The document describes an introduction to quantum computation presented by Ritajit Majumdar and Arunabha Saha. It discusses the EPR paradox, which involves correlated measurements of particles separated in space, and Einstein's view that quantum mechanics was incomplete. It introduces Bell's theorem, which showed that any local hidden variable theory cannot reproduce all the predictions of quantum mechanics. The presentation also covers quantum gates, super dense coding, and mathematical notation used in quantum computation.
Mixin Classes in Odoo 17 How to Extend Models Using Mixin Classes
Introduction to Quantum Computation. Part - 2
1. Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Introduction to Quantum Computation
Part - II
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Ritajit Majumdar, Arunabha Saha
Reference
University of Calcutta
September 25, 2013
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
1 / 54
2. Introduction to Quantum
Computation
1
2
Ritajit Majumdar, Arunabha
Saha
Introduction
EPR Paradox
Outline
Introduction
EPR Paradox
Bell’s Theorem
3
Bell’s Theorem
Mathematical Notation
Quantum Gates
4
Mathematical Notation
Super Dense Coding
Next Presentation
5
6
Super Dense Coding
7
Next Presentation
8
Reference
Quantum Gates
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
2 / 54
3. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
3 / 54
4. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
|ψ does not uniquely determine the outcome of
measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
3 / 54
5. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
|ψ does not uniquely determine the outcome of
measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
It provides the statistical distribution of all possible
outcomes.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
3 / 54
6. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
|ψ does not uniquely determine the outcome of
measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
It provides the statistical distribution of all possible
outcomes.
Next Presentation
Reference
Here arises the confusion
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
3 / 54
7. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
|ψ does not uniquely determine the outcome of
measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
It provides the statistical distribution of all possible
outcomes.
Next Presentation
Reference
Here arises the confusion
? Does the physical system ‘actually have’ the attributes
prior to measurement (realist viewpoint)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
3 / 54
8. Introduction to Quantum
Computation
The Confusion
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What exactly a quantum state means??!!
EPR Paradox
Bell’s Theorem
|ψ does not uniquely determine the outcome of
measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
It provides the statistical distribution of all possible
outcomes.
Next Presentation
Reference
Here arises the confusion
? Does the physical system ‘actually have’ the attributes
prior to measurement (realist viewpoint)
OR
? The properties are ‘created’ by measurement (orthodox
position)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
3 / 54
9. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
EPR Paradox
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
4 / 54
10. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
0
−
Pi-meson decay: π → e + e
+
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Singlet state: |ψ =
|01 −|10
√
2
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
5 / 54
11. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
0
−
Pi-meson decay: π → e + e
+
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Singlet state: |ψ =
|01 −|10
√
2
Quantum Gates
Super Dense Coding
Next Presentation
If the electron (e − ) is found spin up (or |0 ) then the
positron (e + ) will be found to be spin down (or |1 ) and
vice-versa.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
September 25, 2013
5 / 54
12. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
0
−
Pi-meson decay: π → e + e
+
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Singlet state: |ψ =
|01 −|10
√
2
Quantum Gates
Super Dense Coding
Next Presentation
If the electron (e − ) is found spin up (or |0 ) then the
positron (e + ) will be found to be spin down (or |1 ) and
vice-versa.
Reference
Quantum mechanics does not ensure which combination
will be obtained but it is observed that the measurement
is correlated.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
5 / 54
13. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
0
−
Pi-meson decay: π → e + e
+
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Singlet state: |ψ =
|01 −|10
√
2
Quantum Gates
Super Dense Coding
Next Presentation
If the electron (e − ) is found spin up (or |0 ) then the
positron (e + ) will be found to be spin down (or |1 ) and
vice-versa.
Reference
Quantum mechanics does not ensure which combination
will be obtained but it is observed that the measurement
is correlated.
1
Each combination is obtained with probability 2 . But if
the electron is found to be in state |0 then the positron is
definitely in state |1 and vice versa.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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14. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
6 / 54
15. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
One school of thought claimed that the particle has
neither spin up nor spin down prior to measurement, it is
just created by the act of measurement.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
6 / 54
16. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
One school of thought claimed that the particle has
neither spin up nor spin down prior to measurement, it is
just created by the act of measurement.
Einstein mentioned this as “spooky action at a
distance”.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
September 25, 2013
6 / 54
17. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
One school of thought claimed that the particle has
neither spin up nor spin down prior to measurement, it is
just created by the act of measurement.
Einstein mentioned this as “spooky action at a
distance”.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
EPR argument is based on principle of locality.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
6 / 54
18. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
One school of thought claimed that the particle has
neither spin up nor spin down prior to measurement, it is
just created by the act of measurement.
Einstein mentioned this as “spooky action at a
distance”.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
EPR argument is based on principle of locality.
Principle of Locality
No influence can propagate faster than light.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
6 / 54
19. Introduction to Quantum
Computation
EPR Paradox
Ritajit Majumdar, Arunabha
Saha
Outline
This correlation of measurement is independent of spatial
distance.
Introduction
EPR Paradox
Bell’s Theorem
One school of thought claimed that the particle has
neither spin up nor spin down prior to measurement, it is
just created by the act of measurement.
Einstein mentioned this as “spooky action at a
distance”.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
EPR argument is based on principle of locality.
Principle of Locality
No influence can propagate faster than light.
