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Introduction to Quantum Computation. Part - 1

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Arunabha Saha
Arunabha Saha

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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Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Introduction to Quantum Computation Part - I Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Ritajit Majumdar, Arunabha Saha Postulates of Quantum Mechanics Next Presentation Reference University of Calcutta September 9, 2013 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 1 / 70
Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha 1 Introduction 2 Motivations for Quantum Computation Outline Introduction Motivations for Quantum Computation 3 Qubit Qubit Linear Algebra 4 5 Uncertainty Principle Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 6 Postulates of Quantum Mechanics 7 Next Presentation 8 Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 2 / 70
Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70
Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Moore’s law roughly stated that computer power will double for constant cost approximately once every two years. This worked well for a long time. However, at present, quantum effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 3 / 70
Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Moore’s law roughly stated that computer power will double for constant cost approximately once every two years. This worked well for a long time. However, at present, quantum effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference One possible solution is to move to a different computing paradigm. One such paradigm is provided by the theory of quantum computation, which is based on the idea of using quantum mechanics to perform computations, instead of classical physics. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70
Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum systems are exponentially powerful. A system of 500 particles has 2500 ”computing power”. Quantum Computers provide a neat shortcut for solving a range of mathematical tasks known as NP-complete problems. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 4 / 70
Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum systems are exponentially powerful. A system of 500 particles has 2500 ”computing power”. Quantum Computers provide a neat shortcut for solving a range of mathematical tasks known as NP-complete problems. For example, factorisation is an exponential time task for classical computers. But Shor’s quantum algorithm for factorisation is a polynomial time algorithm. It has successfully broken RSA cryptosystem. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 4 / 70
Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Quantum Cryptography. Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Quantum Cryptography. Reference and many more... Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
Qubits: The building blocks of Quantum Computer Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit In classical computer, bits of digital information are either 0 or 1. In a quantum computer, these bits are replaced by a ”superposition” of both 0 and 1. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 or their linear combination. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70
Qubits: The building blocks of Quantum Computer Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit In classical computer, bits of digital information are either 0 or 1. In a quantum computer, these bits are replaced by a ”superposition” of both 0 and 1. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 Qubits are represented as |0 and |1 . Qubits have been created in the laboratory using photons, ions and certain sorts of atomic nuclei. 1 or their linear combination. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70
Superposition Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Suppose we have a k-level system. So there are k distinguishable or classical states for the system. The possible classical states for the system: 0, 1, ..., k − 1. Outline Introduction Motivations for Quantum Computation Qubit Superposition Principle Linear Algebra If a quantum system can be in one of k states, it can also be in any linear superposition of those k states. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 7 / 70
Superpostition Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha |0 , |1 , ..., |k − 1 are called the basis states. The superposition is denoted as a linear combination of these basis. Outline Introduction Motivations for Quantum Computation α0 |0 + α1 |1 + ... + αk−1 |k − 1 Qubit Linear Algebra where, Uncertainty Principle αi ∈ C Postulates of Quantum Mechanics Next Presentation Reference |αi |2 = 1 i (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
Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Consider a Hydrogen atom. This atom may be treated as a qubit. To do so, we define the ground energy state of the electron as |0 and the first energy state as |1 . Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Consider a Hydrogen atom. This atom may be treated as a qubit. To do so, we define the ground energy state of the electron as |0 and the first energy state as |1 . The electron dwells in some linear superposition of these two energy levels. But during measurement, we shall find the electron in any one of the energy states. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
Other examples of Qubits Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Figure: Photon Polarization: The orientation of electrical field oscillation is either horizontal or vertical. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70
Other examples of Qubits Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Figure: Photon Polarization: The orientation of electrical field oscillation is either horizontal or vertical. Postulates of Quantum Mechanics Next Presentation Reference Figure: Electron spin: The spin is either up or down Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70
Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Mathematically, a quantum state (which, as we shall see later, is a vector) is represented by a column matrix. The two fundamental states that we introduced before, |0 and |1 form an orthonormal basis. We shall see more of orthonormality when we see inner products. The matrix representation of |0 and |1 : |0 = Ritajit Majumdar, Arunabha Saha (CU) Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 0 |1 = Motivations for Quantum Computation 0 1 Introduction to Quantum Computation September 9, 2013 11 / 70
Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline So a general quantum state |ψ = α |0 + β |1 is represented as Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 12 / 70
Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline So a general quantum state |ψ = α |0 + β |1 is represented as Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference The matrix notation will be |ψ = = Ritajit Majumdar, Arunabha Saha (CU) α+0 0+β α β Introduction to Quantum Computation September 9, 2013 12 / 70
Introduction to Quantum Computation Qubit: Sign Basis |0 and |1 are called bit basis since they can be thought of as the quantum counter-parts of classical bits 0 and 1 respectively. However, they are not the only possible basis. We may have infinitely many orthonormal basis for a given space. Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Another basis, called the sign basis, is denoted as |+ and |− . |+ = |− = 1 √ 2 1 √ 2 |0 + |0 − 1 √ 2 1 √ 2 |1 |1 Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Figure: Geometrical model of bit basis and sign basis Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 13 / 70
Qubit: Change of Basis Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70
Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = Ritajit Majumdar, Arunabha Saha (CU) Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 |− |− Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 14 / 70
Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = |ψ = 1 2 Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 |− |− Postulates of Quantum Mechanics Next Presentation Reference √ |0 + 3 2 |1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70
Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 Postulates of Quantum Mechanics |− |− Next Presentation Reference √ |ψ = 1 |0 + 23 |1 2 1 1 1 = 2 ( √2 |+ + √2 |− ) + Ritajit Majumdar, Arunabha Saha (CU) √ 3 √ 1 2 ( 2 |+ + 1 √ 2 |− ) Introduction to Quantum Computation September 9, 2013 14 / 70
Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: Uncertainty Principle |0 = |1 = 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 Postulates of Quantum Mechanics |− |− Next Presentation Reference √ |ψ = 1 |0 + 23 |1 2 1 1 1 = 2 ( √2 |+ + √2 |− ) + = 1 ( 2√2 + √ 3 √ ) |+ 2 2 Ritajit Majumdar, Arunabha Saha (CU) + √ 3 √ 1 1 √ 2 ( 2 |+ + 2 √ 1 ( 2√2 − 2√32 ) |− |− ) Introduction to Quantum Computation September 9, 2013 14 / 70
