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Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Introduction to Quantum Computation Part - I Ritajit Majumdar, Arunabha Saha University of Calcutta September 9, 2013 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 1 / 70

Introduction to Quantum Computation. Part - 1

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Introduction to quantum computation. Here the very basic maths described needed for quantum information theory as well as computation. Postulates of quantum mechanics and the Hisenberg`s Uncertainty principle. Basic operator theories are described here.

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Page 1: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Introduction to Quantum ComputationPart - I

Ritajit Majumdar, Arunabha Saha

University of Calcutta

September 9, 2013

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 1 / 70

Page 2: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

1 Introduction

2 Motivations for Quantum Computation

3 Qubit

4 Linear Algebra

5 Uncertainty Principle

6 Postulates of Quantum Mechanics

7 Next Presentation

8 Reference

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 2 / 70

Page 3: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Classical Computation vs QuantumComputation

It may be tempting to say that a quantum computer isone whose operation is governed by the laws of quantummechanics. But since the laws of quantum mechanicsgovern the behaviour of all physical phenomena, thistemptation must be resisted.

Moore’s law roughly stated that computer power willdouble for constant cost approximately once every twoyears. This worked well for a long time. However, atpresent, quantum effects are beginning to interfere in thefunctioning of electronic devices as they are made smallerand smaller.

One possible solution is to move to a different computingparadigm. One such paradigm is provided by the theory ofquantum computation, which is based on the idea of usingquantum mechanics to perform computations, instead ofclassical physics.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70

Page 4: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Classical Computation vs QuantumComputation

It may be tempting to say that a quantum computer isone whose operation is governed by the laws of quantummechanics. But since the laws of quantum mechanicsgovern the behaviour of all physical phenomena, thistemptation must be resisted.

Moore’s law roughly stated that computer power willdouble for constant cost approximately once every twoyears. This worked well for a long time. However, atpresent, quantum effects are beginning to interfere in thefunctioning of electronic devices as they are made smallerand smaller.

One possible solution is to move to a different computingparadigm. One such paradigm is provided by the theory ofquantum computation, which is based on the idea of usingquantum mechanics to perform computations, instead ofclassical physics.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70

Page 5: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Classical Computation vs QuantumComputation

It may be tempting to say that a quantum computer isone whose operation is governed by the laws of quantummechanics. But since the laws of quantum mechanicsgovern the behaviour of all physical phenomena, thistemptation must be resisted.

Moore’s law roughly stated that computer power willdouble for constant cost approximately once every twoyears. This worked well for a long time. However, atpresent, quantum effects are beginning to interfere in thefunctioning of electronic devices as they are made smallerand smaller.

One possible solution is to move to a different computingparadigm. One such paradigm is provided by the theory ofquantum computation, which is based on the idea of usingquantum mechanics to perform computations, instead ofclassical physics.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70

Page 6: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Classical Computation vs QuantumComputation

Quantum systems are exponentially powerful. A system of500 particles has 2500 ”computing power”. QuantumComputers provide a neat shortcut for solving a range ofmathematical tasks known as NP-complete problems.

For example, factorisation is an exponential time task forclassical computers. But Shor’s quantum algorithm forfactorisation is a polynomial time algorithm. It hassuccessfully broken RSA cryptosystem.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 4 / 70

Page 7: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Classical Computation vs QuantumComputation

Quantum systems are exponentially powerful. A system of500 particles has 2500 ”computing power”. QuantumComputers provide a neat shortcut for solving a range ofmathematical tasks known as NP-complete problems.

For example, factorisation is an exponential time task forclassical computers. But Shor’s quantum algorithm forfactorisation is a polynomial time algorithm. It hassuccessfully broken RSA cryptosystem.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 4 / 70

Page 8: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Motivations for Quantum Computation

Faster than light (?) communication.

Highly parallel and efficient quantum algorithms.

Quantum Cryptography.

and many more...

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70

Page 9: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Motivations for Quantum Computation

Faster than light (?) communication.

Highly parallel and efficient quantum algorithms.

Quantum Cryptography.

and many more...

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70

Page 10: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Motivations for Quantum Computation

Faster than light (?) communication.

Highly parallel and efficient quantum algorithms.

Quantum Cryptography.

and many more...

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70

Page 11: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Motivations for Quantum Computation

Faster than light (?) communication.

Highly parallel and efficient quantum algorithms.

Quantum Cryptography.

and many more...

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70

Page 12: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubits: The building blocks of QuantumComputer

In classical computer, bits of digital information are either 0 or1. In a quantum computer, these bits are replaced by a”superposition” of both 0 and 1.

Qubits are represented as |0〉 and |1〉 1. Qubits have beencreated in the laboratory using photons, ions and certain sortsof atomic nuclei.

1or their linear combination.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70

Page 13: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubits: The building blocks of QuantumComputer

In classical computer, bits of digital information are either 0 or1. In a quantum computer, these bits are replaced by a”superposition” of both 0 and 1.

Qubits are represented as |0〉 and |1〉 1. Qubits have beencreated in the laboratory using photons, ions and certain sortsof atomic nuclei.

1or their linear combination.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70

Page 14: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Superposition Principle

Suppose we have a k-level system. So there are kdistinguishable or classical states for the system.The possible classical states for the system: 0, 1, ..., k − 1.

Superposition Principle

If a quantum system can be in one of k states, it can also be inany linear superposition of those k states.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 7 / 70

Page 15: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Superpostition Principle

|0〉, |1〉, ..., |k − 1〉 are called the basis states. Thesuperposition is denoted as a linear combination of these basis.

α0 |0〉+ α1 |1〉+ ...+ αk−1 |k − 1〉

where,

αi ∈ C

∑i

|αi |2 = 1

(more on this later)

Two level systems are called qubits. (k = 2)

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 8 / 70

Page 16: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Physical Interpretation

We may have various interpretations of qubits.

Consider a Hydrogen atom. This atom may be treated asa qubit. To do so, we define the ground energy state ofthe electron as |0〉 and the first energy state as |1〉.

The electron dwells in some linear superposition of thesetwo energy levels. But during measurement, we shall findthe electron in any one of the energy states.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70

Page 17: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Physical Interpretation

We may have various interpretations of qubits.

