Introduction to Quantum Computation. Part - 2

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

EPR Paradox

Bell’s Theorem

Mathematical Notation

Quantum Gates

Super Dense Coding

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Reference

Introduction to Quantum ComputationPart - II

Ritajit Majumdar, Arunabha Saha

University of Calcutta

September 25, 2013

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 1 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

1 Introduction

2 EPR Paradox

3 Bell’s Theorem

4 Mathematical Notation

5 Quantum Gates

6 Super Dense Coding

7 Next Presentation

8 Reference




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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The Confusion

What exactly a quantum state means??!!

|ψ〉 does not uniquely determine the outcome ofmeasurement.

It provides the statistical distribution of all possibleoutcomes.

Here arises the confusion

? Does the physical system ‘actually have’ the attributesprior to measurement (realist viewpoint)

OR

? The properties are ‘created’ by measurement (orthodoxposition)




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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The Confusion






OR





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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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The Confusion






OR





Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

The Confusion






OR





Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

Next Presentation

Reference

The Confusion






OR





Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

Next Presentation

Reference

The Confusion






OR





Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

EPR Paradox

EPR Paradox




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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EPR Paradox

Pi-meson decay: π0 → e− + e+

Singlet state: |ψ〉 = |01〉−|10〉√2

If the electron (e−) is found spin up (or |0〉) then thepositron (e+) will be found to be spin down (or |1〉) andvice-versa.

Quantum mechanics does not ensure which combinationwill be obtained but it is observed that the measurementis correlated.

Each combination is obtained with probability 12 . But if

the electron is found to be in state |0〉 then the positron isdefinitely in state |1〉 and vice versa.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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EPR Paradox










Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

EPR Paradox










Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

EPR Paradox










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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

EPR Paradox

This correlation of measurement is independent of spatialdistance.

One school of thought claimed that the particle hasneither spin up nor spin down prior to measurement, it isjust created by the act of measurement.

Einstein mentioned this as “spooky action at adistance”.

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 momentumconservation.(Pi-meson decay)




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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EPR Paradox











Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

EPR Paradox











Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

Next Presentation

Reference

EPR Paradox











Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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EPR Paradox











Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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EPR Paradox











Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Local Hidden Variable Theory (LHVT)

EPR paper (Phys. Rev. 47, 777 (1935))

In quantum mechanics, in the case of two physical quantitiesdescribed by non-commutating operators, the knowledge of oneprecludes the knowledge of another. Then either (1) thedescription of reality given by the wave function in quantummechanics is not complete, or (2) these two quantities cannothave simultaneous reality.

The wavefunction, ψ, does not describe the system fully.

Another quantity (say λ) addition to ψ is needed todescribe the system completely.

This λ is called “hidden variable” - we have no idea howto calculate or measure it.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

Next Presentation

Reference










Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

Next Presentation

Reference










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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem

In 1964 Bell proved that any LHVT is incompatible withquantum mechanics. [J.S. Bell, physics 1, 195(1964)]

Charlie prepares two particles and sends one to Alice andthe other to Bob.

On getting the particle, Alice can measure physicalproperties PQ and PR .

Alice flips a coin and decides which measurement has tobe done i.e. measuring randomly.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem








Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem








Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

Bell‘s Theorem








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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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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 canperform measurements at the same time (more preciselyin a causally disconnected manner).

By locality principle Alice’s measurement cannot disturbBob‘s measurement.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem









Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem









Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bell‘s Theorem









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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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CHSH Inequality

Now perform simple algebra with the quantity:

QS + RS + RT - QT

QS + RS + RT − QT = (Q + R)S + (R − Q)T

Since R,Q = ±1, it follows that eitherQS + RS + RT − QT = 0In either case (Q + R)S + (R − Q)T = ±2

Let p(q, r , s, t) is the probability that, before themeasurements 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 ) = Σqrstp(q, r , s, t)(qs + rs + rt − qt)6 Σqrstp(q, r , s, t)x2= 2 .......(1)




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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CHSH Inequality

E(QS + RS + RT − QT ) = Σqrstp(q, r , s, t)qs +Σqrstp(q, r , s, t)rs + Σqrstp(q, r , s, t)rt − Σqrstp(q, r , s, t)qt

= 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 ) 6 2

This result also term as CHSH inequality.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