But if we claim that the collapse is not instantaneous,
then it leads to violation of angular momentum
conservation.(Pi-meson decay)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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20. Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Bell‘s Theorem
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
7 / 54
21. Local Hidden Variable Theory (LHVT)
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR paper (Phys. Rev. 47, 777 (1935))
In quantum mechanics, in the case of two physical quantities
described by non-commutating operators, the knowledge of one
precludes the knowledge of another. Then either (1) the
description of reality given by the wave function in quantum
mechanics is not complete, or (2) these two quantities cannot
have simultaneous reality.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
September 25, 2013
8 / 54
22. Local Hidden Variable Theory (LHVT)
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR paper (Phys. Rev. 47, 777 (1935))
In quantum mechanics, in the case of two physical quantities
described by non-commutating operators, the knowledge of one
precludes the knowledge of another. Then either (1) the
description of reality given by the wave function in quantum
mechanics is not complete, or (2) these two quantities cannot
have simultaneous reality.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
The wavefunction, ψ, does not describe the system fully.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
8 / 54
23. Local Hidden Variable Theory (LHVT)
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR paper (Phys. Rev. 47, 777 (1935))
In quantum mechanics, in the case of two physical quantities
described by non-commutating operators, the knowledge of one
precludes the knowledge of another. Then either (1) the
description of reality given by the wave function in quantum
mechanics is not complete, or (2) these two quantities cannot
have simultaneous reality.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
The wavefunction, ψ, does not describe the system fully.
Another quantity (say λ) addition to ψ is needed to
describe the system completely.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
8 / 54
24. Local Hidden Variable Theory (LHVT)
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR paper (Phys. Rev. 47, 777 (1935))
In quantum mechanics, in the case of two physical quantities
described by non-commutating operators, the knowledge of one
precludes the knowledge of another. Then either (1) the
description of reality given by the wave function in quantum
mechanics is not complete, or (2) these two quantities cannot
have simultaneous reality.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
The wavefunction, ψ, does not describe the system fully.
Another quantity (say λ) addition to ψ is needed to
describe the system completely.
This λ is called “hidden variable” - we have no idea how
to calculate or measure it.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
8 / 54
25. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
In 1964 Bell proved that any LHVT is incompatible with
quantum mechanics. [J.S. Bell, physics 1, 195(1964)]
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
9 / 54
26. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
In 1964 Bell proved that any LHVT is incompatible with
quantum mechanics. [J.S. Bell, physics 1, 195(1964)]
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Charlie prepares two particles and sends one to Alice and
the other to Bob.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
9 / 54
27. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
In 1964 Bell proved that any LHVT is incompatible with
quantum mechanics. [J.S. Bell, physics 1, 195(1964)]
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Charlie prepares two particles and sends one to Alice and
the other to Bob.
On getting the particle, Alice can measure physical
properties PQ and PR .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
9 / 54
28. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
In 1964 Bell proved that any LHVT is incompatible with
quantum mechanics. [J.S. Bell, physics 1, 195(1964)]
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Charlie prepares two particles and sends one to Alice and
the other to Bob.
On getting the particle, Alice can measure physical
properties PQ and PR .
Alice flips a coin and decides which measurement has to
be done i.e. measuring randomly.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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29. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Q and R is the value for the property PQ and PR
respectively. Each have one of the two outcomes +1 or -1.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
10 / 54
30. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Q and R is the value for the property PQ and PR
respectively. Each have one of the two outcomes +1 or -1.
Similar is for Bob (Hence PS , PT ).
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
10 / 54
31. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Q and R is the value for the property PQ and PR
respectively. Each have one of the two outcomes +1 or -1.
Similar is for Bob (Hence PS , PT ).
The experiment is arranged so that Alice and Bob can
perform measurements at the same time (more precisely
in a causally disconnected manner).
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
September 25, 2013
10 / 54
32. Introduction to Quantum
Computation
Bell‘s Theorem
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Q and R is the value for the property PQ and PR
respectively. Each have one of the two outcomes +1 or -1.
Similar is for Bob (Hence PS , PT ).
The experiment is arranged so that Alice and Bob can
perform measurements at the same time (more precisely
in a causally disconnected manner).
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
By locality principle Alice’s measurement cannot disturb
Bob‘s measurement.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
10 / 54
33. Introduction to Quantum
Computation
CHSH Inequality
Ritajit Majumdar, Arunabha
Saha
Now perform simple algebra with the quantity:
Outline
Introduction
QS + RS + RT - QT
EPR Paradox
Bell’s Theorem
QS + RS + RT − QT = (Q + R)S + (R − Q)T
Since R, Q = ±1, it follows that either
QS + RS + RT − QT = 0
In either case (Q + R)S + (R − Q)T = ±2
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Let p(q, r , s, t) is the probability that, before the
measurements are performed, the system is in state
Q = q, R = r , S = s, T = t
Let E(.) denote the mean value.