Consider a Hydrogen atom. This atom may be treated asa qubit. To do so, we define the ground energy state ofthe electron as |0〉 and the first energy state as |1〉.

The electron dwells in some linear superposition of thesetwo energy levels. But during measurement, we shall findthe electron in any one of the energy states.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70

Page 18: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Physical Interpretation

We may have various interpretations of qubits.

Consider a Hydrogen atom. This atom may be treated asa qubit. To do so, we define the ground energy state ofthe electron as |0〉 and the first energy state as |1〉.

The electron dwells in some linear superposition of thesetwo energy levels. But during measurement, we shall findthe electron in any one of the energy states.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70

Page 19: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Other examples of Qubits

Figure: Photon Polarization: The orientation of electrical fieldoscillation is either horizontal or vertical.

Figure: Electron spin: The spin is either up or down

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70

Page 20: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Other examples of Qubits

Figure: Photon Polarization: The orientation of electrical fieldoscillation is either horizontal or vertical.

Figure: Electron spin: The spin is either up or down

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70

Page 21: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Mathematical model

Mathematically, a quantum state (which, as we shall see later,is a vector) is represented by a column matrix. The twofundamental states that we introduced before, |0〉 and |1〉 forman orthonormal basis. We shall see more of orthonormalitywhen we see inner products.

The matrix representation of |0〉 and |1〉:

|0〉 =

[10

]

|1〉 =

[01

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 11 / 70

Page 22: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Mathematical model

So a general quantum state |ψ〉 = α |0〉+ β |1〉 is representedas

The matrix notation will be

|ψ〉 =

[α + 00 + β

]

=

[αβ

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 12 / 70

Page 23: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Mathematical model

So a general quantum state |ψ〉 = α |0〉+ β |1〉 is representedas

The matrix notation will be

|ψ〉 =

[α + 00 + β

]

=

[αβ

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 12 / 70

Page 24: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Sign Basis

|0〉 and |1〉 are called bit basis since they can be thought of asthe quantum counter-parts of classical bits 0 and 1respectively. However, they are not the only possible basis. Wemay have infinitely many orthonormal basis for a given space.

Another basis, called the sign basis, is denoted as |+〉 and |−〉.|+〉 = 1√

2|0〉+ 1√

2|1〉

|−〉 = 1√2|0〉 − 1√

2|1〉

Figure: Geometrical model of bit basis and sign basis

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 13 / 70

Page 25: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Change of Basis

Measure |ψ〉 = 12 |0〉+

√3

2 |1〉 in |+〉/|−〉 basis.

It can be checked that:

|0〉 = 1√2|+〉+ 1√

2|−〉

|1〉 = 1√2|+〉 − 1√

2|−〉

|ψ〉 = 12 |0〉+

√3

2 |1〉= 1

2 ( 1√2|+〉+ 1√

2|−〉) +

√3

2 ( 1√2|+〉+ 1√

2|−〉)

= ( 12√

2+√

32√

2) |+〉+ ( 1

2√

2−√

32√

2) |−〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70

Page 26: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Change of Basis

Measure |ψ〉 = 12 |0〉+

√3

2 |1〉 in |+〉/|−〉 basis.

It can be checked that:

|0〉 = 1√2|+〉+ 1√

2|−〉

|1〉 = 1√2|+〉 − 1√

2|−〉

|ψ〉 = 12 |0〉+

√3

2 |1〉= 1

2 ( 1√2|+〉+ 1√

2|−〉) +

√3

2 ( 1√2|+〉+ 1√

2|−〉)

= ( 12√

2+√

32√

2) |+〉+ ( 1

2√

2−√

32√

2) |−〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70

Page 27: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Change of Basis

Measure |ψ〉 = 12 |0〉+

√3

2 |1〉 in |+〉/|−〉 basis.

It can be checked that:

|0〉 = 1√2|+〉+ 1√

2|−〉

|1〉 = 1√2|+〉 − 1√

2|−〉

|ψ〉 = 12 |0〉+

√3

2 |1〉

= 12 ( 1√

2|+〉+ 1√

2|−〉) +

√3

2 ( 1√2|+〉+ 1√

2|−〉)

= ( 12√

2+√

32√

2) |+〉+ ( 1

2√

2−√

32√

2) |−〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70

Page 28: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Change of Basis

Measure |ψ〉 = 12 |0〉+

√3

2 |1〉 in |+〉/|−〉 basis.

It can be checked that:

|0〉 = 1√2|+〉+ 1√

2|−〉

|1〉 = 1√2|+〉 − 1√

2|−〉

|ψ〉 = 12 |0〉+

√3

2 |1〉= 1

2 ( 1√2|+〉+ 1√

2|−〉) +

√3

2 ( 1√2|+〉+ 1√

2|−〉)

= ( 12√

2+√

32√

2) |+〉+ ( 1

2√

2−√

32√

2) |−〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70

Page 29: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Qubit: Change of Basis

Measure |ψ〉 = 12 |0〉+

√3

2 |1〉 in |+〉/|−〉 basis.

It can be checked that:

|0〉 = 1√2|+〉+ 1√

2|−〉

|1〉 = 1√2|+〉 − 1√

2|−〉

|ψ〉 = 12 |0〉+

√3

2 |1〉= 1

2 ( 1√2|+〉+ 1√

2|−〉) +

√3

2 ( 1√2|+〉+ 1√

2|−〉)

= ( 12√

2+√

32√

2) |+〉+ ( 1

2√

2−√

32√

2) |−〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70

Page 30: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Linear Algebra

Linear Algebraa very short introduction

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 15 / 70

Page 31: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Vector Space

A vector space consists of vectors(|α〉 , |β〉 , |γ〉), togetherwith a set of scalars(a, b, c,....)2, which is closed under twooperations:

Vector addition

Scalar multiplication

2the scalars can be complex numbersRitajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 16 / 70

Page 32: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Vector Addition

The sum of any two vectors is another vector

|α〉+ |β〉 = |γ〉

Vector addition is commutative

|α〉+ |β〉 = |β〉+ |α〉

it is associative also

|α〉+ (|β〉+ |γ〉) = (|α〉+ |β〉) + |γ〉

There exists a zero(or null) vector3 with the property

|α〉+ |0〉 = |α〉 , ∀ |α〉

For every vector |α〉 there is an associative inversevector(|−α〉) such that

|α〉+ |−α〉 = |0〉3|0〉 and 0 are different

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 17 / 70

Page 33: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Scalar Multiplication

The product of any scalar with any vector is another vector

a |α〉 = |γ〉

Scalar multiplication is distributive w.r.t vector addition

a(|α〉+ |β〉) = a |α〉+ a |β〉

and with respect to scalar addition also

(a + b) |α〉 = a |α〉+ b |α〉

It is also associative w.r.t ordinary scalar multiplication

a(b |α〉) = (ab) |α〉

Multiplication by scalars 0 and 1 has the effect

0 |α〉 = |0〉 ; 1 |α〉 = |α〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 18 / 70

Page 34: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Basis Vectors

Linear combination of vectors |α〉, |β〉, |γ〉,... is of the form

|α〉+ |β〉+ |γ〉+ . . .