CHSH Inequality

E(QS + RS + RT − QT ) = Σqrstp(q, r , s, t)qs +Σqrstp(q, r , s, t)rs + Σqrstp(q, r , s, t)rt − Σqrstp(q, r , s, t)qt

= 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 ) 6 2

This result also term as CHSH inequality.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

Nature Does Not follow Bell Inequality

Let Charlie prepares a quantum system of two qubits andshared to Alice and Bob

|ψ〉 = |01〉−|10〉√2

After measurement the following observableQ = Z1 S = −Z2−X2√

2

R = X1 T = Z2−X2√2

〈QS〉 = 1√2

; 〈RS〉 = 1√2

; 〈RT 〉 = 1√2

; 〈QT 〉 = 1√2

Therefore,

〈QS〉+ 〈RS〉+ 〈RT 〉 − 〈QT 〉 = 2√

2




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Hence it is observed that the previous result violates Bellinequality.

Bell inequality is not obeyed by Nature.

This indicates that may be some of the basic assumptionsare incorrect.

Assumptions of local realism

(1) Physical properties PQ ,PR ,PS ,PT have definite valuesQ,R,S ,T which exist independent of observation.

(Assumption of realism)

(2) Alice measurement does not influence the result of Bob‘smeasurement.

(Assumption of locality)




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference













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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Reference

Bipartite System

Suppose we have two parties, Alice and Bob, each having aqubit. Alice’s qubit is denoted as |ψ1〉 = α0 |0〉+ α1 |1〉 andthat of Bob is denoted as |ψ2〉 = β0 |0〉+ β1 |1〉.

So what will be the state of the composite system of Alice andBob?

It is given as -

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)

= α0β0 |00〉1 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

1|0〉 ⊗ |0〉 ≡ |0〉 |0〉 ≡ |00〉Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 15 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bipartite System

Suppose we have two parties, Alice and Bob, each having aqubit. Alice’s qubit is denoted as |ψ1〉 = α0 |0〉+ α1 |1〉 andthat of Bob is denoted as |ψ2〉 = β0 |0〉+ β1 |1〉.

So what will be the state of the composite system of Alice andBob?

It is given as -

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)

= α0β0 |00〉1 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

1|0〉 ⊗ |0〉 ≡ |0〉 |0〉 ≡ |00〉Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 15 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Bipartite System (Contd.)

Similar is the reverse case. If we have a bipartite systemdenoted as

α0β0 |00〉+ α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

then we can factorise it to get the corresponding qubitsinvolved.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Entangled State

A pure state of two systems is entangled if it cannot bewritten as a product of two states -

|ψAB〉 6= |ψA〉 ⊗ |ψB〉

We have four such entangled states called Bell States or EPRStates.

|φ+〉 = 1√2

(|00〉+ |11〉)|φ−〉 = 1√

2(|00〉 − |11〉)

|ψ+〉 = 1√2

(|01〉+ |10〉)|ψ−〉 = 1√

2(|01〉 − |10〉)

These 4 states are called Maximally Entangled States.

Any state of the form a |00〉 ± b |11〉 or a |01〉 ± b |10〉, wherea 6= b, is called Pure Entangled State.




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Why Can’t be Factorised?

Let us consider the state |ψ〉 = 1√2

(|00〉+ |11〉)

And suppose we can factorise the state in the form

(α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉).

So we should have

α0β0 = 1√2

α0β1 = 0α1β0 = 0α1β1 = 1√

2




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Why Can’t be Factorised?

From the 2nd and the 3rd equations we have -

either

α0 = 0 or β1 = 0

AND

either

α1 = 0 or β0 = 0.

But if any two of those have 0 values, then the 1st and the lastequations are not satisfied.

Hence the state cannot be factorised.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Measurement of Entangled bits.

For entangled states, |ψAB〉 6= |ψA〉 ⊗ |ψB〉

Consider the state |ψ〉 = 1√2

(|01〉+ |10〉)

What happens if we measure the 1st qubit only?

Suppose after measurement, we found the 1st qubit to be instate |0〉. Then without measurement, we can knowimmediately that the 2nd qubit is in state |1〉.

So irrespective of the spatial distance between the twoentangled qubits, measuring one of them disturbs the state ofthe other.