E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)(qs + rs + rt − qt)
Σqrst p(q, r , s, t)x2
=2
.......(1)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
11 / 54
34. Introduction to Quantum
Computation
CHSH Inequality
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)qs +
Σqrst p(q, r , s, t)rs + Σqrst p(q, r , s, t)rt − Σqrst p(q, r , s, t)qt
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
= E(QS) + E(RS) + E(RT ) − E(QT )
.......(2)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
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35. Introduction to Quantum
Computation
CHSH Inequality
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
E(QS + RS + RT − QT ) = Σqrst p(q, r , s, t)qs +
Σqrst p(q, r , s, t)rs + Σqrst p(q, r , s, t)rt − Σqrst p(q, r , s, t)qt
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
= E(QS) + E(RS) + E(RT ) − E(QT )
.......(2)
Comparing eqn.(1)and eqn.(2) we obtain Bell Inequality
E(QS) + E(RS) + E(RT ) − E(QT )
Next Presentation
Reference
2
This result also term as CHSH inequality.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
12 / 54
36. Nature Does Not follow Bell Inequality
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Let Charlie prepares a quantum system of two qubits and
shared to Alice and Bob
|ψ =
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
|01 −|10
√
2
Quantum Gates
Super Dense Coding
After measurement the following observable
2
Q = Z1
S = −Z√−X2
2
R = X1
QS =
Next Presentation
Reference
Z2√ 2
−X
2
1
√ ;
2
T =
1
RS = √2 ; RT =
1
√
2
; QT =
1
√
2
Therefore,
√
QS + RS + RT − QT = 2 2
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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37. Nature Does Not follow Bell Inequality
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Hence it is observed that the previous result violates Bell
inequality.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
14 / 54
38. Nature Does Not follow Bell Inequality
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Hence it is observed that the previous result violates Bell
inequality.
Bell inequality is not obeyed by Nature.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
14 / 54
39. Nature Does Not follow Bell Inequality
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Hence it is observed that the previous result violates Bell
inequality.
Bell inequality is not obeyed by Nature.
This indicates that may be some of the basic assumptions
are incorrect.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
14 / 54
40. Nature Does Not follow Bell Inequality
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Hence it is observed that the previous result violates Bell
inequality.
Bell inequality is not obeyed by Nature.
This indicates that may be some of the basic assumptions
are incorrect.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Assumptions of local realism
Reference
(1) Physical properties PQ , PR , PS , PT have definite values
Q, R, S, T which exist independent of observation.
(Assumption of realism)
(2) Alice measurement does not influence the result of Bob‘s
measurement.
(Assumption of locality )
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
14 / 54
41. Introduction to Quantum
Computation
Bipartite System
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Suppose we have two parties, Alice and Bob, each having a
qubit. Alice’s qubit is denoted as |ψ1 = α0 |0 + α1 |1 and
that of Bob is denoted as |ψ2 = β0 |0 + β1 |1 .
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
So what will be the state of the composite system of Alice and
Bob?
Super Dense Coding
Next Presentation
Reference
1 |0
⊗ |0 ≡ |0 |0 ≡ |00
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
15 / 54
42. Introduction to Quantum
Computation
Bipartite System
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Suppose we have two parties, Alice and Bob, each having a
qubit. Alice’s qubit is denoted as |ψ1 = α0 |0 + α1 |1 and
that of Bob is denoted as |ψ2 = β0 |0 + β1 |1 .
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
So what will be the state of the composite system of Alice and
Bob?
Super Dense Coding
Next Presentation
Reference
It is given as |ψ12 = (α0 |0 + α1 |1 ) ⊗ (β0 |0 + β1 |1 )
= α0 β0 |00
1 |0
1
+α0 β1 |01 + α1 β0 |10 + α1 β1 |11
⊗ |0 ≡ |0 |0 ≡ |00
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
15 / 54
43. Bipartite System (Contd.)
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Similar is the reverse case. If we have a bipartite system
denoted as
α0 β0 |00 + α0 β1 |01 + α1 β0 |10 + α1 β1 |11
then we can factorise it to get the corresponding qubits
involved.
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
16 / 54
44. Introduction to Quantum
Computation
Entangled State
Ritajit Majumdar, Arunabha
Saha
A pure state of two systems is entangled if it cannot be
written as a product of two states -
Outline
Introduction
EPR Paradox
Bell’s Theorem
|ψAB = |ψA ⊗ |ψB
Mathematical Notation
We have four such entangled states called Bell States or EPR
States.
Quantum Gates
Super Dense Coding
Next Presentation
Reference
|φ+
|φ−
|ψ+
|ψ−
=
=
=
=
1
√ (|00
2
1
√ (|00
2
1
√ (|01
2
1
√ (|01
2
+ |11
− |11
+ |10
− |10
)
)
)
)
These 4 states are called Maximally Entangled States.
Any state of the form a |00 ± b |11 or a |01 ± b |10 , where
a = b, is called Pure Entangled State.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
17 / 54
45. Introduction to Quantum
Computation
Why Can’t be Factorised?
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Let us consider the state |ψ =
1
√ (|00
2
+ |11 )
And suppose we can factorise the state in the form
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
(α0 |0 + α1 |1 ) ⊗ (β0 |0 + β1 |1 ).
Super Dense Coding
Next Presentation
Reference
So we should have
1
α0 β0 = √2
α0 β1 = 0
α1 β0 = 0
1
α1 β1 = √2
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
18 / 54
46. Why Can’t be Factorised?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
From the 2
nd
rd
and the 3
equations we have -
Introduction
EPR Paradox
Bell’s Theorem
either
Mathematical Notation
α0 = 0 or β1 = 0
AND
Quantum Gates
Super Dense Coding
Next Presentation
Reference
either
α1 = 0 or β0 = 0.
But if any two of those have 0 values, then the 1st and the last
equations are not satisfied.
Hence the state cannot be factorised.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
19 / 54
47. Measurement of Entangled bits.
For entangled states, |ψAB = |ψA ⊗ |ψB
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Consider the state |ψ =
1
√ (|01
2
+ |10 )
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
20 / 54
48. Measurement of Entangled bits.
For entangled states, |ψAB = |ψA ⊗ |ψB
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Consider the state |ψ =
1
√ (|01
2
+ |10 )
Introduction
EPR Paradox
What happens if we measure the 1st qubit only?