Introduction to Quantum Computation Linear Algebra Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Linear Algebra a very short introduction Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 15 / 70
Introduction to Quantum Computation Vector Space Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation A vector space consists of vectors(|α , |β , |γ ), together with a set of scalars(a, b, c,....)2 , which is closed under two operations: Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Vector addition Reference Scalar multiplication 2 the scalars can be complex numbers Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 16 / 70
Introduction to Quantum Computation Vector Addition The sum of any two vectors is another vector Ritajit Majumdar, Arunabha Saha Outline |α + |β = |γ Introduction Motivations for Quantum Computation Vector addition is commutative Qubit |α + |β = |β + |α Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics it is associative also Next Presentation |α + (|β + |γ ) = (|α + |β ) + |γ Reference There exists a zero(or null) vector3 with the property |α + |0 = |α , ∀ |α For every vector |α there is an associative inverse vector(|−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
Introduction to Quantum Computation Scalar Multiplication Ritajit Majumdar, Arunabha Saha The product of any scalar with any vector is another vector Outline Introduction a |α = |γ Scalar multiplication is distributive w.r.t vector addition Motivations for Quantum Computation Qubit Linear Algebra a(|α + |β ) = a |α + a |β and with respect to scalar addition also (a + b) |α = a |α + b |α Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 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 ; Ritajit Majumdar, Arunabha Saha (CU) 1 |α = |α Introduction to Quantum Computation September 9, 2013 18 / 70
Introduction to Quantum Computation Basis Vectors Ritajit Majumdar, Arunabha Saha Linear combination of vectors |α , |β , |γ ,... is of the form Outline Introduction |α + |β + |γ + . . . Motivations for Quantum Computation Qubit A vector |λ is said to be linearly independent of the set of vectors |α , |β , |γ ,. . . ,if it cannot be written as a linear combination of them. A set of vectors is linearly independent if each one is independent of all the rest. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference If every vector can be written as a linear combination of members of this set then the collection of vectors said to span the space. A set of linearly independent vectors that spans the space is called a basis. The number of vectors in any basis is called the dimension of space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 19 / 70
Introduction to Quantum Computation Inner product Ritajit Majumdar, Arunabha Saha Outline An inner product is a function which takes two vectors as input an gives a complex number as output. The dual(or complex conjugate) of any vector |α is α| Introduction Motivations for Quantum Computation Qubit Linear Algebra |α ∗ = α| Uncertainty Principle The inner product of two vectors (|α , |β ) written as α|β which has the properties: α|β = β|α α|α 4 Postulates of Quantum Mechanics Next Presentation Reference ∗ 0, and α|α = 0 ⇔ |α = |0 α| (b |β + c |γ ) = b α|β + c α|γ 4 This is a complex number; α|β ∈ C Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 20 / 70
Introduction to Quantum Computation Inner Product Space Ritajit Majumdar, Arunabha Saha A vector space with an inner product is called inner product space. i.e. the above conditions satisfied for any vectors |α , |β , |γ ∈ V and for any scalar c Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra n Uncertainty Principle e.g. C has an inner product defined by Postulates of Quantum Mechanics Next Presentation   b1 ∗ ∗  .  ai∗ bi = [a1 . . . an ]  .  . α|β ≡ i Reference bn     a1 b1 . . where |α =  .  and |β =  .  . . an Ritajit Majumdar, Arunabha Saha (CU) bn Introduction to Quantum Computation September 9, 2013 21 / 70
Introduction to Quantum Computation Orthonormal Set Inner product of any vector with itself gives a non-negative number — its square-root of is real which is called norm Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation α = Qubit α|α Linear Algebra It also termed as length of the vector. A unit vector one whose norm is 1 is said to be normalized5 . Uncertainty Principle Two vectors whose inner product is zero is said to be orthogonal Next Presentation Postulates of Quantum Mechanics Reference α|α = 0 A mutually collection of orthogonal normalized vectors is called an orthonormal set αi |αj = δij , where δij 5 Normalization: |ek = Ritajit Majumdar, Arunabha Saha (CU) = 0, i = j = 0, i = j |k k Introduction to Quantum Computation September 9, 2013 22 / 70
Introduction to Quantum Computation Orthonormal Set(Contd.) Ritajit Majumdar, Arunabha Saha Outline Introduction If an orthonormal basis is chosen then the inner product of two vectors can be written as Motivations for Quantum Computation Qubit Linear Algebra α|β = ∗ a1 b1 + ∗ a2 b2 + ... + ∗ an bn Uncertainty Principle Postulates of Quantum Mechanics hence the norm(squared) Next Presentation 2 2 α|α = |a1 | + |a2 | + . . . + |an | 2 Reference each components are aj = ej |α , Ritajit Majumdar, Arunabha Saha (CU) where ej ‘s are basis vector Introduction to Quantum Computation September 9, 2013 23 / 70
Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70
Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 24 / 70
Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Postulates of Quantum Mechanics Next Presentation Reference Zero operator which maps all vectors to zero vector, 0 |υ ≡ |0 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70
Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Postulates of Quantum Mechanics Next Presentation Reference Zero operator which maps all vectors to zero vector, 0 |υ ≡ |0 . Let V, W, X are vector spaces, and A : V → W and B : W → X are linear operators. Then the composition of operators 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
Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle yes it has..!! Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle yes it has..!! say, A : V → W is a linear operator between vector spaces V and W. Let the basis set for V and W are (|υ1 , . . . , |υm ) and (|ω1 , . . . , |ωn ) respectively. Then we can say For each k in 1,2,....,m, there exist complex numbers A1k through Ank such that A |υk = Postulates of Quantum Mechanics Next Presentation Reference Aik |ωi i The matrix whose entries are Aik is the matrix representation of the linear operator. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = AIn = A Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = AIn = A compact notation: (In )ij = δij Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = 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
Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70
Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra These are a set of 2 x 2 complex matrices which are Hermitian and unitary. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70
Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra These are a set of 2 x 2 complex matrices which are Hermitian and unitary. They look like Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference σ0 ≡ I ≡ 10 , 01 σ2 ≡ σy ≡ Y ≡ Ritajit Majumdar, Arunabha Saha (CU) σ1 ≡ σx ≡ X ≡ 0 −i , i 0 01 10 σ3 ≡ σz ≡ Z ≡ 1 0 0 −1 Introduction to Quantum Computation September 9, 2013 27 / 70
Introduction to Quantum Computation Pauli Matrices(properties) Ritajit Majumdar, Arunabha Saha Outline Some properties of Pauli matrices 2 σ1 = 2 σ2 = 2 σ3 = −iσ1 σ2 σ3 = Introduction 1 0 0 1 =I det(σi ) = −1 Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Tr (σi ) = 0 Each Pauli matrices has two eigenvalues +1 and -1. Postulates of Quantum Mechanics Next Presentation Normalized eigenvectors are Reference ψx+ = 1 √ 2 1 , 1 ψx− = 1 √ 2 1 −1 ψy + = 1 √ 2 1 , i ψy − = 1 √ 2 1 −i ψz+ = 1 , 0 Ritajit Majumdar, Arunabha Saha (CU) ψz− = 0 1 Introduction to Quantum Computation September 9, 2013 28 / 70
Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli matrices are used here as rotation6 operators. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle On the basis of Pauli matrices the X, Y, Z quantum gates7 are designed. The Pauli-X gate is the quantum equivalent of NOT gate. It maps |0 to |1 and |1 to |0 . Postulates of Quantum Mechanics Next Presentation Reference 6 rotation 7 all of Bloch sphere acts on single qubit Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 29 / 70
Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 . ˙ ı Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 30 / 70
Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 . ˙ ı Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Pauli-Z gate leaves the basis state |0 unchanged and maps |1 to -|1 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 30 / 70