A vector |λ〉 is said to be linearly independent of the setof vectors |α〉, |β〉, |γ〉,. . . ,if it cannot be written as alinear combination of them.

A set of vectors is linearly independent if each one isindependent of all the rest.

If every vector can be written as a linear combination ofmembers of this set then the collection of vectors said tospan the space.

A set of linearly independent vectors that spans the spaceis called a basis.

The number of vectors in any basis is called thedimension of space.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 19 / 70

Page 35: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Inner product

An inner product is a function which takes two vectors as inputan gives a complex number as output.The dual(or complex conjugate) of any vector |α〉 is 〈α|

|α〉∗ = 〈α|

The inner product of two vectors (|α〉, |β〉) written as 〈α|β〉4which has the properties:

〈α|β〉 = 〈β|α〉∗

〈α|α〉 > 0, and 〈α|α〉 = 0⇔ |α〉 = |0〉

〈α| (b |β〉+ c |γ〉) = b 〈α|β〉+ c 〈α|γ〉

4This is a complex number;〈α|β〉 ∈ CRitajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 20 / 70

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Linear Algebra

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Inner Product Space

A vector space with an inner product is called inner productspace.i.e. the above conditions satisfied for any vectors|α〉, |β〉, |γ〉 ∈ V and for any scalar c

e.g. Cn has an inner product defined by

〈α|β〉 ≡∑i

a∗i bi = [a∗1 . . . a∗n]

b1

...bn

where |α〉 =

a1

...an

and |β〉 =

b1

...bn

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 21 / 70

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Motivations for QuantumComputation

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Linear Algebra

Uncertainty Principle

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Orthonormal Set

Inner product of any vector with itself gives a non-negativenumber — its square-root of is real which is called norm

‖ α ‖=√〈α|α〉

It also termed as length of the vector.A unit vector one whose norm is 1 is said to be normalized5.

Two vectors whose inner product is zero is said to beorthogonal

〈α|α〉 = 0

A mutually collection of orthogonal normalized vectors iscalled an orthonormal set

〈αi |αj〉 = δij , where δij

{= 0, i = j

6= 0, i 6= j

5Normalization: |ek 〉 = |k〉‖k‖

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 22 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Orthonormal Set(Contd.)

If an orthonormal basis is chosen then the inner product oftwo vectors can be written as

〈α|β〉 = a∗1b1 + a∗2b2 + . . .+ a∗nbn

hence the norm(squared)

〈α|α〉 = |a1|2 + |a2|2 + . . .+ |an|2

each components are

aj = 〈ej |α〉 , where ej ‘s are basis vector

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 23 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Linear Operator and Matrices

A linear operator between vector spaces V and W isdefined to be any function A : V → W which is linear inits inputs

A

(∑i

ai |υi 〉

)=∑i

aiA(|υi 〉)

another linear operator on any vector space V is Identityoperator, Iv defined as Iv |υ〉 ≡ |υ〉.Zero operator which maps all vectors to zero vector,0 |υ〉 ≡ |0〉.Let V, W, X are vector spaces, and A : V →W andB : W → X are linear operators. Then the composition ofoperators B and A denoted by BA and defined by(BA)(|υ〉) ≡ B(A(|υ〉))

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Linear Operator and Matrices

A linear operator between vector spaces V and W isdefined to be any function A : V → W which is linear inits inputs

A

(∑i

ai |υi 〉

)=∑i

aiA(|υi 〉)

another linear operator on any vector space V is Identityoperator, Iv defined as Iv |υ〉 ≡ |υ〉.

Zero operator which maps all vectors to zero vector,0 |υ〉 ≡ |0〉.Let V, W, X are vector spaces, and A : V →W andB : W → X are linear operators. Then the composition ofoperators B and A denoted by BA and defined by(BA)(|υ〉) ≡ B(A(|υ〉))

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Linear Operator and Matrices

A linear operator between vector spaces V and W isdefined to be any function A : V → W which is linear inits inputs

A

(∑i

ai |υi 〉

)=∑i

aiA(|υi 〉)

another linear operator on any vector space V is Identityoperator, Iv defined as Iv |υ〉 ≡ |υ〉.Zero operator which maps all vectors to zero vector,0 |υ〉 ≡ |0〉.

Let V, W, X are vector spaces, and A : V →W andB : W → X are linear operators. Then the composition ofoperators B and A denoted by BA and defined by(BA)(|υ〉) ≡ B(A(|υ〉))

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Linear Operator and Matrices

A linear operator between vector spaces V and W isdefined to be any function A : V → W which is linear inits inputs

A

(∑i

ai |υi 〉

)=∑i

aiA(|υi 〉)

another linear operator on any vector space V is Identityoperator, Iv defined as Iv |υ〉 ≡ |υ〉.Zero operator which maps all vectors to zero vector,0 |υ〉 ≡ |0〉.Let V, W, X are vector spaces, and A : V →W andB : W → X are linear operators. Then the composition ofoperators B and A denoted by BA and defined by(BA)(|υ〉) ≡ B(A(|υ〉))

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Matrix Representation of Linear Operators

already we have seen that matrices can be regarded aslinear operators..!!

does linear operators has a matrix representation..??!

yes it has..!!

say, A : V →W is a linear operator between vector spacesV and W. Let the basis set for V and W are(|υ1〉 , . . . , |υm〉) and (|ω1〉 , . . . , |ωn〉) respectively. Thenwe can say For each k in 1,2,....,m, there exist complexnumbers A1k through Ank such that