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
20 / 54
49. Measurement of Entangled bits.
For entangled states, |ψAB = |ψA ⊗ |ψB
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Consider the state |ψ =
1
√ (|01
2
+ |10 )
Introduction
EPR Paradox
What happens if we measure the 1st qubit only?
Suppose after measurement, we found the 1st qubit to be in
state |0 . Then without measurement, we can know
immediately that the 2nd qubit is in state |1 .
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
20 / 54
50. Measurement of Entangled bits.
For entangled states, |ψAB = |ψA ⊗ |ψB
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Consider the state |ψ =
1
√ (|01
2
+ |10 )
Introduction
EPR Paradox
What happens if we measure the 1st qubit only?
Suppose after measurement, we found the 1st qubit to be in
state |0 . Then without measurement, we can know
immediately that the 2nd qubit is in state |1 .
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
So irrespective of the spatial distance between the two
entangled qubits, measuring one of them disturbs the state of
the other.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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51. Introduction to Quantum
Computation
Quantum Gates
Ritajit Majumdar, Arunabha
Saha
“When we get to the very, very small world - say circuits of
seven atoms - we have a lot of new things that would happen
that represent completely new opportunities for design. Atoms
on small scale behave like nothing on a large scale, for they
satisfy the laws of quantum mechanics. So, as we go down and
fiddle around with the atoms there, we are working with
different laws, and we can expect to do different things” -
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Richard Feynman
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
September 25, 2013
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52. Evolution of Quantum Systems
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Quantum systems evolve by Unitary transformations. If |ψ(t1 )
is the state of the system at time t1 and |ψ(t2 ) is the state of
the system at time t2 , then -
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
|ψ(t2 ) = U(t1 , t2 ) |ψ(t1 )
Next Presentation
Reference
where U(t1 , t2 ) is a Unitary Matrix.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
22 / 54
53. Evolution of Quantum Systems
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Quantum systems evolve by Unitary transformations. If |ψ(t1 )
is the state of the system at time t1 and |ψ(t2 ) is the state of
the system at time t2 , then -
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
|ψ(t2 ) = U(t1 , t2 ) |ψ(t1 )
Next Presentation
Reference
where U(t1 , t2 ) is a Unitary Matrix.
Analogous to Classical AND, OR, NOT etc. gates, Quantum
Computation has its own “Gates” which are represented as
Unitary Matrices.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
22 / 54
54. Introduction to Quantum
Computation
Classical Gates
Ritajit Majumdar, Arunabha
Saha
Outline
Classical Computers consist of wires that carry bits of
informtion and Gates that transform these bits in some way.
Introduction
EPR Paradox
Bell’s Theorem
Some of the famous classical logic gates are -
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
23 / 54
55. Irreversibility of Classical Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Classical Gates are irreversible 2 , i.e., one cannot determine
unique inputs for all outputs.
Introduction
EPR Paradox
Bell’s Theorem
For example, in an AND gate if the output is 0, it cannot be
determined whether the input values were 00, 01 or 10. Similar
is the case for output 1 in OR gate.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
2 The simplest example of a Classical Reversible Logic Gate is a NOT
gate.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
24 / 54
56. Irreversibility of Classical Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Classical Gates are irreversible 2 , i.e., one cannot determine
unique inputs for all outputs.
Introduction
EPR Paradox
Bell’s Theorem
For example, in an AND gate if the output is 0, it cannot be
determined whether the input values were 00, 01 or 10. Similar
is the case for output 1 in OR gate.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Unfortunately, logical irreversibility comes at a price.
Fundamental Physics states that energy must be dissipated
when information is erased.
And this dissipation is kTln2 per bit erased where k is the
Boltzmann constant (k = 1.3805 × 10−23 JK −1 ) and T is the
temperature in Absolute Scale.
2 The simplest example of a Classical Reversible Logic Gate is a NOT
gate.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
24 / 54
57. Introduction to Quantum
Computation
Quantum Logic Gates
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Just as any classical computation can be broken into a
sequence of classical logic gates that act on only a few classical
bits at a time, quantum computation too can be broken down
into a sequence of quantum logic gates that act on only a few
quantum bits at a time.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
3 Operation of Quantum Gates on a superposition takes same time as
the operation on a basis state.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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58. Introduction to Quantum
Computation
Quantum Logic Gates
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Just as any classical computation can be broken into a
sequence of classical logic gates that act on only a few classical
bits at a time, quantum computation too can be broken down
into a sequence of quantum logic gates that act on only a few
quantum bits at a time.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
The main difference is that where classical logic gates act on
classical bits 0 or 1, quantum gates can manipulate arbitrary
multi-partite quantum states including arbitrary superpositions3
of the computational basis states, which may also be entangled.
3 Operation of Quantum Gates on a superposition takes same time as
the operation on a basis state.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
25 / 54
59. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
60. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
61. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
−1
Introduction
Quantum Gates
is unitary.
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
62. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
Bell’s Theorem
Quantum Gates
is unitary.
Super Dense Coding
3. U −1 = U † (which is the criterion for determining unitarity.)
Ritajit Majumdar, Arunabha Saha (CU)
EPR Paradox
Mathematical Notation
1. U † is unitary.
−1
Introduction
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
26 / 54
63. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
−1
Introduction
Quantum Gates
is unitary.
Super Dense Coding
3. U −1 = U † (which is the criterion for determining unitarity.)
Next Presentation
Reference
†
4. UU = I
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
64. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
−1
Introduction
Quantum Gates
is unitary.