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EPR Paradox

Bell’s Theorem


Quantum Gates

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(|01〉+ |10〉)







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EPR Paradox

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(|01〉+ |10〉)







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EPR Paradox

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(|01〉+ |10〉)



So irrespective of the spatial distance between the twoentangled qubits, measuring one of them disturbs the state ofthe other.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 20 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Quantum Gates

“When we get to the very, very small world - say circuits ofseven atoms - we have a lot of new things that would happenthat represent completely new opportunities for design. Atomson small scale behave like nothing on a large scale, for theysatisfy the laws of quantum mechanics. So, as we go down andfiddle around with the atoms there, we are working withdifferent laws, and we can expect to do different things” -

Richard Feynman




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Evolution of Quantum Systems

Quantum systems evolve by Unitary transformations. If |ψ(t1)〉is the state of the system at time t1 and |ψ(t2)〉 is the state ofthe system at time t2, then -

|ψ(t2)〉 = U(t1, t2) |ψ(t1)〉

where U(t1, t2) is a Unitary Matrix.

Analogous to Classical AND, OR, NOT etc. gates, QuantumComputation has its own “Gates” which are represented asUnitary Matrices.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Evolution of Quantum Systems

Quantum systems evolve by Unitary transformations. If |ψ(t1)〉is the state of the system at time t1 and |ψ(t2)〉 is the state ofthe system at time t2, then -

|ψ(t2)〉 = U(t1, t2) |ψ(t1)〉

where U(t1, t2) is a Unitary Matrix.

Analogous to Classical AND, OR, NOT etc. gates, QuantumComputation has its own “Gates” which are represented asUnitary Matrices.




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Classical Gates

Classical Computers consist of wires that carry bits ofinformtion and Gates that transform these bits in some way.

Some of the famous classical logic gates are -




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Irreversibility of Classical Gates

Classical Gates are irreversible 2, i.e., one cannot determineunique inputs for all outputs.

For example, in an AND gate if the output is 0, it cannot bedetermined whether the input values were 00, 01 or 10. Similaris the case for output 1 in OR gate.

Unfortunately, logical irreversibility comes at a price.Fundamental Physics states that energy must be dissipatedwhen information is erased.

And this dissipation is kTln2 per bit erased where k is theBoltzmann constant (k = 1.3805× 10−23JK−1) and T is thetemperature in Absolute Scale.

2The simplest example of a Classical Reversible Logic Gate is a NOTgate.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 24 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Irreversibility of Classical Gates

Classical Gates are irreversible 2, i.e., one cannot determineunique inputs for all outputs.

For example, in an AND gate if the output is 0, it cannot bedetermined whether the input values were 00, 01 or 10. Similaris the case for output 1 in OR gate.

Unfortunately, logical irreversibility comes at a price.Fundamental Physics states that energy must be dissipatedwhen information is erased.

And this dissipation is kTln2 per bit erased where k is theBoltzmann constant (k = 1.3805× 10−23JK−1) and T is thetemperature in Absolute Scale.

2The simplest example of a Classical Reversible Logic Gate is a NOTgate.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 24 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Quantum Logic Gates

Just as any classical computation can be broken into asequence of classical logic gates that act on only a few classicalbits at a time, quantum computation too can be broken downinto a sequence of quantum logic gates that act on only a fewquantum bits at a time.

The main difference is that where classical logic gates act onclassical bits 0 or 1, quantum gates can manipulate arbitrarymulti-partite quantum states including arbitrary superpositions3

of the computational basis states, which may also be entangled.

3Operation of Quantum Gates on a superposition takes same time asthe operation on a basis state.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 25 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Quantum Logic Gates

Just as any classical computation can be broken into asequence of classical logic gates that act on only a few classicalbits at a time, quantum computation too can be broken downinto a sequence of quantum logic gates that act on only a fewquantum bits at a time.

The main difference is that where classical logic gates act onclassical bits 0 or 1, quantum gates can manipulate arbitrarymulti-partite quantum states including arbitrary superpositions3

of the computational basis states, which may also be entangled.