A |υk〉 =∑i

Aik |ωi 〉

The matrix whose entries are Aik is the matrixrepresentation of the linear operator.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Matrix Representation of Linear Operators

already we have seen that matrices can be regarded aslinear operators..!!

does linear operators has a matrix representation..??!

yes it has..!!

say, A : V →W is a linear operator between vector spacesV and W. Let the basis set for V and W are(|υ1〉 , . . . , |υm〉) and (|ω1〉 , . . . , |ωn〉) respectively. Thenwe can say For each k in 1,2,....,m, there exist complexnumbers A1k through Ank such that

A |υk〉 =∑i

Aik |ωi 〉

The matrix whose entries are Aik is the matrixrepresentation of the linear operator.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Matrix Representation of Linear Operators

already we have seen that matrices can be regarded aslinear operators..!!

does linear operators has a matrix representation..??!

yes it has..!!

say, A : V →W is a linear operator between vector spacesV and W. Let the basis set for V and W are(|υ1〉 , . . . , |υm〉) and (|ω1〉 , . . . , |ωn〉) respectively. Thenwe can say For each k in 1,2,....,m, there exist complexnumbers A1k through Ank such that

A |υk〉 =∑i

Aik |ωi 〉

The matrix whose entries are Aik is the matrixrepresentation of the linear operator.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Matrix Representation of Linear Operators

already we have seen that matrices can be regarded aslinear operators..!!

does linear operators has a matrix representation..??!

yes it has..!!

say, A : V →W is a linear operator between vector spacesV and W. Let the basis set for V and W are(|υ1〉 , . . . , |υm〉) and (|ω1〉 , . . . , |ωn〉) respectively. Thenwe can say For each k in 1,2,....,m, there exist complexnumbers A1k through Ank such that

A |υk〉 =∑i

Aik |ωi 〉

The matrix whose entries are Aik is the matrixrepresentation of the linear operator.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Identity Matrix

The exotic way to express 1

DefinitionIdentity matrix is one with the main diagonals and zeroseverywhere. it is denoted by In or I

Matrix representation of Identity operator1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

It satisfies the property InA = AIn = A

compact notation: (In)ij = δij

It satisfies Idempotent law, I.I = I

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Identity Matrix

The exotic way to express 1

DefinitionIdentity matrix is one with the main diagonals and zeroseverywhere. it is denoted by In or I

Matrix representation of Identity operator1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

It satisfies the property InA = AIn = A

compact notation: (In)ij = δij

It satisfies Idempotent law, I.I = I

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Identity Matrix

The exotic way to express 1

DefinitionIdentity matrix is one with the main diagonals and zeroseverywhere. it is denoted by In or I

Matrix representation of Identity operator1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

It satisfies the property InA = AIn = A

compact notation: (In)ij = δij

It satisfies Idempotent law, I.I = I

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Identity Matrix

The exotic way to express 1

DefinitionIdentity matrix is one with the main diagonals and zeroseverywhere. it is denoted by In or I

Matrix representation of Identity operator1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

It satisfies the property InA = AIn = A

compact notation: (In)ij = δij

It satisfies Idempotent law, I.I = I

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Pauli Matrices

Pauli matrices are named after the physicist WolfgangPauli.

These are a set of 2 x 2 complex matrices which areHermitian and unitary.

They look like

σ0 ≡ I ≡[

1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Pauli Matrices

Pauli matrices are named after the physicist WolfgangPauli.

These are a set of 2 x 2 complex matrices which areHermitian and unitary.

They look like

σ0 ≡ I ≡[

1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Pauli Matrices

Pauli matrices are named after the physicist WolfgangPauli.

These are a set of 2 x 2 complex matrices which areHermitian and unitary.

They look like

σ0 ≡ I ≡[

1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

]

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Pauli Matrices(properties)

Some properties of Pauli matrices

σ21 = σ2

2 = σ23 = −iσ1σ2σ3 =

(1 00 1

)= I

det(σi ) = −1

Tr(σi ) = 0

Each Pauli matrices has two eigenvalues +1 and -1.

Normalized eigenvectors are

ψx+ = 1√2

(11

), ψx− = 1√

2

(1−1

)ψy+ = 1√

2

(1i

), ψy− = 1√

2

(1−i

)ψz+ =

(10

), ψz− =

(01

)

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 28 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Pauli Matrices and Quantum Computation

Pauli matrices are used here as rotation6 operators.

On the basis of Pauli matrices the X, Y, Z quantum gates7 aredesigned.

The Pauli-X gate is the quantum equivalent of NOTgate. It maps |0〉 to |1〉 and |1〉 to |0〉.

6rotation of Bloch sphere7all acts on single qubit

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 29 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Pauli Matrices and Quantum Computation

Pauli-Y gate maps |0〉 to ı̇ |1〉 and |1〉 to −ı̇ |0〉.

Pauli-Z gate leaves the basis state |0〉 unchanged andmaps |1〉 to -|1〉.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 30 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Pauli Matrices and Quantum Computation

Pauli-Y gate maps |0〉 to ı̇ |1〉 and |1〉 to −ı̇ |0〉.

Pauli-Z gate leaves the basis state |0〉 unchanged andmaps |1〉 to -|1〉.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 30 / 70

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Introduction to QuantumComputation

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Hilbert Space

Hilbert Space

Wave functions live in Hilbert space

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Hilbert Space: Few Basics

The mathematical concept of Hilbert space named afterDavid Hilbert, but this term coined by John von Neumann.

Basically this is the generalization of the notion ofEuclidean space i.e. it extends the methods of algebraand calculus of 2D Euclidean plane and 3D space to spaceof any finite or infinite dimensions.

Hilbert space is an abstract vector space with innerproduct defined in it, which allows length and angle to bemeasured.

Hilbert space must be complete.8

8any Cauchy sequence of functions in Hilbert space converges to afunction that is also in the space.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Hilbert Space: Few Basics

The mathematical concept of Hilbert space named afterDavid Hilbert, but this term coined by John von Neumann.

Basically this is the generalization of the notion ofEuclidean space i.e. it extends the methods of algebraand calculus of 2D Euclidean plane and 3D space to spaceof any finite or infinite dimensions.

Hilbert space is an abstract vector space with innerproduct defined in it, which allows length and angle to bemeasured.