Super Dense Coding
3. U −1 = U † (which is the criterion for determining unitarity.)
Next Presentation
Reference
†
4. UU = I
5. |det(U)| = 1
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
65. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
−1
Introduction
Quantum Gates
is unitary.
Super Dense Coding
3. U −1 = U † (which is the criterion for determining unitarity.)
Next Presentation
Reference
†
4. UU = I
5. |det(U)| = 1
6. The columns (rows) of U form an orthonormal set of
vectors.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
26 / 54
66. Properties of Quantum Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The properties of quantum logic gates are the direct
consequence that they are described by Unitary Matrices. If U
is a Unitary , then the following facts holds -
2. U
EPR Paradox
Bell’s Theorem
Mathematical Notation
1. U † is unitary.
−1
Introduction
Quantum Gates
is unitary.
Super Dense Coding
3. U −1 = U † (which is the criterion for determining unitarity.)
Next Presentation
Reference
†
4. UU = I
5. |det(U)| = 1
6. The columns (rows) of U form an orthonormal set of
vectors.
The fact that for any quantum gate U, UU † = I , ensures that
we can always undo a quantum gate, i.e, a quantum gate is
logically reversible.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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67. Notation of Quantum Logic Gates
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Just like Classical computation, in quantum we consider a
quantum input wire that carries a qubit, a quantum gate that
performs some transformation on it, and a qauntum output
wire 4 that carries the qubit out.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
4 It may be noted that unlike classical computation, creating a quantum
wire is extremely difficult since a qubit evolves with time according to the
Schrodinger Equation.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
27 / 54
68. Introduction to Quantum
Computation
Bit Flip
Ritajit Majumdar, Arunabha
Saha
The simplest quantum gate is the “bit flip” gate, which is
analogous to the classical NOT gate.
Bit Flip is given by the pauli X matrix -
Outline
Introduction
EPR Paradox
Bell’s Theorem
0
1
1
0
Mathematical Notation
Quantum Gates
Super Dense Coding
The operation of the Bit Flip gate is -
Next Presentation
Reference
X |0 = |1
X |1 = |0
Figure : Working of a Quantum Bit Flip gate
It may be checked that X † = X and XX † = I
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
28 / 54
69. Introduction to Quantum
Computation
Phase Shift
Ritajit Majumdar, Arunabha
Saha
The pauli matrix Z is called the “Phase Shift” gate. The
operation of this gate is -
Outline
Introduction
EPR Paradox
Z |0 = |0
Z |1 = − |1
Bell’s Theorem
Mathematical Notation
Quantum Gates
The matrix notation of the Phase Flip gate is -
Super Dense Coding
Next Presentation
1
0
0
−1
Reference
Figure : Working of a Quantum Phase Flip gate
It may be checked that Z † = Z and ZZ † = I
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
29 / 54
70. Introduction to Quantum
Computation
Hadamard Gate
Ritajit Majumdar, Arunabha
Saha
One of the most important single qubit gate is the Hadamard
Gate. The matrix notation of this gate is 1
√
2
1
√
2
1
√
2
1
− √2
Outline
Introduction
EPR Paradox
=
1
√
2
1
1
1
−1
Bell’s Theorem
Mathematical Notation
Quantum Gates
Hadamard Gate switches between the bit basis and the sign
basis.
Super Dense Coding
Next Presentation
Reference
H |0 =
H |1 =
1
√
2
1
√
2
|0 +
|0 −
1
√
2
1
√
2
|1 = |+
|1 = |−
H |+ = |0
H |− = |1
Note that, starting with only the state |0 , Hadamard Trasform
can produce an equal superposition of both |0 and |1 . This is
an extremely powerful property and is the source of quantum
parallelism.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
30 / 54
71. Introduction to Quantum
Computation
Taking Stock
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Figure : A relative view of Bit Flip, Phase Flip and Hadamard
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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72. Geometrical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
A unitary transformation is mathematically a rotation.
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
32 / 54
73. Geometrical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
A unitary transformation is mathematically a rotation.
For the Bit Flip gate X, we have X |0 = |1 and X |1 = |0 .
So the bit flip gate can be geometrically thought of as a
rotation about an axis which is at an angle of π/4 from the
two orthonormal axes |0 and |1 .
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
32 / 54
74. Geometrical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
A unitary transformation is mathematically a rotation.
For the Bit Flip gate X, we have X |0 = |1 and X |1 = |0 .
So the bit flip gate can be geometrically thought of as a
rotation about an axis which is at an angle of π/4 from the
two orthonormal axes |0 and |1 .
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Figure : Geometrical interpretation of Bit Flip gate
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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75. Geometrical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
The geometrical picture of the 3 single qubit gates discussed
earlier -
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Figure : The Geometric Interpretation of Bit Flip, Phase Flip and
Hadamard Gates
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
33 / 54
76. Introduction to Quantum
Computation
Multiple Qubit Gates
Ritajit Majumdar, Arunabha
Saha
Outline
“In natural science, Nature has given us a world and we’re just
to discover its laws. In computers, we can stuff laws into it and
create a world.” - Alan Kay
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
The last few gates were single qubit quantum gates. Now we
look into few multiple qubit quantum gates.
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
34 / 54
77. Introduction to Quantum
Computation
CNOT Gate
A reversible gate of considerable importance in quantum
computation is the 2-bit Controlled-NOT (CNOT) gate. The
effect of CNOT gate is to flip the 2nd bit if and only if the 1st
bit is set to 1.
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
That is, the decision to negate is controlled by the value of
the 1st bit. Hence the name.
