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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. The Qubit - PowerPoint PPT Presentation
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IPQI-2010-Anu Venugopalan 1
qubits, quantum registers and gates
Anu Venugopalan
Guru Gobind Singh Indraprastha UniveristyDelhi
_______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-
2010) Institute of Physics (IOP), Bhubaneswar
January 2010
IPQI-2010-Anu Venugopalan 2
The Qubit______________________________________
0
2 2| | | | 1Normalization
Physical implementations - Photons, electron, spin, nuclear spin
1
‘Bit’ : fundamental concept of classical computation & info. - 0
or 1
‘Qubit’ : fundamental concept of quantum computation &
info 0 1
- can be thought of mathematical objects having some specific properties
IPQI-2010-Anu Venugopalan 3
The Qubit- computational basis______________________________________
10
General state of a qubit
and are complex coefficients that can
take any possible values and satisfy the
normalization condition:
122
Computational basis
1
01 ;
0
10
State space : C2
orthonormal basis
IPQI-2010-Anu Venugopalan 4
The Qubit- measurement in the computational basis
_________________________________
•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
10
IPQI-2010-Anu Venugopalan 5
The Qubit- measurement in the computational basis
_________________________________
-
•A qubit can exist in a continuum of states between and until it is observed
•Measuring on the qubit:
10 measurement
0
1
with prob
with prob2
2
0 1
IPQI-2010-Anu Venugopalan 6
How much information does a qubit hold?_________________________________
10 Geometric representation in terms of a Bloch sphere:
1
2sin0
2cos
ieA point on a unit 3D sphere
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
IPQI-2010-Anu Venugopalan 7
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 : 1; 0
computational basis for C4 :
11;01;10; 00
11;10;01; 00alternate representation :
IPQI-2010-Anu Venugopalan 8
Two qubit register – computational basis_________________________________
11;10;01; 00computational basis :
The state vector for two qubits:212,1
10
10
21
20
2
11
10
1
state of qubit 1
state of qubit 2
11100100 111001002,1
IPQI-2010-Anu Venugopalan 9
Two qubit register – matrix representation
_________________________________
1
0
0
0
11;
0
1
0
0
10;
0
0
1
0
01;
0
0
0
1
00
computational basis :
21
202
120
2
11
101
110
1
10
10
state of qubit 1
state of qubit 2
IPQI-2010-Anu Venugopalan 10
Two qubit register – matrix representation
_________________________________
The state vector for two qubits:212,1
11100100 111001002,1
11
10
01
00
21
11
20
11
21
10
20
10
IPQI-2010-Anu Venugopalan 11
n-qubit register_________________________________
Two-qubit register- state space C2 ; 4 basis states
4 terms in the superposition for the state vector
11100100 111001002,1
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
IPQI-2010-Anu Venugopalan 12
n-qubit register_________________________________
n-qubit register- 2n basis states 2n terms in the superposition for the state vector
n............3,2,1 : 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.
IPQI-2010-Anu Venugopalan 13
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.
IPQI-2010-Anu Venugopalan 14
Computing –gates – quantum analogs_________________________________
example: single qubit quantum gates:
The NOT Gate
01ˆ ;10ˆ
ˆ01
10
XX
Xx
0
1
1
0ˆ ; 1
0
0
1ˆ XX
ˆ
X 0110ˆ X
IPQI-2010-Anu Venugopalan 15
Computing –gates – quantum analogs_________________________________
example: single qubit quantum gates:
The phase flip Gate
11ˆ ;00ˆ
ˆ10
01
ZZ
Zz
0
1
1
0ˆ ; 1
0
0
1ˆ ZZ
ˆ
X 0010ˆ Z
IPQI-2010-Anu Venugopalan 16
Computing –gates – quantum analogs_________________________________
example: single qubit quantum gates:
The Hadamard Gate
11
11
2
1ˆ
H
1
1
2
1
1
0ˆ ; 1
1
2
1
0
1ˆ HH
102
11ˆ
102
10ˆ
H
H
This gate is uniquely quantum-mechanical with no classical counterpart
IPQI-2010-Anu Venugopalan 17
Computing –gates – quantum analogs_________________________________
example: single qubit quantum gates: The Hadamard Gate
11
11
2
1ˆ
H
102
110ˆ H
IPQI-2010-Anu Venugopalan 18
Computing –gates – quantum analogs_________________________________example: two-qubit quantum gates: The quantum controlled NOT gate
The classical
C-NOT gate
0 1 1
1 0 1
1 1 0
0 0 0
cba
The Quantum C-NOT gate
(reversible)
a
b
a
ba
control qubit
target
qubit
1011 ;1110
0101 ;0000
action
IPQI-2010-Anu Venugopalan 19
Quantum Gates – the C-NOT gate_________________________________Quantum C-NOT gate is reversible- it corresponds to a Unitary operator,
a
b
a
ba 1011 ˆ ;1110ˆ
0101 ˆ ;0000ˆ
CNCN
CNCN
UU
UU
CNU
matrix representation
0100
1000
0010
0001
ˆCNU
IPQI-2010-Anu Venugopalan 20
Quantum Gates – the swap circuit_________________________________
Three quantum C-NOT gates
a
b
b
a
abbabbbba
baababaaba
,)(,,
)(,,,
CNU CNU
CNU
IPQI-2010-Anu Venugopalan 21
A quantum circuit for producing Bell states
_________________________________
2
110000
2
100101
2
110010
2
100111
xyHx
y
These are very useful states
IPQI-2010-Anu Venugopalan 22
The No-Cloning Theorem_________________________________
Copying/cloning a single classical bit
Use a C- NOT gate
x
0
x
y
x
xy
x
x
Two bits of the same input x
The C-NOT operationCan we have a similar quantum circuit that can clone/copy a qubit?
IPQI-2010-Anu Venugopalan 23
The No-Cloning Theorem_________________________________
A quantum C-NOT gate to clone/copy a qubit?
X= 10
0
010
Input state
Action of C-NOT
inputCNU ˆ
1100 output state
IPQI-2010-Anu Venugopalan 24
The No-Cloning Theorem_________________________________
Can a quantum C-NOT gate clone/copy a qubit?
X= 10
0
010
Input state
1100 output state
If the circuit had cloned the input state
x as in the case of the classical circuit,
the output state should be
XX
IPQI-2010-Anu Venugopalan 25
The No-Cloning Theorem_________________________________
X= 10
0
1000 Input state:
1100 output state:
If the circuit had cloned the input state x as in the
case of the classical circuit, the output state should
be
XX 11100100
101022
IPQI-2010-Anu Venugopalan 26
The No-Cloning Theorem_________________________________
1000 Input state:
1100
output state:
Output state expected of a cloning machine
11100100 22 clearly
11100100 22
1100
We have not managed to clone the state
IPQI-2010-Anu Venugopalan 27
The No-Cloning Theorem_________________________________
1100 11100100 22
We have not managed to clone the state
0,1or 1,0 if
11or 00RHS = LHS
Cloning happens only if the input state is either or 0 1
- These are like classical bits!
IPQI-2010-Anu Venugopalan 28
The No-Cloning Theorem_________________________________
Cloning happens only if the input state is either or 0 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 = 10
The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state.
IPQI-2010-Anu Venugopalan 29
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 (11) (2001), pp. 25–29.
IPQI-2010-Anu Venugopalan 30
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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