Introduction to Quantum Computation Hilbert Space Ritajit Majumdar, Arunabha Saha Outline Introduction Hilbert Space Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Wave functions live in Hilbert space Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 31 / 70
Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Hilbert space is an abstract vector space with inner product defined in it, which allows length and angle to be measured. Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Hilbert space is an abstract vector space with inner product defined in it, which allows length and angle to be measured. Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Hilbert space must be complete.8 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
Hilbert Space: Formal Approach Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The set of all functions of x constitute a vector space. To represent a possible physical state,the wave function needed to be normalized |ψ|2 dx ≡ ψ|ψ = 1 Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra The set of all square-integrable functions on a specified interval,9 b a f (x) such that |f (x)|2 dx < ∞, Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference constitutes a smaller vector space. It is known to mathematician as L2 (a, b); physicists call it Hilbert space. Definition A Euclidean space Rn is a vector space endowed with the inner product 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. 9 The limits(a and b) can be ±∞ Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 33 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction Observables Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference woow..! It looks good..!! Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 34 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction In quantum mechanics the measurement process affects the state in a non-deterministic, but in a statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. Linear Algebra Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Motivations for Quantum Computation Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 35 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction In quantum mechanics the measurement process affects the state in a non-deterministic, but in a statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. Linear Algebra Motivations for Quantum Computation Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 36 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α This equation states that if a measurement of the observable A is made while the system of interest is in state |α , then the observed value of the particular measurement must return the eigenvalue a with certainty. If the system is in the general state |φ ∈ H then the eigenvalue a return with probability | α|φ |2 (Born rule). Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 36 / 70
Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α This equation states that if a measurement of the observable A is made while the system of interest is in state |α , then the observed value of the particular measurement must return the eigenvalue a with certainty. If the system is in the general state |φ ∈ H then the eigenvalue a return with probability | α|φ |2 (Born rule). Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference More precisely, the observables are Hermitian operator so its represented by Hermitian matrix. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 36 / 70
Introduction to Quantum Computation Hermitian Operator Ritajit Majumdar, Arunabha Saha Outline Introduction Hermitian Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference hmmm...sounds like me!! is it a new breed ??!! Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 37 / 70
Introduction to Quantum Computation Hermitian Operator Ritajit Majumdar, Arunabha Saha Outline Introduction No, its nothing new. Its the self-adjoint operator. Motivations for Quantum Computation Qubit Definition Linear Algebra Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose i.e. the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference ∗ mathematically aij = aji or in matrix notation, A = (AT )∗ In compact notation, A = A† Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 38 / 70
Hermitian Operators: Expectation Value Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction As previously said that the measurements are non-deterministic, so we can get a probabilistic measure of any observable, that is known as expectation value Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ˆ Q = ˆ ˆ ψ ∗ Qψ = ψ Qψ Operators representing observables have the very special property that, ˆ ˆ ∀f (x) f Qf = Qf f Postulates of Quantum Mechanics Next Presentation Reference More strong condition for hermiticity, ˆ ˆ f Qg = Qf g ∀f (x) and g (x) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 39 / 70
Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 40 / 70
Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Eigenfunctions belonging to distinct eigenvalues are orthogonal. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 40 / 70
Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Eigenfunctions belonging to distinct eigenvalues are orthogonal. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation The eigenfunctions of a Hermitian operator is complete. Any function(in Hilbert space) can be expressed as the linear combination of them. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 40 / 70
Introduction to Quantum Computation Uncertainty Principle Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 41 / 70
Introduction to Quantum Computation Wave Particle Duality Ritajit Majumdar, Arunabha Saha According to de Broglie hypothesis: Outline Introduction p= h λ = 2π λ Motivations for Quantum Computation Qubit where p is the momentum, λ is the wavelength and h is called Plank’s constant. It has a value of 6.63 × 10−34 Joule-sec. h = 2π is called the reduced Plank constant. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 42 / 70
Introduction to Quantum Computation Wave Particle Duality Ritajit Majumdar, Arunabha Saha According to de Broglie hypothesis: Outline Introduction p= h λ = 2π λ Motivations for Quantum Computation Qubit where p is the momentum, λ is the wavelength and h is called Plank’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 wave nature and vice versa. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 42 / 70
Heisenberg Uncertainty Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The Uncertainty Principle is a direct consequence of the wave particle duality. The wavelength of a wave is well defined, while asking for its position is absurd. Vice versa is the case for a particle. And from de Broglie hypothesis, we get that momentum is inversely proportional to wavelength. Since every substance has both wave and particle nature - Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Uncertainty Principle One can never know with perfect accuracy both the position and the momentum of a particle. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 43 / 70
Uncertainty Principle: Mathematical Notation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha According to Heisenberg, the uncertainty in the position and momentum of a substance must be at least as big as 2 . So we can write the mathematical notation of the uncertainty principle: Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra δx.δp ≥ 2 Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 44 / 70
Introduction to Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha ’Not only is the Universe stranger than we think, it is stranger than we can think’ - Warner Heisenberg Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 45 / 70
Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 46 / 70
Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Postulates of Quantum Mechanics Next Presentation Reference The problems with classical physics led to the development of Quantum Mechanics and Special Relativity. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Postulates of Quantum Mechanics Next Presentation Reference The problems with classical physics led to the development of Quantum Mechanics and Special Relativity. Some of the problems leading to the development of Quantum Mechanics are Black Body Radiation Photoelectric Effect Double Slit Experiment Compton Scattering Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
Double Slit Experiment: Setup Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha This experiment shows an aberrant result which cannot be explained using classical laws of physics. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 47 / 70
Double Slit Experiment: Setup Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha This experiment shows an aberrant result which cannot be explained using classical laws of physics. The experiment setup consists of a monochromatic source of light and two extremely small slits, big enough for only one photon particle to pass through it. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 47 / 70