Quantum Gates
Super Dense Coding
Next Presentation
The symbolic representation of CNOT is -
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
September 25, 2013
35 / 54
78. Introduction to Quantum
Computation
CNOT Gate
Ritajit Majumdar, Arunabha
Saha
The matrix representation of CNOT gate is
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
The truth table of CNOT gate is given as a
0
0
1
1
b
0
1
0
1
a’
0
0
1
1
Reference
b’
0
1
1
0
CNOT gate is extremely important because it can create
Entanglement.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
36 / 54
79. Introduction to Quantum
Computation
Fredkin Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Fredkin gate, also called the Controlled-Swap (C-SWAP) gate,
is a multiple qubit gate. The action of this gate swaps the 2nd
and 3rd qubits only if the 1st qubit i.e. the control bit is 1.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
37 / 54
80. Introduction to Quantum
Computation
Fredkin Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Fredkin gate, also called the Controlled-Swap (C-SWAP) gate,
is a multiple qubit gate. The action of this gate swaps the 2nd
and 3rd qubits only if the 1st qubit i.e. the control bit is 1.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
The matrix representation of the Fredkin Gate is
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Super Dense Coding
Next Presentation
Reference
September 25, 2013
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81. Introduction to Quantum
Computation
Fredkin Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Fredkin Gate may be treated like a reversible AND gate.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
38 / 54
82. Introduction to Quantum
Computation
Fredkin Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Fredkin Gate may be treated like a reversible AND gate.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
From the figure, it is clear that if C = 0, then the last output
gives AB. The 2nd output is then a junk.
The 1st input, A, is retained and by operating the gate again,
the 2nd input, B, is retrieved.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
38 / 54
83. Introduction to Quantum
Computation
Fredkin Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Fredkin Gate may be treated like a reversible AND gate.
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
From the figure, it is clear that if C = 0, then the last output
gives AB. The 2nd output is then a junk.
The 1st input, A, is retained and by operating the gate again,
the 2nd input, B, is retrieved.
Hence it operates like a reversible AND gate.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
38 / 54
84. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Toffoli Gate, or the Controlled Controlled NOT (CCNOT)
Gate, has 3 input bits and 3 output bits. Two of the bits are
control bits that are unaffected by the action of the Toffoli
Gate. The 3rd bit is the target bit which is flipped if both the
control bits are set to 1, otherwise left unchanged.
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
39 / 54
85. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Toffoli Gate, or the Controlled Controlled NOT (CCNOT)
Gate, has 3 input bits and 3 output bits. Two of the bits are
control bits that are unaffected by the action of the Toffoli
Gate. The 3rd bit is the target bit which is flipped if both the
control bits are set to 1, otherwise left unchanged.
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
The matrix representation of Toffoli gate is
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
39 / 54
86. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Figure : Toffoli Gate: a = a, b = b, c = c ⊕ ab
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
40 / 54
87. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Figure : Toffoli Gate: a = a, b = b, c = c ⊕ ab
Next Presentation
Reference
From the figure, it is clear that if c = 1, then
c = 1 ⊕ ab = ¬(ab)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
40 / 54
88. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Figure : Toffoli Gate: a = a, b = b, c = c ⊕ ab
Next Presentation
Reference
From the figure, it is clear that if c = 1, then
c = 1 ⊕ ab = ¬(ab)
Thus Toffoli Gate acts as a reversible NAND gate.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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89. Introduction to Quantum
Computation
Toffoli Gate
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Figure : Toffoli Gate: a = a, b = b, c = c ⊕ ab
Next Presentation
Reference
From the figure, it is clear that if c = 1, then
c = 1 ⊕ ab = ¬(ab)
Thus Toffoli Gate acts as a reversible NAND gate.
Since NAND is a universal gate in Classical Computation, and
it can be realised in Quantum Computers by the Toffoli Gate,
it can be claimed that Any computation that is possible in Classical Computers
is also possible in Quantum Computers.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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90. Introduction to Quantum
Computation
Super Dense Coding
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Breaking news!
A single qubit can transmit two full classical bits of
information.
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
41 / 54
91. Quantum Measurement Revisited
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
From the measurement principle of Quantum Mechanics, we
know that if we have a state
|ψ = α |0 + β |1
then after measurement the state of the system collapses to |0
with probability |α|2 or to |1 with probability |β|2 .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
September 25, 2013
42 / 54
92. Quantum Measurement Revisited
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
From the measurement principle of Quantum Mechanics, we
know that if we have a state
|ψ = α |0 + β |1
then after measurement the state of the system collapses to |0
with probability |α|2 or to |1 with probability |β|2 .
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Hence, only one classical bit of information can be stored in
one qubit.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
42 / 54
93. What is Superdense Coding?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Superdense Coding 5 is a protocol proposed by Charles H.
Bennett and Stephen J. Wiesner.
Introduction
EPR Paradox
Bell’s Theorem
This is a simple protocol which enables the transportation of 2
cbits using only one ebit 6 .
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
5 PRL
1992 Vol 69 Number 20
we shall be using ”cbit” for classical bit and ”ebit” for
entangled bit
6 Henceforth,
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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94. Superdense Coding: Initial Setup
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
The initial step involves preparing an Entangled State. Alice
and Bob prepare a Bell State, say
Outline
Introduction
EPR Paradox
|φ+ =
1
√ (|00
2
+ |11 )
Bell’s Theorem
Mathematical Notation
Quantum Gates
After preparation, Alice keeps one of the two entangled qubits
with herself and sends the other one to Bob.