3Operation of Quantum Gates on a superposition takes same time asthe operation on a basis state.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 25 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Properties of Quantum Gates

The properties of quantum logic gates are the directconsequence that they are described by Unitary Matrices. If Uis a Unitary , then the following facts holds -

1. U† is unitary.

2. U−1 is unitary.

3. U−1 = U† (which is the criterion for determining unitarity.)

4. UU† = I

5. |det(U)| = 1

6. The columns (rows) of U form an orthonormal set ofvectors.

The fact that for any quantum gate U, UU† = I , ensures thatwe can always undo a quantum gate, i.e, a quantum gate islogically reversible.




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1. U† is unitary.



4. UU† = I

5. |det(U)| = 1






Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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Notation of Quantum Logic Gates

Just like Classical computation, in quantum we consider aquantum input wire that carries a qubit, a quantum gate thatperforms some transformation on it, and a qauntum outputwire 4 that carries the qubit out.

4It may be noted that unlike classical computation, creating a quantumwire is extremely difficult since a qubit evolves with time according to theSchrodinger Equation.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 27 / 54



Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Bit Flip

The simplest quantum gate is the “bit flip” gate, which isanalogous to the classical NOT gate.

Bit Flip is given by the pauli X matrix -[0 11 0

]The operation of the Bit Flip gate is -

X |0〉 = |1〉X |1〉 = |0〉

Figure : Working of a Quantum Bit Flip gate

It may be checked that X † = X and XX † = I




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Phase Shift

The pauli matrix Z is called the “Phase Shift” gate. Theoperation of this gate is -

Z |0〉 = |0〉Z |1〉 = − |1〉

The matrix notation of the Phase Flip gate is -[1 00 −1

]

Figure : Working of a Quantum Phase Flip gate

It may be checked that Z † = Z and ZZ † = I




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Hadamard Gate

One of the most important single qubit gate is the HadamardGate. The matrix notation of this gate is -[

1√2

1√2

1√2− 1√

2

]= 1√

2

[1 11 −1

]Hadamard Gate switches between the bit basis and the signbasis.

H |0〉 = 1√2|0〉+ 1√

2|1〉 = |+〉

H |1〉 = 1√2|0〉 − 1√

2|1〉 = |−〉

H |+〉 = |0〉H |−〉 = |1〉

Note that, starting with only the state |0〉, Hadamard Trasformcan produce an equal superposition of both |0〉 and |1〉. This isan extremely powerful property and is the source of quantumparallelism.




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Taking Stock

Figure : A relative view of Bit Flip, Phase Flip and Hadamard




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EPR Paradox

Bell’s Theorem


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Geometrical Interpretation

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 arotation about an axis which is at an angle of π/4 from thetwo orthonormal axes |0〉 and |1〉.

Figure : Geometrical interpretation of Bit Flip gate




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Introduction

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The geometrical picture of the 3 single qubit gates discussedearlier -

Figure : The Geometric Interpretation of Bit Flip, Phase Flip andHadamard Gates




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Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Multiple Qubit Gates

“In natural science, Nature has given us a world and we’re justto discover its laws. In computers, we can stuff laws into it andcreate a world.” - Alan Kay

The last few gates were single qubit quantum gates. Now welook into few multiple qubit quantum gates.




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CNOT Gate

A reversible gate of considerable importance in quantumcomputation is the 2-bit Controlled-NOT (CNOT) gate. Theeffect of CNOT gate is to flip the 2nd bit if and only if the 1st

bit is set to 1.

That is, the decision to negate is controlled by the value ofthe 1st bit. Hence the name.

The symbolic representation of CNOT is -




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CNOT Gate

The matrix representation of CNOT gate is -1 0 0 00 1 0 00 0 0 10 0 1 0

The truth table of CNOT gate is given as -

a b a’ b’0 0 0 00 1 0 11 0 1 11 1 1 0

CNOT gate is extremely important because it can createEntanglement.




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Fredkin Gate

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.

The matrix representation of the Fredkin Gate is -

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 1 0 00 0 0 0 0 0 0 1




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Fredkin Gate

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.

The matrix representation of the Fredkin Gate is -

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 1 0 00 0 0 0 0 0 0 1




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Fredkin Gate

Fredkin Gate may be treated like a reversible AND gate.

From the figure, it is clear that if C = 0, then the last outputgives 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.