Hilbert space must be complete.8

8any Cauchy sequence of functions in Hilbert space converges to afunction that is also in the space.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Hilbert Space: Few Basics

The mathematical concept of Hilbert space named afterDavid Hilbert, but this term coined by John von Neumann.

Basically this is the generalization of the notion ofEuclidean space i.e. it extends the methods of algebraand calculus of 2D Euclidean plane and 3D space to spaceof any finite or infinite dimensions.

Hilbert space is an abstract vector space with innerproduct defined in it, which allows length and angle to bemeasured.

Hilbert space must be complete.8

8any Cauchy sequence of functions in Hilbert space converges to afunction that is also in the space.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70

Page 62: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Hilbert Space: Few Basics

The mathematical concept of Hilbert space named afterDavid Hilbert, but this term coined by John von Neumann.

Basically this is the generalization of the notion ofEuclidean space i.e. it extends the methods of algebraand calculus of 2D Euclidean plane and 3D space to spaceof any finite or infinite dimensions.

Hilbert space is an abstract vector space with innerproduct defined in it, which allows length and angle to bemeasured.

Hilbert space must be complete.8

8any Cauchy sequence of functions in Hilbert space converges to afunction that is also in the space.Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Hilbert Space: Formal Approach

The set of all functions of x constitute a vector space. Torepresent a possible physical state,the wave function needed tobe normalized ∫

|ψ|2dx ≡ 〈ψ|ψ〉 = 1

The set of all square-integrable functions on a specifiedinterval,9

f (x) such that∫ b

a|f (x)|2dx <∞,

constitutes a smaller vector space. It is known tomathematician as L2(a, b); physicists call it Hilbert space.

DefinitionA Euclidean space Rn is a vector space endowed with the innerproduct 〈x |y〉 = 〈y |x〉∗ norm ‖ x ‖=

√〈x |x〉 and associated metric

‖ x − y ‖, such that every Cauchy sequence takes a limit in Rn.This makes Rn a Hilbert space.

9The limits(a and b) can be ±∞Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 33 / 70

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Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Observables

Observables

woow..! It looks good..!!

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 34 / 70

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

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Reference

Observables

A system observable is a measurable operator, where theproperty of the system state can be determined by somesequence of physical operations.

In quantum mechanics the measurement process affectsthe state in a non-deterministic, but in a statisticallypredictable way. In particular, after a measurement isapplied, the state description by a single vector may bedestroyed, being replaced by a statistical ensemble.

In quantum mechanics each dynamical variable (e.g.position, translational momentum, orbital angularmomentum, spin, total angular momentum, energy, etc.)is associated with a Hermitian operator that acts on thestate of the quantum system and whose eigenvaluescorrespond to the possible values of the dynamicalvariable.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70

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Introduction to QuantumComputation

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Observables

A system observable is a measurable operator, where theproperty of the system state can be determined by somesequence of physical operations.

In quantum mechanics the measurement process affectsthe state in a non-deterministic, but in a statisticallypredictable way. In particular, after a measurement isapplied, the state description by a single vector may bedestroyed, being replaced by a statistical ensemble.

In quantum mechanics each dynamical variable (e.g.position, translational momentum, orbital angularmomentum, spin, total angular momentum, energy, etc.)is associated with a Hermitian operator that acts on thestate of the quantum system and whose eigenvaluescorrespond to the possible values of the dynamicalvariable.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70

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Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

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Observables

A system observable is a measurable operator, where theproperty of the system state can be determined by somesequence of physical operations.

In quantum mechanics the measurement process affectsthe state in a non-deterministic, but in a statisticallypredictable way. In particular, after a measurement isapplied, the state description by a single vector may bedestroyed, being replaced by a statistical ensemble.

In quantum mechanics each dynamical variable (e.g.position, translational momentum, orbital angularmomentum, spin, total angular momentum, energy, etc.)is associated with a Hermitian operator that acts on thestate of the quantum system and whose eigenvaluescorrespond to the possible values of the dynamicalvariable.

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Observables

e.g. let |α〉 is an eigenvector of the observable A, witheigenvalue a and exits in a d-dimensional Hilbert space,then

A |α〉 = a |α〉

This equation states that if a measurement of theobservable A is made while the system of interest is instate |α〉, then the observed value of the particularmeasurement must return the eigenvalue a with certainty.If the system is in the general state |φ〉 ∈ H then theeigenvalue a return with probability | 〈α|φ〉 |2(Born rule).

More precisely, the observables are Hermitian operatorso its represented by Hermitian matrix.

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Observables

e.g. let |α〉 is an eigenvector of the observable A, witheigenvalue a and exits in a d-dimensional Hilbert space,then

A |α〉 = a |α〉This equation states that if a measurement of theobservable A is made while the system of interest is instate |α〉, then the observed value of the particularmeasurement must return the eigenvalue a with certainty.If the system is in the general state |φ〉 ∈ H then theeigenvalue a return with probability | 〈α|φ〉 |2(Born rule).

More precisely, the observables are Hermitian operatorso its represented by Hermitian matrix.

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Observables

e.g. let |α〉 is an eigenvector of the observable A, witheigenvalue a and exits in a d-dimensional Hilbert space,then

A |α〉 = a |α〉This equation states that if a measurement of theobservable A is made while the system of interest is instate |α〉, then the observed value of the particularmeasurement must return the eigenvalue a with certainty.If the system is in the general state |φ〉 ∈ H then theeigenvalue a return with probability | 〈α|φ〉 |2(Born rule).

More precisely, the observables are Hermitian operatorso its represented by Hermitian matrix.

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Hermitian Operator

Hermitian

hmmm...sounds like me!! is it a new breed ??!!

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Hermitian Operator

No, its nothing new. Its the self-adjoint operator.