Double Slit Experiment: One Slit Open Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha If initially only one slit is open, we get a probability distribution as shown in figure. So if the two slits are opened individually, the two distinct probability distributations are obtained in the screen. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 48 / 70
Double Slit Experiment: The Classical Expectation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Since the opening of two slits individually are independent events, classically we expect that if the two slits are opened together, the two probability distributions should add up giving the new probability distribution. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 49 / 70
Double Slit Experiment: The Anomaly Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha What we observe in reality when both slits are opened together, is an interference pattern. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 50 / 70
Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha There is no classical explanation to this observation. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha There is no classical explanation to this observation. However, using quantum mechanics, it can be explained. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha 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 responsible for such aberrant observation. The wave nature of light is responsible for the interference pattern observed. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
Double Slit Experiment: Complete Picture 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 Figure: Double Slit Experiment showing the anomaly - deviation of the observed result from the one predicted by Classical Physics. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 52 / 70
Nobody understands Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum mechanics is a very counter-intuitive theory. The results of quantum mechanics is nothing like what we experience in everyday life. It is just that nature behaves very strangely at the level of elementary particles. And this strange way is described by quantum mechanics. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 53 / 70
Nobody understands Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum mechanics is a very counter-intuitive theory. The results of quantum mechanics is nothing like what we experience in everyday life. It is just that nature behaves very strangely at the level of elementary particles. And this strange way is described by quantum mechanics. In fact, so strange is the theory that Richard Feynman once said Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 53 / 70
1st Postulate: State Space Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Postulate 1 Associated to any isolated system is a Hilbert Space called the state space. The system is completely defined by the state vector, which is a unit vector in the state space. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 54 / 70
State Vector: Mathematical Realisation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Let us consider a quantum state |ψ = α |0 + β |1 Motivations for Quantum Computation Since a state is a unit vector, the norm of the vector must be unity, or in mathematical notation, Qubit ψ|ψ = 1 Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Hence, the condition that |ψ is a unit vector is equivalent to Reference |α|2 + |β|2 = 1 This condition is called the normalization condition of the state vector. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 55 / 70
State: Why unit vector? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Let |ψ = α0 |0 + α1 |1 + . . . + αk−1 |k − 1 be a quantum state. As we shall see later, the superposition is not observable. When a state is observed, it collapses into one of the basis states. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 56 / 70
State: Why unit vector? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation The square of the amplitude |αi |2 gives the probability that the system collapses to the state |i . Qubit Linear Algebra Uncertainty Principle Since the total probability is always 1, we must have: Postulates of Quantum Mechanics Next Presentation |αi |2 = 1 Reference This condition is satisfied only when the state vector is a unit vector. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 57 / 70
2nd Postulate: Evolution Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Postulate 2 Outline The evolution of a closed quantum system is described by a unitary transformation. That is, if |ψ1 is the state of the system at time t1 and |ψ2 at time t2 , then: |ψ2 = U(t1 , t2 ) |ψ1 where U(t1 , t2 ) is a unitary operator. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 58 / 70
2nd Postulate: Evolution Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Postulate 2 Outline The evolution of a closed quantum system is described by a unitary transformation. That is, if |ψ1 is the state of the system at time t1 and |ψ2 at time t2 , then: |ψ2 = U(t1 , t2 ) |ψ1 where U(t1 , t2 ) is a unitary operator. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Figure: Quantum systems evolve by the rotation of the Hilbert Space Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 58 / 70
Time Evolution: Schrodinger Equation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The time evolution of a closed quantum system is given by the Schrodinger Equation: Introduction Motivations for Quantum Computation Qubit i d|ψ dt = H |ψ Linear Algebra Uncertainty Principle where H is the hamiltonian and it is defined as the total energy (kinetic + potential) of the system. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 59 / 70
Time Evolution: Schrodinger Equation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The time evolution of a closed quantum system is given by the Schrodinger Equation: Introduction Motivations for Quantum Computation Qubit i d|ψ dt = H |ψ Linear Algebra Uncertainty Principle where H is the hamiltonian and it is defined as the total energy (kinetic + potential) of the system. Postulates of Quantum Mechanics Next Presentation Reference The connection between the hamitonian picture and the unitary 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 ) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 59 / 70
Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 60 / 70
Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Postulates of Quantum Mechanics Next Presentation Reference After measurement, the state of the system collapses to either |0 or |1 with the said probability. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Postulates of Quantum Mechanics Next Presentation Reference After measurement, the state of the system collapses to either |0 or |1 with the said probability. However, after measurement if the new state of the system is |0 (say), then further measurements in the same basis gives outcome |0 with probability 1. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
3rd Postulate: Measurement Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Postulate 3 Motivations for Quantum Computation Quantum measurements are described by a collection {Mm } of measurement operators. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ before experiment, then the probability that result m occurs is given by, Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference p(m) = ψ| Mm † .Mm |ψ And the state of the system after measurement is √ Ritajit Majumdar, Arunabha Saha (CU) Mm |ψ ψ|Mm †.Mm |ψ Introduction to Quantum Computation September 9, 2013 61 / 70
So what is the Big Deal? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The postulate states that a quantum system stays in a superposition when it is not observed. When a measurement is done, it immediately collapses to one of its eigenstates. Hence we can never observe what the original superposition of the system was. We merely observe the state after it collapses. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra This inherent ambiguity provides an excellent security in Quantum Cryptography. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 62 / 70
Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Without opening the box, an observer cannot know whether the cat is alive or dead. So the cat may be said to be in a superposition of the two states. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 63 / 70
Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Without opening the box, an observer cannot know whether the cat is alive or dead. So the cat may be said to be in a superposition of the two states. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference However, when the box is opened, we observe deterministically that the cat is either dead or alive. We can, by no means, observe the superposition. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