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Fredkin Gate








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Introduction

EPR Paradox

Bell’s Theorem


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Fredkin Gate








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Toffoli Gate

Toffoli Gate, or the Controlled Controlled NOT (CCNOT)Gate, has 3 input bits and 3 output bits. Two of the bits arecontrol bits that are unaffected by the action of the ToffoliGate. The 3rd bit is the target bit which is flipped if both thecontrol bits are set to 1, otherwise left unchanged.

The matrix representation of Toffoli gate is -

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0




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Toffoli Gate

Toffoli Gate, or the Controlled Controlled NOT (CCNOT)Gate, has 3 input bits and 3 output bits. Two of the bits arecontrol bits that are unaffected by the action of the ToffoliGate. The 3rd bit is the target bit which is flipped if both thecontrol bits are set to 1, otherwise left unchanged.

The matrix representation of Toffoli gate is -

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0




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Toffoli Gate

Figure : Toffoli Gate: a′ = a, b′ = b, c ′ = c ⊕ ab

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, andit can be realised in Quantum Computers by the Toffoli Gate,it can be claimed that -

Any computation that is possible in Classical Computersis also possible in Quantum Computers.




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Toffoli Gate



c ′ = 1⊕ ab = ¬(ab)







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Toffoli Gate



c ′ = 1⊕ ab = ¬(ab)







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Toffoli Gate



c ′ = 1⊕ ab = ¬(ab)







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Super Dense Coding

Breaking news!

A single qubit can transmit two full classical bits ofinformation.




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Quantum Measurement Revisited

From the measurement principle of Quantum Mechanics, weknow 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.

Hence, only one classical bit of information can be stored inone qubit.




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Quantum Measurement Revisited

From the measurement principle of Quantum Mechanics, weknow 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.

Hence, only one classical bit of information can be stored inone qubit.




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What is Superdense Coding?

Superdense Coding 5 is a protocol proposed by Charles H.Bennett and Stephen J. Wiesner.

This is a simple protocol which enables the transportation of 2cbits using only one ebit 6.

5PRL 1992 Vol 69 Number 206Henceforth, we shall be using ”cbit” for classical bit and ”ebit” for

entangled bitRitajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 43 / 54



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Superdense Coding: Initial Setup

The initial step involves preparing an Entangled State. Aliceand Bob prepare a Bell State, say

|φ+〉 = 1√2

(|00〉+ |11〉)

After preparation, Alice keeps one of the two entangled qubitswith herself and sends the other one to Bob.




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How to prepare the Bell State

Before proceeding to the main protocol, the question thatneeds to be asked is -

How to prepare the Bell State?

Consider Alice and Bob both start with one qubit each, both instate |0〉. Alice operates her qubit only with a Hadamard Gate.

This operation is followed by a CNOT Gate on the two qubits.




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EPR Paradox

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EPR Paradox

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Quantum Gates

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Introduction

EPR Paradox

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What happens when Alice operates a Hadamard Gate on herqubit?

H |0〉 = [1√2

1√2

1√2− 1√

2

] [10

]=

[1√2

1√2

]

= 1√2

[11

]= 1√

2

[10

]+ 1√

2

[01

]

= 1√2|0〉+ 1√

2|1〉




Outline

Introduction

EPR Paradox

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H |0〉 = [1√2

1√2

1√2− 1√

2

] [10

]=

[1√2

1√2

]

= 1√2

[11

]= 1√

2

[10

]+ 1√

2

[01

]

= 1√2|0〉+ 1√

2|1〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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H |0〉 = [1√2

1√2

1√2− 1√

2

] [10

]=

[1√2

1√2

]

= 1√2

[11

]

= 1√2

[10

]+ 1√

2

[01

]

= 1√2|0〉+ 1√

2|1〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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H |0〉 = [1√2

1√2

1√2− 1√

2

] [10

]=

[1√2

1√2

]

= 1√2

[11

]= 1√

2

[10

]+ 1√

2

[01

]

= 1√2|0〉+ 1√

2|1〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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H |0〉 = [1√2

1√2

1√2− 1√

2

] [10

]=

[1√2

1√2

]

= 1√2

[11

]= 1√

2

[10

]+ 1√

2

[01

]

= 1√2|0〉+ 1√

2|1〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


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Now we have Alice’s qubit as -

1√2|0〉+ 1√

2|1〉

while Bob’s qubit is |0〉 as before.