DefinitionHermitian matrix is a square matrix with complex entries thatis equal to its own conjugate transpose i.e. the element inthe i-th row and j-th column is equal to the complex conjugateof the element in the j-th row and i-th column, for all indices iand j.

mathematically aij = a∗ji or in matrix notation, A = (AT )∗

In compact notation, A = A†

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Hermitian Operators: Expectation Value

As previously said that the measurements arenon-deterministic, so we can get a probabilistic measure of anyobservable, that is known as expectation value

〈Q̂〉 =∫ψ∗Q̂ψ =

⟨ψ∣∣∣Q̂ψ⟩

Operators representing observables have the very specialproperty that,⟨

f∣∣∣Q̂f

⟩=⟨Q̂f∣∣∣f ⟩ ∀f (x)

More strong condition for hermiticity,⟨f∣∣∣Q̂g

⟩=⟨Q̂f∣∣∣g⟩ ∀f (x) and g(x)

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Hermitian Operators:Properties

Eigenvalues of Hermitian operators are real.

Eigenfunctions belonging to distinct eigenvalues areorthogonal.

The eigenfunctions of a Hermitian operator is complete.Any function(in Hilbert space) can be expressed as thelinear combination of them.

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Hermitian Operators:Properties

Eigenvalues of Hermitian operators are real.

Eigenfunctions belonging to distinct eigenvalues areorthogonal.

The eigenfunctions of a Hermitian operator is complete.Any function(in Hilbert space) can be expressed as thelinear combination of them.

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Hermitian Operators:Properties

Eigenvalues of Hermitian operators are real.

Eigenfunctions belonging to distinct eigenvalues areorthogonal.

The eigenfunctions of a Hermitian operator is complete.Any function(in Hilbert space) can be expressed as thelinear combination of them.

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Uncertainty Principle

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Wave Particle Duality

According to de Broglie hypothesis:

p = hλ = 2π~

λ

where p is the momentum, λ is the wavelength and h is calledPlank’s constant. It has a value of 6.63× 10−34 Joule-sec.~ = h

2π is called the reduced Plank constant.

This formula essentially states that every particle has a wavenature and vice versa.

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Wave Particle Duality

According to de Broglie hypothesis:

p = hλ = 2π~

λ

where p is the momentum, λ is the wavelength and h is calledPlank’s constant. It has a value of 6.63× 10−34 Joule-sec.~ = h

2π is called the reduced Plank constant.

This formula essentially states that every particle has a wavenature and vice versa.

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Heisenberg Uncertainty Principle

The Uncertainty Principle is a direct consequence of the waveparticle duality. The wavelength of a wave is well defined, whileasking for its position is absurd. Vice versa is the case for aparticle. And from de Broglie hypothesis, we get thatmomentum is inversely proportional to wavelength. Since everysubstance has both wave and particle nature -

Uncertainty Principle

One can never know with perfect accuracy both the positionand the momentum of a particle.

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Uncertainty Principle: Mathematical Notation

According to Heisenberg, the uncertainty in the position andmomentum of a substance must be at least as big as ~

2 . So wecan write the mathematical notation of the uncertaintyprinciple:

δx .δp ≥ ~2

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Introduction to Quantum Mechanics

’Not only is the Universe stranger than wethink, it is stranger than we can think’

- Warner Heisenberg

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Quantum Mechanics

By the late nineteenth century the laws of physics werebased on Mechanics and laws of Gravitation from Newton,Maxwell’s equations describing Electricity and Magnetismand on Statistical Mechanics describing the state of largecollection of matter.

These laws of physics described nature very well undermost conditions. However, some experiments of the late19th and early 20th century could not be explained.

The problems with classical physics led to the developmentof Quantum Mechanics and Special Relativity.

Some of the problems leading to the development ofQuantum Mechanics are

I Black Body RadiationI Photoelectric EffectI Double Slit ExperimentI Compton Scattering

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Quantum Mechanics

By the late nineteenth century the laws of physics werebased on Mechanics and laws of Gravitation from Newton,Maxwell’s equations describing Electricity and Magnetismand on Statistical Mechanics describing the state of largecollection of matter.

These laws of physics described nature very well undermost conditions. However, some experiments of the late19th and early 20th century could not be explained.

The problems with classical physics led to the developmentof Quantum Mechanics and Special Relativity.

Some of the problems leading to the development ofQuantum Mechanics are

I Black Body RadiationI Photoelectric EffectI Double Slit ExperimentI Compton Scattering

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Quantum Mechanics

By the late nineteenth century the laws of physics werebased on Mechanics and laws of Gravitation from Newton,Maxwell’s equations describing Electricity and Magnetismand on Statistical Mechanics describing the state of largecollection of matter.

These laws of physics described nature very well undermost conditions. However, some experiments of the late19th and early 20th century could not be explained.

The problems with classical physics led to the developmentof Quantum Mechanics and Special Relativity.

Some of the problems leading to the development ofQuantum Mechanics are

I Black Body RadiationI Photoelectric EffectI Double Slit ExperimentI Compton Scattering

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Introduction

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Quantum Mechanics

By the late nineteenth century the laws of physics werebased on Mechanics and laws of Gravitation from Newton,Maxwell’s equations describing Electricity and Magnetismand on Statistical Mechanics describing the state of largecollection of matter.

These laws of physics described nature very well undermost conditions. However, some experiments of the late19th and early 20th century could not be explained.

The problems with classical physics led to the developmentof Quantum Mechanics and Special Relativity.

Some of the problems leading to the development ofQuantum Mechanics are

I Black Body RadiationI Photoelectric EffectI Double Slit ExperimentI Compton Scattering

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Double Slit Experiment: Setup

This experiment shows an aberrant result which cannot beexplained using classical laws of physics.

The experiment setup consists of a monochromatic source oflight and two extremely small slits, big enough for only onephoton particle to pass through it.

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Double Slit Experiment: Setup

This experiment shows an aberrant result which cannot beexplained using classical laws of physics.

The experiment setup consists of a monochromatic source oflight and two extremely small slits, big enough for only onephoton particle to pass through it.

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Double Slit Experiment: One Slit Open

If initially only one slit is open, we get a probability distributionas shown in figure.

So if the two slits are opened individually, the two distinctprobability distributations are obtained in the screen.

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Double Slit Experiment: The ClassicalExpectation

Since the opening of two slits individually are independentevents, classically we expect that if the two slits are openedtogether, the two probability distributions should add up givingthe new probability distribution.

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Double Slit Experiment: The Anomaly

What we observe in reality when both slits are openedtogether, is an interference pattern.

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Double Slit Experiment: Quantum Explanation

There is no classical explanation to this observation.

However, using quantum mechanics, it can be explained.