4th Postulate: Composite System Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction What we have seen so far was a single qubit system. What happens when there are multiple qubits? This is given by the last postulate: Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Postulate 4 Next Presentation The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have n systems, and the system number i is prepared in state |ψi , then the joint state of the total system is Reference |ψ1 ⊗ |ψ2 ⊗ ... ⊗ |ψn Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 64 / 70
Introduction to Quantum Computation Two Qubit System Ritajit Majumdar, Arunabha Saha Outline Introduction Let us consider a two qubit system. Classically, with two bits, we can have 4 states - 00, 01, 10, 11. A quantum system is a linear superposition of all these four states. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle So, a general two qubit quantum state can be represented as |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 10 Postulates of Quantum Mechanics Next Presentation Reference 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
Measurement in Two Qubit System Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Measurement is similar to single qubit system. When we measure the two qubit system we get outcome j with probability |αj |2 and the new state will be |j . Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics That is for the system Next Presentation Reference |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 , we get outcome |00 with probability |α00 |2 and the new state of the system will be |00 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 66 / 70
Introduction to Quantum Computation Partial Measurement Ritajit Majumdar, Arunabha Saha Outline So what if we want to measure only the first qubit? Or maybe only the second one? Introduction Motivations for Quantum Computation Qubit We take the same two qubit system, |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 , Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation If only the first qubit is measured then we get the outcome |0 for the first qubit with probability |α00 |2 + |α01 |2 Reference 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
Coming up in next talk... Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Einstein-Polosky-Rosen (EPR) Paradox Qubit Bell State Linear Algebra Uncertainty Principle Quantum Entanglement Density Matrix Notation of Quantum Mechanics Quantum Gates Postulates of Quantum Mechanics Next Presentation Reference and many more... Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 68 / 70
Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Michael A. Nielsen, Isaac Chuang Quantum Computation and Quantum Information Cambridge University Press David J. Griffiths Introduction to Quantum Mechanics Prentice Hall, 2nd Edition Umesh Vazirani, University of California Berkeley Quantum Mechanics and Quantum Computation https://class.coursera.org/qcomp-2012-001/ Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Michael A. Nielsen, University of Queensland Quantum Computing for the determined http://michaelnielsen.org/blog/ quantum-computing-for-the-determined/ James Branson, University of California San Diego Quantum Physics (UCSD Physics 130) http://quantummechanics.ucsd.edu/ph130a/130_ notes/130_notes.html Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 69 / 70
Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle N. David Mermin, Cornell University Lecture Notes on Quantum Computation Postulates of Quantum Mechanics Next Presentation http://www.wikipedia.org Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 70 / 70

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Introduction to Quantum Computation. Part - 1

  • 1. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Introduction to Quantum Computation Part - I Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Ritajit Majumdar, Arunabha Saha Postulates of Quantum Mechanics Next Presentation Reference University of Calcutta September 9, 2013 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 1 / 70
  • 2. Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha 1 Introduction 2 Motivations for Quantum Computation Outline Introduction Motivations for Quantum Computation 3 Qubit Qubit Linear Algebra 4 5 Uncertainty Principle Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 6 Postulates of Quantum Mechanics 7 Next Presentation 8 Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 2 / 70
  • 3. Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70
  • 4. Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Moore’s law roughly stated that computer power will double for constant cost approximately once every two years. This worked well for a long time. However, at present, quantum effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 3 / 70
  • 5. Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline It may be tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behaviour of all physical phenomena, this temptation must be resisted. Moore’s law roughly stated that computer power will double for constant cost approximately once every two years. This worked well for a long time. However, at present, quantum effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference One possible solution is to move to a different computing paradigm. One such paradigm is provided by the theory of quantum computation, which is based on the idea of using quantum mechanics to perform computations, instead of classical physics. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 3 / 70
  • 6. Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum systems are exponentially powerful. A system of 500 particles has 2500 ”computing power”. Quantum Computers provide a neat shortcut for solving a range of mathematical tasks known as NP-complete problems. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 4 / 70
  • 7. Classical Computation vs Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum systems are exponentially powerful. A system of 500 particles has 2500 ”computing power”. Quantum Computers provide a neat shortcut for solving a range of mathematical tasks known as NP-complete problems. For example, factorisation is an exponential time task for classical computers. But Shor’s quantum algorithm for factorisation is a polynomial time algorithm. It has successfully broken RSA cryptosystem. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 4 / 70
  • 8. Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
  • 9. Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
  • 10. Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Quantum Cryptography. Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
  • 11. Motivations for Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Faster than light (?) communication. Uncertainty Principle Highly parallel and efficient quantum algorithms. Postulates of Quantum Mechanics Next Presentation Quantum Cryptography. Reference and many more... Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 5 / 70
  • 12. Qubits: The building blocks of Quantum Computer Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit In classical computer, bits of digital information are either 0 or 1. In a quantum computer, these bits are replaced by a ”superposition” of both 0 and 1. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 or their linear combination. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70
  • 13. Qubits: The building blocks of Quantum Computer Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit In classical computer, bits of digital information are either 0 or 1. In a quantum computer, these bits are replaced by a ”superposition” of both 0 and 1. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 Qubits are represented as |0 and |1 . Qubits have been created in the laboratory using photons, ions and certain sorts of atomic nuclei. 1 or their linear combination. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 6 / 70
  • 14. Superposition Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Suppose we have a k-level system. So there are k distinguishable or classical states for the system. The possible classical states for the system: 0, 1, ..., k − 1. Outline Introduction Motivations for Quantum Computation Qubit Superposition Principle Linear Algebra If a quantum system can be in one of k states, it can also be in any linear superposition of those k states. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 7 / 70
  • 15. Superpostition Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha |0 , |1 , ..., |k − 1 are called the basis states. The superposition is denoted as a linear combination of these basis. Outline Introduction Motivations for Quantum Computation α0 |0 + α1 |1 + ... + αk−1 |k − 1 Qubit Linear Algebra where, Uncertainty Principle αi ∈ C Postulates of Quantum Mechanics Next Presentation Reference |αi |2 = 1 i (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
  • 16. Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