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
44 / 54
95. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Before proceeding to the main protocol, the question that
needs to be asked is -
Mathematical Notation
Quantum Gates
Super Dense Coding
How to prepare the Bell State?
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
45 / 54
96. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Before proceeding to the main protocol, the question that
needs to be asked is -
Mathematical Notation
Quantum Gates
Super Dense Coding
How to prepare the Bell State?
Next Presentation
Reference
Consider Alice and Bob both start with one qubit each, both in
state |0 . Alice operates her qubit only with a Hadamard Gate.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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97. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Before proceeding to the main protocol, the question that
needs to be asked is -
Mathematical Notation
Quantum Gates
Super Dense Coding
How to prepare the Bell State?
Next Presentation
Reference
Consider Alice and Bob both start with one qubit each, both in
state |0 . Alice operates her qubit only with a Hadamard Gate.
This operation is followed by a CNOT Gate on the two qubits.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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98. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
What happens when Alice operates a Hadamard Gate on her
qubit?
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
46 / 54
99. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
What happens when Alice operates a Hadamard Gate on her
qubit?
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
H |0 =
Quantum Gates
Super Dense Coding
1
√
2
1
√
2
Ritajit Majumdar, Arunabha Saha (CU)
1
√
2
1
− √2
1
=
0
1
√
2
1
√
2
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
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100. How to prepare the Bell State
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
What happens when Alice operates a Hadamard Gate on her
qubit?
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
H |0 =
Quantum Gates
Super Dense Coding
1
√
2
1
√
2
=
1
√
2
Ritajit Majumdar, Arunabha Saha (CU)
1
√
2
1
− √2
1
=
0
1
√
2
1
√
2
Next Presentation
Reference
1
1
Introduction to Quantum Computation
September 25, 2013
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101. Introduction to Quantum
Computation
How to prepare the Bell State
Ritajit Majumdar, Arunabha
Saha
Outline
What happens when Alice operates a Hadamard Gate on her
qubit?
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
H |0 =
Quantum Gates
Super Dense Coding
1
√
2
1
√
2
=
1
√
2
Ritajit Majumdar, Arunabha Saha (CU)
1
√
2
1
− √2
1
=
1
1
√
2
1
=
0
1
+
0
1
√
2
1
√
2
1
√
2
Next Presentation
Reference
0
1
Introduction to Quantum Computation
September 25, 2013
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102. Introduction to Quantum
Computation
How to prepare the Bell State
Ritajit Majumdar, Arunabha
Saha
Outline
What happens when Alice operates a Hadamard Gate on her
qubit?
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
H |0 =
Quantum Gates
Super Dense Coding
1
√
2
1
√
2
=
1
√
2
1
√
2
1
− √2
1
=
1
=
Ritajit Majumdar, Arunabha Saha (CU)
1
√
2
1
=
0
1
√
2
|0 +
1
+
0
1
√
2
1
√
2
1
√
2
1
√
2
Next Presentation
Reference
0
1
|1
Introduction to Quantum Computation
September 25, 2013
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103. How to prepare the Bell State?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Now we have Alice’s qubit as 1
√
2
|0 +
EPR Paradox
1
√
2
|1
while Bob’s qubit is |0 as before.
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
47 / 54
104. How to prepare the Bell State?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Now we have Alice’s qubit as 1
√
2
|0 +
EPR Paradox
1
√
2
|1
Bell’s Theorem
Mathematical Notation
Quantum Gates
while Bob’s qubit is |0 as before.
Super Dense Coding
So the 2 qubit system takes the form -
Next Presentation
Reference
1
√
2
Ritajit Majumdar, Arunabha Saha (CU)
|00 +
1
√
2
|10
Introduction to Quantum Computation
September 25, 2013
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105. How to prepare the Bell State?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Now we have Alice’s qubit as 1
√
2
EPR Paradox
|0 +
1
√
2
|1
Bell’s Theorem
Mathematical Notation
Quantum Gates
while Bob’s qubit is |0 as before.
Super Dense Coding
So the 2 qubit system takes the form -
Next Presentation
Reference
1
√
2
|00 +
1
√
2
|10
Now applying CNOT Gate, the resultant state is 1
√ (|00
2
+ |11 )
which is the required Bell State.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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106. Super Dense Conding: The Protocol
Alice wants to send 2 cbits of information to Bob using her
qubit.
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
48 / 54
107. Super Dense Conding: The Protocol
Alice wants to send 2 cbits of information to Bob using her
qubit.
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
There are 4 possible combinations of the 2 classical bits that
Alice wants to send -
EPR Paradox
Bell’s Theorem
Mathematical Notation
00, 01, 10, 11
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
48 / 54
108. Super Dense Conding: The Protocol
Alice wants to send 2 cbits of information to Bob using her
qubit.
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
There are 4 possible combinations of the 2 classical bits that
Alice wants to send -
EPR Paradox
Bell’s Theorem
Mathematical Notation
00, 01, 10, 11
Quantum Gates
Super Dense Coding
Depending on which combination Alice wants to send, she
performs the following operations on her qubit -
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Next Presentation
Reference
September 25, 2013
48 / 54
109. Super Dense Conding: The Protocol
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
After performing the required operation, Alice sends her qubit
to Bob.
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
49 / 54
110. Super Dense Conding: The Protocol
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
After performing the required operation, Alice sends her qubit
to Bob.
Mathematical Notation
Quantum Gates
Super Dense Coding
Bob now has both the qubits with him.
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
49 / 54
111. Super Dense Conding: The Protocol
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
After performing the required operation, Alice sends her qubit
to Bob.