It is the wave particle duality of light that is responsiblefor such aberrant observation. The wave nature of light isresponsible for the interference pattern observed.

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Double Slit Experiment: Quantum Explanation

There is no classical explanation to this observation.

However, using quantum mechanics, it can be explained.

It is the wave particle duality of light that is responsiblefor such aberrant observation. The wave nature of light isresponsible for the interference pattern observed.

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Double Slit Experiment: Quantum Explanation

There is no classical explanation to this observation.

However, using quantum mechanics, it can be explained.

It is the wave particle duality of light that is responsiblefor such aberrant observation. The wave nature of light isresponsible for the interference pattern observed.

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Double Slit Experiment: Complete Picture

Figure: Double Slit Experiment showing the anomaly - deviation ofthe observed result from the one predicted by Classical Physics.

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Nobody understands Quantum Mechanics

Quantum mechanics is a very counter-intuitive theory.The results of quantum mechanics is nothing like what weexperience in everyday life. It is just that nature behavesvery strangely at the level of elementary particles. Andthis strange way is described by quantum mechanics.

In fact, so strange is the theory that Richard Feynmanonce said

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Nobody understands Quantum Mechanics

Quantum mechanics is a very counter-intuitive theory.The results of quantum mechanics is nothing like what weexperience in everyday life. It is just that nature behavesvery strangely at the level of elementary particles. Andthis strange way is described by quantum mechanics.

In fact, so strange is the theory that Richard Feynmanonce said

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1st Postulate: State Space

Postulate 1Associated to any isolated system is a Hilbert Space called thestate space. The system is completely defined by the statevector, which is a unit vector in the state space.

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State Vector: Mathematical Realisation

Let us consider a quantum state |ψ〉 = α |0〉+ β |1〉

Since a state is a unit vector, the norm of the vector must beunity, or in mathematical notation,

〈ψ|ψ〉 = 1

Hence, the condition that |ψ〉 is a unit vector is equivalent to

|α|2 + |β|2 = 1

This condition is called the normalization condition of the statevector.

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State: Why unit vector?

Let |ψ〉 = α0 |0〉+ α1 |1〉+ . . .+ αk−1 |k − 1〉 be a quantumstate. As we shall see later, the superposition is not observable.When a state is observed, it collapses into one of the basisstates.

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State: Why unit vector?

The square of the amplitude |αi |2 gives the probability that thesystem collapses to the state |i〉.

Since the total probability is always 1, we must have:∑|αi |2 = 1

This condition is satisfied only when the state vector is a unitvector.

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2nd Postulate: Evolution

Postulate 2The evolution of a closed quantum system is described by aunitary transformation.

That is, if |ψ1〉 is the state of the system at time t1 and |ψ2〉 attime t2, then:

|ψ2〉 = U(t1, t2) |ψ1〉where U(t1, t2) is a unitary operator.

Figure: Quantum systems evolve by the rotation of the HilbertSpace

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2nd Postulate: Evolution

Postulate 2The evolution of a closed quantum system is described by aunitary transformation.

That is, if |ψ1〉 is the state of the system at time t1 and |ψ2〉 attime t2, then:

|ψ2〉 = U(t1, t2) |ψ1〉where U(t1, t2) is a unitary operator.

Figure: Quantum systems evolve by the rotation of the HilbertSpace

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Time Evolution: Schrodinger Equation

The time evolution of a closed quantum system is given by theSchrodinger Equation:

i~ d|ψ〉dt = H |ψ〉

where H is the hamiltonian and it is defined as the total energy(kinetic + potential) of the system.

The connection between the hamitonian picture and theunitary operator picture is given by:

|ψ2〉 = exp −iH(t2−t1)~ |ψ1〉 = U(t1, t2) |ψ1〉

where we define, U(t1, t2) ≡ exp −iH(t2−t1)~

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Time Evolution: Schrodinger Equation

The time evolution of a closed quantum system is given by theSchrodinger Equation:

i~ d|ψ〉dt = H |ψ〉

where H is the hamiltonian and it is defined as the total energy(kinetic + potential) of the system.

The connection between the hamitonian picture and theunitary operator picture is given by:

|ψ2〉 = exp −iH(t2−t1)~ |ψ1〉 = U(t1, t2) |ψ1〉

where we define, U(t1, t2) ≡ exp −iH(t2−t1)~

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Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Attempt at 3rd Postulate

Unlike classical physics, measurement in quantummechanics is not deterministic. Even if we have thecomplete knowledge of a system, we can at most predictthe probability of a certain outcome from a set of possibleoutcomes.

If we have a quantum state |ψ〉 = α |0〉+ β |1〉, then theprobability of getting outcome |0〉 is |α|2 and that of |1〉 is|β|2.

After measurement, the state of the system collapses toeither |0〉 or |1〉 with the said probability.

However, after measurement if the new state of thesystem is |0〉 (say), then further measurements in thesame basis gives outcome |0〉 with probability 1.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70

Page 107: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Attempt at 3rd Postulate

Unlike classical physics, measurement in quantummechanics is not deterministic. Even if we have thecomplete knowledge of a system, we can at most predictthe probability of a certain outcome from a set of possibleoutcomes.

If we have a quantum state |ψ〉 = α |0〉+ β |1〉, then theprobability of getting outcome |0〉 is |α|2 and that of |1〉 is|β|2.

After measurement, the state of the system collapses toeither |0〉 or |1〉 with the said probability.

However, after measurement if the new state of thesystem is |0〉 (say), then further measurements in thesame basis gives outcome |0〉 with probability 1.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70

Page 108: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Attempt at 3rd Postulate

Unlike classical physics, measurement in quantummechanics is not deterministic. Even if we have thecomplete knowledge of a system, we can at most predictthe probability of a certain outcome from a set of possibleoutcomes.

If we have a quantum state |ψ〉 = α |0〉+ β |1〉, then theprobability of getting outcome |0〉 is |α|2 and that of |1〉 is|β|2.

After measurement, the state of the system collapses toeither |0〉 or |1〉 with the said probability.

However, after measurement if the new state of thesystem is |0〉 (say), then further measurements in thesame basis gives outcome |0〉 with probability 1.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70

Page 109: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Attempt at 3rd Postulate

Unlike classical physics, measurement in quantummechanics is not deterministic. Even if we have thecomplete knowledge of a system, we can at most predictthe probability of a certain outcome from a set of possibleoutcomes.