  • 17. Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Consider a Hydrogen atom. This atom may be treated as a qubit. To do so, we define the ground energy state of the electron as |0 and the first energy state as |1 . Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
  • 18. Qubit: Physical Interpretation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha We may have various interpretations of qubits. Outline Consider a Hydrogen atom. This atom may be treated as a qubit. To do so, we define the ground energy state of the electron as |0 and the first energy state as |1 . The electron dwells in some linear superposition of these two energy levels. But during measurement, we shall find the electron in any one of the energy states. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 9 / 70
  • 19. Other examples of Qubits Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Figure: Photon Polarization: The orientation of electrical field oscillation is either horizontal or vertical. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70
  • 20. Other examples of Qubits Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Figure: Photon Polarization: The orientation of electrical field oscillation is either horizontal or vertical. Postulates of Quantum Mechanics Next Presentation Reference Figure: Electron spin: The spin is either up or down Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 10 / 70
  • 21. Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Mathematically, a quantum state (which, as we shall see later, is a vector) is represented by a column matrix. The two fundamental states that we introduced before, |0 and |1 form an orthonormal basis. We shall see more of orthonormality when we see inner products. The matrix representation of |0 and |1 : |0 = Ritajit Majumdar, Arunabha Saha (CU) Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 1 0 |1 = Motivations for Quantum Computation 0 1 Introduction to Quantum Computation September 9, 2013 11 / 70
  • 22. Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline So a general quantum state |ψ = α |0 + β |1 is represented as Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 12 / 70
  • 23. Qubit: Mathematical model Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline So a general quantum state |ψ = α |0 + β |1 is represented as Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference The matrix notation will be |ψ = = Ritajit Majumdar, Arunabha Saha (CU) α+0 0+β α β Introduction to Quantum Computation September 9, 2013 12 / 70
  • 24. Introduction to Quantum Computation Qubit: Sign Basis |0 and |1 are called bit basis since they can be thought of as the quantum counter-parts of classical bits 0 and 1 respectively. However, they are not the only possible basis. We may have infinitely many orthonormal basis for a given space. Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Another basis, called the sign basis, is denoted as |+ and |− . |+ = |− = 1 √ 2 1 √ 2 |0 + |0 − 1 √ 2 1 √ 2 |1 |1 Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Figure: Geometrical model of bit basis and sign basis Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 13 / 70
  • 25. Qubit: Change of Basis Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70
  • 26. Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = Ritajit Majumdar, Arunabha Saha (CU) Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 |− |− Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 14 / 70
  • 27. Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = |ψ = 1 2 Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 |− |− Postulates of Quantum Mechanics Next Presentation Reference √ |0 + 3 2 |1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 14 / 70
  • 28. Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: |0 = |1 = Uncertainty Principle 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 Postulates of Quantum Mechanics |− |− Next Presentation Reference √ |ψ = 1 |0 + 23 |1 2 1 1 1 = 2 ( √2 |+ + √2 |− ) + Ritajit Majumdar, Arunabha Saha (CU) √ 3 √ 1 2 ( 2 |+ + 1 √ 2 |− ) Introduction to Quantum Computation September 9, 2013 14 / 70
  • 29. Introduction to Quantum Computation Qubit: Change of Basis Ritajit Majumdar, Arunabha Saha Outline Measure |ψ = 1 2 √ |0 + 3 2 Introduction |1 in |+ /|− basis. Motivations for Quantum Computation Qubit Linear Algebra It can be checked that: Uncertainty Principle |0 = |1 = 1 √ 2 1 √ 2 |+ + |+ − 1 √ 2 1 √ 2 Postulates of Quantum Mechanics |− |− Next Presentation Reference √ |ψ = 1 |0 + 23 |1 2 1 1 1 = 2 ( √2 |+ + √2 |− ) + = 1 ( 2√2 + √ 3 √ ) |+ 2 2 Ritajit Majumdar, Arunabha Saha (CU) + √ 3 √ 1 1 √ 2 ( 2 |+ + 2 √ 1 ( 2√2 − 2√32 ) |− |− ) Introduction to Quantum Computation September 9, 2013 14 / 70
  • 30. Introduction to Quantum Computation Linear Algebra Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Linear Algebra a very short introduction Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 15 / 70
  • 31. Introduction to Quantum Computation Vector Space Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation A vector space consists of vectors(|α , |β , |γ ), together with a set of scalars(a, b, c,....)2 , which is closed under two operations: Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Vector addition Reference Scalar multiplication 2 the scalars can be complex numbers Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 16 / 70
  • 32. Introduction to Quantum Computation Vector Addition The sum of any two vectors is another vector Ritajit Majumdar, Arunabha Saha Outline |α + |β = |γ Introduction Motivations for Quantum Computation Vector addition is commutative Qubit |α + |β = |β + |α Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics it is associative also Next Presentation |α + (|β + |γ ) = (|α + |β ) + |γ Reference There exists a zero(or null) vector3 with the property |α + |0 = |α , ∀ |α For every vector |α there is an associative inverse vector(|−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
  • 33. Introduction to Quantum Computation Scalar Multiplication Ritajit Majumdar, Arunabha Saha The product of any scalar with any vector is another vector Outline Introduction a |α = |γ Scalar multiplication is distributive w.r.t vector addition Motivations for Quantum Computation Qubit Linear Algebra a(|α + |β ) = a |α + a |β and with respect to scalar addition also (a + b) |α = a |α + b |α Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 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 ; Ritajit Majumdar, Arunabha Saha (CU) 1 |α = |α Introduction to Quantum Computation September 9, 2013 18 / 70
  • 34. Introduction to Quantum Computation Basis Vectors Ritajit Majumdar, Arunabha Saha Linear combination of vectors |α , |β , |γ ,... is of the form Outline Introduction |α + |β + |γ + . . . Motivations for Quantum Computation Qubit A vector |λ is said to be linearly independent of the set of vectors |α , |β , |γ ,. . . ,if it cannot be written as a linear combination of them. A set of vectors is linearly independent if each one is independent of all the rest. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference If every vector can be written as a linear combination of members of this set then the collection of vectors said to span the space. A set of linearly independent vectors that spans the space is called a basis. The number of vectors in any basis is called the dimension of space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 19 / 70
  • 35. Introduction to Quantum Computation Inner product Ritajit Majumdar, Arunabha Saha Outline An inner product is a function which takes two vectors as input an gives a complex number as output. The dual(or complex conjugate) of any vector |α is α| Introduction Motivations for Quantum Computation Qubit Linear Algebra |α ∗ = α| Uncertainty Principle The inner product of two vectors (|α , |β ) written as α|β which has the properties: α|β = β|α α|α 4 Postulates of Quantum Mechanics Next Presentation Reference ∗ 0, and α|α = 0 ⇔ |α = |0 α| (b |β + c |γ ) = b α|β + c α|γ 4 This is a complex number; α|β ∈ C Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 20 / 70
  • 36. Introduction to Quantum Computation Inner Product Space Ritajit Majumdar, Arunabha Saha A vector space with an inner product is called inner product space. i.e. the above conditions satisfied for any vectors |α , |β , |γ ∈ V and for any scalar c Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra n Uncertainty Principle e.g. C has an inner product defined by Postulates of Quantum Mechanics Next Presentation   b1 ∗ ∗  .  ai∗ bi = [a1 . . . an ]  .  . α|β ≡ i Reference bn     a1 b1 . . where |α =  .  and |β =  .  . . an Ritajit Majumdar, Arunabha Saha (CU) bn Introduction to Quantum Computation September 9, 2013 21 / 70