Mathematical Notation
Quantum Gates
Super Dense Coding
Bob now has both the qubits with him.
Next Presentation
Reference
Bob performs the initial operations in reverse order, i.e. first
CNOT gate and then Hadamard Gate on the 1st qubit only.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
49 / 54
112. Super Dense Conding: The Protocol
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
After performing the required operation, Alice sends her qubit
to Bob.
Mathematical Notation
Quantum Gates
Super Dense Coding
Bob now has both the qubits with him.
Next Presentation
Reference
Bob performs the initial operations in reverse order, i.e. first
CNOT gate and then Hadamard Gate on the 1st qubit only.
Bob now has the two classical bits that Alice sent her.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
49 / 54
113. Super Dense Conding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Consider Alice wants to send cbit 01 to Bob. Then she
operates her qubit with the Pauli matrix σx or the bit flip gate.
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
7 NOTE: Alice can perform operation on her qubit only. Hence, the
operation affects only the 1st qubit.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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114. Super Dense Conding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Consider Alice wants to send cbit 01 to Bob. Then she
operates her qubit with the Pauli matrix σx or the bit flip gate.
Bell’s Theorem
So the state of the system after operation is:
Super Dense Coding
Mathematical Notation
Quantum Gates
Next Presentation
σx |φ+ =
1
√ (|10
2
+ |01 7 )
Reference
7 NOTE: Alice can perform operation on her qubit only. Hence, the
operation affects only the 1st qubit.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
50 / 54
115. Super Dense Conding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Consider Alice wants to send cbit 01 to Bob. Then she
operates her qubit with the Pauli matrix σx or the bit flip gate.
Bell’s Theorem
So the state of the system after operation is:
Super Dense Coding
Mathematical Notation
Quantum Gates
Next Presentation
σx |φ+ =
1
√ (|10
2
+ |01 7 )
Reference
Alice now sends her qubit to Bob who has the other half of the
entangled qubit.
7 NOTE: Alice can perform operation on her qubit only. Hence, the
operation affects only the 1st qubit.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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116. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Quantum Gates
Super Dense Coding
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
51 / 54
117. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
September 25, 2013
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118. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Reference
So the final state will be -
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
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119. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Reference
So the final state will be 1
1
√ ( √ (|0
2
2
− |1 ) |1 +
Ritajit Majumdar, Arunabha Saha (CU)
1
√ (|0
2
+ |1 ) |1 )
Introduction to Quantum Computation
September 25, 2013
51 / 54
120. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Reference
So the final state will be 1
1
√ ( √ (|0
2
2
− |1 ) |1 +
1
√ (|0
2
+ |1 ) |1 )
= 1 (|01 − |11 + |01 + |11 )
2
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
51 / 54
121. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Reference
So the final state will be 1
1
√ ( √ (|0
2
2
− |1 ) |1 +
1
√ (|0
2
+ |1 ) |1 )
= 1 (|01 − |11 + |01 + |11 )
2
= 1 (2 |01 )
2
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
51 / 54
122. Super Dense Coding: Case Study
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Bob now performs CNOT operation on the system
1
√ (|10 + |01 ). The resultant state will be 2
Outline
Introduction
EPR Paradox
1
√ (|11
2
+ |01 )
Bell’s Theorem
Mathematical Notation
Now, the Hadamard Gate is applied on the 1st qubit only. We
know -
Quantum Gates
Super Dense Coding
Next Presentation
1
H |0 = √2 (|0 + |1 )
1
and H |1 = √2 (|0 − |1 )
Reference
So the final state will be 1
1
√ ( √ (|0
2
2
− |1 ) |1 +
1
√ (|0
2
+ |1 ) |1 )
= 1 (|01 − |11 + |01 + |11 )
2
= 1 (2 |01 )
2
= |01
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
51 / 54
123. Coming up in next talk...
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
Bell’s Theorem
Mathematical Notation
No Cloning Theorem
Quantum Gates
Quantum Teleportation
Super Dense Coding
Next Presentation
Conclusive Quantum Teleportation
Reference
Quantum Algorithms
and many more...
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
52 / 54
124. Introduction to Quantum
Computation
Reference
Ritajit Majumdar, Arunabha
Saha
Michael A. Nielsen, Isaac Chuang
Quantum Computation and Quantum Information
Cambridge University Press
Outline
David J. Griffiths
Introduction to Quantum Mechanics
Prentice Hall, 2nd Edition
Mathematical Notation
Umesh Vazirani, University of California Berkeley
Quantum Mechanics and Quantum Computation
https://class.coursera.org/qcomp-2012-001/
Reference
Introduction
EPR Paradox
Bell’s Theorem
Quantum Gates
Super Dense Coding
Next Presentation
Michael A. Nielsen, University of Queensland
Quantum Computing for the determined
http://michaelnielsen.org/blog/
quantum-computing-for-the-determined/
www.springer.com
Quantum Gates
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
53 / 54
125. Introduction to Quantum
Computation
Reference
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
EPR Paradox
A. Einstein, B. Podolsky, N. Rosen,
Physical Review 47, 777 (1935)
Bell’s Theorem
Mathematical Notation
Quantum Gates
J.S. Bell
Physics 1, 195 (1964)
Super Dense Coding
Next Presentation
Reference
D. Bohm
Physical Review 85, 166, 180 (1952)
Charles H. Bennett, Stephen J. Wiesner
Physical Review Letters, Nov 16, 1992
Volume 69, Number 20
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 25, 2013
54 / 54