If we have a quantum state |ψ〉 = α |0〉+ β |1〉, then theprobability of getting outcome |0〉 is |α|2 and that of |1〉 is|β|2.

After measurement, the state of the system collapses toeither |0〉 or |1〉 with the said probability.

However, after measurement if the new state of thesystem is |0〉 (say), then further measurements in thesame basis gives outcome |0〉 with probability 1.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70

Page 110: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

3rd Postulate: Measurement

Postulate 3

Quantum measurements are described by a collection {Mm} ofmeasurement operators. The index m refers to themeasurement outcomes that may occur in the experiment. Ifthe state of the quantum system is |ψ〉 before experiment, thenthe probability that result m occurs is given by,

p(m) = 〈ψ|Mm † .Mm |ψ〉

And the state of the system after measurement is

Mm|ψ〉√〈ψ|Mm†.Mm|ψ〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 61 / 70

Page 111: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

So what is the Big Deal?

The postulate states that a quantum system stays in asuperposition when it is not observed. When a measurement isdone, it immediately collapses to one of its eigenstates. Hencewe can never observe what the original superposition of thesystem was. We merely observe the state after it collapses.

This inherent ambiguity provides an excellent security inQuantum Cryptography.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 62 / 70

Page 112: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Schrodinger’s Cat

This is a thought experiment proposed by ErwinSchrodinger.

Place a cat in a steel chamber with a device containing avial of hydrocyanic acid and a radioactive substance. Ifeven a single atom of the substance decays, it will trip ahammer and break the vial which in turn kills the cat.

Without opening the box, an observer cannot knowwhether the cat is alive or dead. So the cat may be saidto be in a superposition of the two states.

However, when the box is opened, we observedeterministically that the cat is either dead or alive. Wecan, by no means, observe the superposition.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70

Page 113: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Schrodinger’s Cat

This is a thought experiment proposed by ErwinSchrodinger.

Place a cat in a steel chamber with a device containing avial of hydrocyanic acid and a radioactive substance. Ifeven a single atom of the substance decays, it will trip ahammer and break the vial which in turn kills the cat.

Without opening the box, an observer cannot knowwhether the cat is alive or dead. So the cat may be saidto be in a superposition of the two states.

However, when the box is opened, we observedeterministically that the cat is either dead or alive. Wecan, by no means, observe the superposition.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70

Page 114: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Schrodinger’s Cat

This is a thought experiment proposed by ErwinSchrodinger.

Place a cat in a steel chamber with a device containing avial of hydrocyanic acid and a radioactive substance. Ifeven a single atom of the substance decays, it will trip ahammer and break the vial which in turn kills the cat.

Without opening the box, an observer cannot knowwhether the cat is alive or dead. So the cat may be saidto be in a superposition of the two states.

However, when the box is opened, we observedeterministically that the cat is either dead or alive. Wecan, by no means, observe the superposition.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70

Page 115: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Schrodinger’s Cat

This is a thought experiment proposed by ErwinSchrodinger.

Place a cat in a steel chamber with a device containing avial of hydrocyanic acid and a radioactive substance. Ifeven a single atom of the substance decays, it will trip ahammer and break the vial which in turn kills the cat.

Without opening the box, an observer cannot knowwhether the cat is alive or dead. So the cat may be saidto be in a superposition of the two states.

However, when the box is opened, we observedeterministically that the cat is either dead or alive. Wecan, by no means, observe the superposition.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70

Page 116: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

4th Postulate: Composite System

What we have seen so far was a single qubit system. Whathappens when there are multiple qubits? This is given by thelast postulate:

Postulate 4The state space of a composite physical system is the tensorproduct of the state spaces of the component physical systems.Moreover, if we have n systems, and the system number i isprepared in state |ψi 〉, then the joint state of the total system is

|ψ1〉 ⊗ |ψ2〉 ⊗ ...⊗ |ψn〉

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 64 / 70

Page 117: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Two Qubit System

Let us consider a two qubit system. Classically, with two bits,we can have 4 states - 00, 01, 10, 11. A quantum system is alinear superposition of all these four states.

So, a general two qubit quantum state can be represented as

|ψ〉 = α00 |00〉+ α01 |01〉+ α10 |10〉+ α11 |11〉 10

where,

|α00|2 + |α01|2 + |α10|2 + |α11|2 = 1

10|0〉 ⊗ |0〉 ≡ |0〉 |0〉 ≡ |00〉Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 65 / 70

Page 118: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Measurement in Two Qubit System

Measurement is similar to single qubit system. When wemeasure the two qubit system we get outcome j withprobability |αj |2 and the new state will be |j〉.

That is for the system

|ψ〉 = α00 |00〉+ α01 |01〉+ α10 |10〉+ α11 |11〉,

we get outcome |00〉 with probability |α00|2 and the new stateof the system will be |00〉.

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 66 / 70

Page 119: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Partial Measurement

So what if we want to measure only the first qubit? Or maybeonly the second one?

We take the same two qubit system,

|ψ〉 = α00 |00〉+ α01 |01〉+ α10 |10〉+ α11 |11〉,

If only the first qubit is measured then we get the outcome |0〉for the first qubit with probability |α00|2 + |α01|2

and the state collapses to

|φ〉 = α00|00〉+α01|01〉√|α00|2+|α01|2

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 67 / 70

Page 120: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Coming up in next talk...

Einstein-Polosky-Rosen (EPR) Paradox

Bell State

Quantum Entanglement

Density Matrix Notation of Quantum Mechanics

Quantum Gates

and many more...

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 68 / 70

Page 121: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

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/

James Branson, University of California San DiegoQuantum Physics (UCSD Physics 130)http://quantummechanics.ucsd.edu/ph130a/130_

notes/130_notes.htmlRitajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 69 / 70

Page 122: Introduction to Quantum Computation. Part - 1

Introduction to QuantumComputation

Ritajit Majumdar, ArunabhaSaha

Outline

Introduction

Motivations for QuantumComputation

Qubit

Linear Algebra

Uncertainty Principle

Postulates of QuantumMechanics

Next Presentation

Reference

Reference

N. David Mermin, Cornell UniversityLecture Notes on Quantum Computation

http://www.wikipedia.org

Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 70 / 70