  • 37. Introduction to Quantum Computation Orthonormal Set Inner product of any vector with itself gives a non-negative number — its square-root of is real which is called norm Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation α = Qubit α|α Linear Algebra It also termed as length of the vector. A unit vector one whose norm is 1 is said to be normalized5 . Uncertainty Principle Two vectors whose inner product is zero is said to be orthogonal Next Presentation Postulates of Quantum Mechanics Reference α|α = 0 A mutually collection of orthogonal normalized vectors is called an orthonormal set αi |αj = δij , where δij 5 Normalization: |ek = Ritajit Majumdar, Arunabha Saha (CU) = 0, i = j = 0, i = j |k k Introduction to Quantum Computation September 9, 2013 22 / 70
  • 38. Introduction to Quantum Computation Orthonormal Set(Contd.) Ritajit Majumdar, Arunabha Saha Outline Introduction If an orthonormal basis is chosen then the inner product of two vectors can be written as Motivations for Quantum Computation Qubit Linear Algebra α|β = ∗ a1 b1 + ∗ a2 b2 + ... + ∗ an bn Uncertainty Principle Postulates of Quantum Mechanics hence the norm(squared) Next Presentation 2 2 α|α = |a1 | + |a2 | + . . . + |an | 2 Reference each components are aj = ej |α , Ritajit Majumdar, Arunabha Saha (CU) where ej ‘s are basis vector Introduction to Quantum Computation September 9, 2013 23 / 70
  • 39. Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70
  • 40. Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 24 / 70
  • 41. Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Postulates of Quantum Mechanics Next Presentation Reference Zero operator which maps all vectors to zero vector, 0 |υ ≡ |0 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 24 / 70
  • 42. Introduction to Quantum Computation Linear Operator and Matrices Ritajit Majumdar, Arunabha Saha A linear operator between vector spaces V and W is defined to be any function A : V → W which is linear in its inputs Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ai |υi A i = ai A(|υi ) i another linear operator on any vector space V is Identity operator, Iv defined as Iv |υ ≡ |υ . Postulates of Quantum Mechanics Next Presentation Reference Zero operator which maps all vectors to zero vector, 0 |υ ≡ |0 . Let V, W, X are vector spaces, and A : V → W and B : W → X are linear operators. Then the composition of operators 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
  • 43. Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
  • 44. Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
  • 45. Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle yes it has..!! Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
  • 46. Matrix Representation of Linear Operators Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha already we have seen that matrices can be regarded as linear operators..!! does linear operators has a matrix representation..??! Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle yes it has..!! say, A : V → W is a linear operator between vector spaces V and W. Let the basis set for V and W are (|υ1 , . . . , |υm ) and (|ω1 , . . . , |ωn ) respectively. Then we can say For each k in 1,2,....,m, there exist complex numbers A1k through Ank such that A |υk = Postulates of Quantum Mechanics Next Presentation Reference Aik |ωi i The matrix whose entries are Aik is the matrix representation of the linear operator. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 25 / 70
  • 47. Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
  • 48. Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = AIn = A Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
  • 49. Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = AIn = A compact notation: (In )ij = δij Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 26 / 70
  • 50. Introduction to Quantum Computation Identity Matrix Ritajit Majumdar, Arunabha Saha Outline The exotic way to express 1 Introduction Definition Motivations for Quantum Computation Identity matrix is one with the main diagonals and zeros everywhere. it is denoted by In or I Qubit Matrix representation of Identity operator   1 0 ... 0 0 1 . . . 0    . . .. .  .. . . .. . Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 0 0 ... 1 It satisfies the property In A = 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
  • 51. Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70
  • 52. Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra These are a set of 2 x 2 complex matrices which are Hermitian and unitary. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 27 / 70
  • 53. Introduction to Quantum Computation Pauli Matrices Ritajit Majumdar, Arunabha Saha Outline Introduction Pauli matrices are named after the physicist Wolfgang Pauli. Motivations for Quantum Computation Qubit Linear Algebra These are a set of 2 x 2 complex matrices which are Hermitian and unitary. They look like Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference σ0 ≡ I ≡ 10 , 01 σ2 ≡ σy ≡ Y ≡ Ritajit Majumdar, Arunabha Saha (CU) σ1 ≡ σx ≡ X ≡ 0 −i , i 0 01 10 σ3 ≡ σz ≡ Z ≡ 1 0 0 −1 Introduction to Quantum Computation September 9, 2013 27 / 70
  • 54. Introduction to Quantum Computation Pauli Matrices(properties) Ritajit Majumdar, Arunabha Saha Outline Some properties of Pauli matrices 2 σ1 = 2 σ2 = 2 σ3 = −iσ1 σ2 σ3 = Introduction 1 0 0 1 =I det(σi ) = −1 Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Tr (σi ) = 0 Each Pauli matrices has two eigenvalues +1 and -1. Postulates of Quantum Mechanics Next Presentation Normalized eigenvectors are Reference ψx+ = 1 √ 2 1 , 1 ψx− = 1 √ 2 1 −1 ψy + = 1 √ 2 1 , i ψy − = 1 √ 2 1 −i ψz+ = 1 , 0 Ritajit Majumdar, Arunabha Saha (CU) ψz− = 0 1 Introduction to Quantum Computation September 9, 2013 28 / 70
  • 55. Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli matrices are used here as rotation6 operators. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle On the basis of Pauli matrices the X, Y, Z quantum gates7 are designed. The Pauli-X gate is the quantum equivalent of NOT gate. It maps |0 to |1 and |1 to |0 . Postulates of Quantum Mechanics Next Presentation Reference 6 rotation 7 all of Bloch sphere acts on single qubit Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 29 / 70
  • 56. Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 . ˙ ı Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 30 / 70
  • 57. Pauli Matrices and Quantum Computation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 . ˙ ı Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Pauli-Z gate leaves the basis state |0 unchanged and maps |1 to -|1 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 30 / 70
  • 58. Introduction to Quantum Computation Hilbert Space Ritajit Majumdar, Arunabha Saha Outline Introduction Hilbert Space Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Wave functions live in Hilbert space Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 31 / 70
  • 59. Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
  • 60. Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
  • 61. Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Hilbert space is an abstract vector space with inner product defined in it, which allows length and angle to be measured. Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
  • 62. Hilbert Space: Few Basics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The mathematical concept of Hilbert space named after David Hilbert, but this term coined by John von Neumann. Motivations for Quantum Computation Basically this is the generalization of the notion of Euclidean space i.e. it extends the methods of algebra and calculus of 2D Euclidean plane and 3D space to space of any finite or infinite dimensions. Linear Algebra Hilbert space is an abstract vector space with inner product defined in it, which allows length and angle to be measured. Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Hilbert space must be complete.8 8 any Cauchy sequence of functions in Hilbert space converges to a function that is also in the space. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 32 / 70
  • 63. Hilbert Space: Formal Approach Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The set of all functions of x constitute a vector space. To represent a possible physical state,the wave function needed to be normalized |ψ|2 dx ≡ ψ|ψ = 1 Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra The set of all square-integrable functions on a specified interval,9 b a f (x) such that |f (x)|2 dx < ∞, Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference constitutes a smaller vector space. It is known to mathematician as L2 (a, b); physicists call it Hilbert space. Definition A Euclidean space Rn is a vector space endowed with the inner product 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. 9 The limits(a and b) can be ±∞ Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 33 / 70
  • 64. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction Observables Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference woow..! It looks good..!! Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 34 / 70
  • 65. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70
