C-NOT gate
a
0
0
1
1
b
0
1
0
1
c
0
1
1
0
The Quantum C-NOT gate
(reversible)
control qubit
a
a
target qubit
b
a b
action
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00  00 ;
01  01
10  11 ;
11  10
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Quantum Gates – the C-NOT gate
_________________________________
Quantum C-NOT gate is reversible- it corresponds
to a Unitary operator,
Û CN
a
a
a b
b
Uˆ CN 00  00 ; Uˆ CN 01  01
Uˆ CN 10  11 ; Uˆ CN 11  10
matrix representation Uˆ CN
IPQI-2010-Anu Venugopalan
1

0

0

0

0
1
0
0
0
0
0
1
0

0
1

0 
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Quantum Gates – the swap circuit
_________________________________
Three quantum C-NOT gates
a
b
b
a
Û CN
Û CN
a, b  a, a  b  a  b, a  (a  b)
Û CN
 a  b, b  b, b  (a  b)  b, a
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A quantum circuit for producing Bell states
_________________________________
00  11
 00 
2
Ĥ
01  10
01 
 xy
y
2
10 
11 
x
00  11
2
These are very useful states
01  10
2
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The No-Cloning Theorem
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Copying/cloning a single classical bit
Use a C- NOT gate
x
0
x
x
x
y
xy
x

Two bits of the same
input x
Can we have a similar
The C-NOT operation quantum circuit that can
clone/copy a qubit?
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The No-Cloning Theorem
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A quantum C-NOT gate to clone/copy
a qubit?
X=
 0  1
0
Input state
 0
  1  0
IPQI-2010-Anu Venugopalan
Action of C-NOT
Uˆ CN  input
output state
 00   11
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The No-Cloning Theorem
_________________________________
Can a quantum C-NOT gate clone/copy
a qubit?
X=
 0  1
output state
 00   11
0
Input state
 0
  1  0
IPQI-2010-Anu Venugopalan
If the circuit had cloned the input state
x as in the case of the classical circuit,
the output state should be
XX
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The No-Cloning Theorem
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X=  0   1
Input state:
 00   10
output state: 
0
00   11
If the circuit had cloned the input state x as in the
case of the classical circuit, the output state should be
XX
  0   1    0   1 
  2 00   01   10   2 11
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The No-Cloning Theorem
_________________________________
Input state:
output state:
Output
state
expected of a
cloning machine
 00   11
 00   10
 00   11
 00   01   10   11
2
2
clearly

 2 00   01   10   2 11
We have not managed to clone the state
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The No-Cloning Theorem
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 00   11
 
2
00   01   10   11
2
We have not managed to clone the state
if
  0,   1 or   1,   0
RHS = LHS
 00 or 11
Cloning happens only if the input state is either
0 or 1
- These are like classical bits!
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The No-Cloning Theorem
_________________________________
Cloning happens only if the input state is either 0
or 1
Only orthogonal states (classical bits) can be cloned
It is impossible to clone an unknown quantum state like an
input state of the qubit =  0   1
The no cloning theorem is a result of quantum mechanics
which forbids the creation of identical copies of an
arbitrary unknown quantum state.
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The No-Cloning Theorem
_________________________________
The no cloning theorem was stated by Wootters, Zurek,
and Dieks in 1982, and has profound implications in
quantum computing and related fields. The theorem
follows from the fact that all quantum operations must
be unitary linear transformation on the state
•W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be
Cloned, Nature 299 (1982), pp. 802–803.
•D. Dieks, Communication by EPR devices, Physics Letters A,
vol. 92(6) (1982), pp. 271–272.
•V. Buzek and M. Hillery, Quantum cloning, Physics World 14
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(11) (2001), pp. 25–29.
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Some consequences of the no-cloning theorem
___________________________________
•The no cloning theorem prevents us from using classical
error correction techniques on quantum states.
•the no cloning theorem is a vital ingredient in quantum
cryptography, as it forbids eavesdroppers from creating
copies of a transmitted quantum cryptographic key.
•Fundamentally, the no-cloning theorem protects the
uncertainty principle in quantum mechanics
•More fundamentally, the no cloning theorem prevents
superluminal communication via quantum entanglement.
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