So the 2 qubit system takes the form -

1√2|00〉+ 1√

2|10〉

Now applying CNOT Gate, the resultant state is -

1√2

(|00〉+ |11〉)

which is the required Bell State.




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EPR Paradox

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1√2|0〉+ 1√

2|1〉



1√2|00〉+ 1√

2|10〉


1√2

(|00〉+ |11〉)





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EPR Paradox

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1√2|0〉+ 1√

2|1〉



1√2|00〉+ 1√

2|10〉


1√2

(|00〉+ |11〉)





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EPR Paradox

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Super Dense Conding: The Protocol

Alice wants to send 2 cbits of information to Bob using herqubit.

There are 4 possible combinations of the 2 classical bits thatAlice wants to send -

00, 01, 10, 11

Depending on which combination Alice wants to send, sheperforms the following operations on her qubit -




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00, 01, 10, 11





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00, 01, 10, 11





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After performing the required operation, Alice sends her qubitto Bob.

Bob now has both the qubits with him.

Bob performs the initial operations in reverse order, i.e. firstCNOT gate and then Hadamard Gate on the 1st qubit only.

Bob now has the two classical bits that Alice sent her.




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Super Dense Conding: Case Study

Consider Alice wants to send cbit 01 to Bob. Then sheoperates her qubit with the Pauli matrix σx or the bit flip gate.

So the state of the system after operation is:

σx |φ+〉 = 1√2

(|10〉+ |01〉 7)

Alice now sends her qubit to Bob who has the other half of theentangled qubit.

7NOTE: Alice can perform operation on her qubit only. Hence, theoperation affects only the 1st qubit.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 25, 2013 50 / 54



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σx |φ+〉 = 1√2

(|10〉+ |01〉 7)





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σx |φ+〉 = 1√2

(|10〉+ |01〉 7)





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Super Dense Coding: Case Study

Bob now performs CNOT operation on the system1√2

(|10〉+ |01〉). The resultant state will be -

1√2

(|11〉+ |01〉)

Now, the Hadamard Gate is applied on the 1st qubit only. Weknow -

H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)

So the final state will be -

1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




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EPR Paradox

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

Super Dense Coding

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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1√2

(|11〉+ |01〉)


H |0〉 = 1√2

(|0〉+ |1〉)and H |1〉 = 1√

2(|0〉 − |1〉)


1√2

( 1√2

(|0〉 − |1〉) |1〉+ 1√2

(|0〉+ |1〉) |1〉)

= 12 (|01〉 − |11〉+ |01〉+ |11〉)

= 12 (2 |01〉)

= |01〉




Outline

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EPR Paradox

Bell’s Theorem


Quantum Gates

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Coming up in next talk...

No Cloning Theorem

Quantum Teleportation

Conclusive Quantum Teleportation

Quantum Algorithms

and many more...




Outline

Introduction

EPR Paradox

Bell’s Theorem


Quantum Gates

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Reference

Michael A. Nielsen, Isaac ChuangQuantum Computation and Quantum InformationCambridge University Press

David J. GriffithsIntroduction to Quantum MechanicsPrentice Hall, 2nd Edition

Umesh Vazirani, University of California BerkeleyQuantum Mechanics and Quantum Computationhttps://class.coursera.org/qcomp-2012-001/

Michael A. Nielsen, University of QueenslandQuantum Computing for the determinedhttp://michaelnielsen.org/blog/

quantum-computing-for-the-determined/

www.springer.com

Quantum Gates


https://class.coursera.org/qcomp-2012-001/

http://michaelnielsen.org/blog/quantum-computing-for-the-determined/

http://michaelnielsen.org/blog/quantum-computing-for-the-determined/

www.springer.com



Outline

Introduction

EPR Paradox

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Reference

A. Einstein, B. Podolsky, N. Rosen,Physical Review 47, 777 (1935)

J.S. BellPhysics 1, 195 (1964)

D. BohmPhysical Review 85, 166, 180 (1952)

Charles H. Bennett, Stephen J. WiesnerPhysical Review Letters, Nov 16, 1992Volume 69, Number 20


Education

Introduction to Quantum Computation. Part - 2