  • 66. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction In quantum mechanics the measurement process affects the state in a non-deterministic, but in a statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. Linear Algebra Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Motivations for Quantum Computation Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 35 / 70
  • 67. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline A system observable is a measurable operator, where the property of the system state can be determined by some sequence of physical operations. Introduction In quantum mechanics the measurement process affects the state in a non-deterministic, but in a statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. Linear Algebra Motivations for Quantum Computation Qubit Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 35 / 70
  • 68. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 36 / 70
  • 69. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α This equation states that if a measurement of the observable A is made while the system of interest is in state |α , then the observed value of the particular measurement must return the eigenvalue a with certainty. If the system is in the general state |φ ∈ H then the eigenvalue a return with probability | α|φ |2 (Born rule). Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 36 / 70
  • 70. Introduction to Quantum Computation Observables Ritajit Majumdar, Arunabha Saha Outline Introduction e.g. let |α is an eigenvector of the observable A, with eigenvalue a and exits in a d-dimensional Hilbert space, then A |α = a |α This equation states that if a measurement of the observable A is made while the system of interest is in state |α , then the observed value of the particular measurement must return the eigenvalue a with certainty. If the system is in the general state |φ ∈ H then the eigenvalue a return with probability | α|φ |2 (Born rule). Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference More precisely, the observables are Hermitian operator so its represented by Hermitian matrix. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 36 / 70
  • 71. Introduction to Quantum Computation Hermitian Operator Ritajit Majumdar, Arunabha Saha Outline Introduction Hermitian Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference hmmm...sounds like me!! is it a new breed ??!! Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 37 / 70
  • 72. Introduction to Quantum Computation Hermitian Operator Ritajit Majumdar, Arunabha Saha Outline Introduction No, its nothing new. Its the self-adjoint operator. Motivations for Quantum Computation Qubit Definition Linear Algebra Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose i.e. the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference ∗ mathematically aij = aji or in matrix notation, A = (AT )∗ In compact notation, A = A† Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 38 / 70
  • 73. Hermitian Operators: Expectation Value Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction As previously said that the measurements are non-deterministic, so we can get a probabilistic measure of any observable, that is known as expectation value Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle ˆ Q = ˆ ˆ ψ ∗ Qψ = ψ Qψ Operators representing observables have the very special property that, ˆ ˆ ∀f (x) f Qf = Qf f Postulates of Quantum Mechanics Next Presentation Reference More strong condition for hermiticity, ˆ ˆ f Qg = Qf g ∀f (x) and g (x) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 39 / 70
  • 74. Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 40 / 70
  • 75. Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Eigenfunctions belonging to distinct eigenvalues are orthogonal. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 40 / 70
  • 76. Hermitian Operators:Properties Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Eigenvalues of Hermitian operators are real. Eigenfunctions belonging to distinct eigenvalues are orthogonal. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation The eigenfunctions of a Hermitian operator is complete. Any function(in Hilbert space) can be expressed as the linear combination of them. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 40 / 70
  • 77. Introduction to Quantum Computation Uncertainty Principle Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 41 / 70
  • 78. Introduction to Quantum Computation Wave Particle Duality Ritajit Majumdar, Arunabha Saha According to de Broglie hypothesis: Outline Introduction p= h λ = 2π λ Motivations for Quantum Computation Qubit where p is the momentum, λ is the wavelength and h is called Plank’s constant. It has a value of 6.63 × 10−34 Joule-sec. h = 2π is called the reduced Plank constant. Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 42 / 70
  • 79. Introduction to Quantum Computation Wave Particle Duality Ritajit Majumdar, Arunabha Saha According to de Broglie hypothesis: Outline Introduction p= h λ = 2π λ Motivations for Quantum Computation Qubit where p is the momentum, λ is the wavelength and h is called Plank’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 wave nature and vice versa. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 42 / 70
  • 80. Heisenberg Uncertainty Principle Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction The Uncertainty Principle is a direct consequence of the wave particle duality. The wavelength of a wave is well defined, while asking for its position is absurd. Vice versa is the case for a particle. And from de Broglie hypothesis, we get that momentum is inversely proportional to wavelength. Since every substance has both wave and particle nature - Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Uncertainty Principle One can never know with perfect accuracy both the position and the momentum of a particle. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 43 / 70
  • 81. Uncertainty Principle: Mathematical Notation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha According to Heisenberg, the uncertainty in the position and momentum of a substance must be at least as big as 2 . So we can write the mathematical notation of the uncertainty principle: Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra δx.δp ≥ 2 Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 44 / 70
  • 82. Introduction to Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha ’Not only is the Universe stranger than we think, it is stranger than we can think’ - Warner Heisenberg Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 45 / 70
  • 83. Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
  • 84. Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 46 / 70
  • 85. Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Postulates of Quantum Mechanics Next Presentation Reference The problems with classical physics led to the development of Quantum Mechanics and Special Relativity. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
  • 86. Introduction to Quantum Computation Quantum Mechanics Ritajit Majumdar, Arunabha Saha By the late nineteenth century the laws of physics were based on Mechanics and laws of Gravitation from Newton, Maxwell’s equations describing Electricity and Magnetism and on Statistical Mechanics describing the state of large collection of matter. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle These laws of physics described nature very well under most conditions. However, some experiments of the late 19th and early 20th century could not be explained. Postulates of Quantum Mechanics Next Presentation Reference The problems with classical physics led to the development of Quantum Mechanics and Special Relativity. Some of the problems leading to the development of Quantum Mechanics are Black Body Radiation Photoelectric Effect Double Slit Experiment Compton Scattering Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 46 / 70
  • 87. Double Slit Experiment: Setup Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha This experiment shows an aberrant result which cannot be explained using classical laws of physics. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 47 / 70
  • 88. Double Slit Experiment: Setup Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha This experiment shows an aberrant result which cannot be explained using classical laws of physics. The experiment setup consists of a monochromatic source of light and two extremely small slits, big enough for only one photon particle to pass through it. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 47 / 70
  • 89. Double Slit Experiment: One Slit Open Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha If initially only one slit is open, we get a probability distribution as shown in figure. So if the two slits are opened individually, the two distinct probability distributations are obtained in the screen. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 48 / 70
  • 90. Double Slit Experiment: The Classical Expectation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Since the opening of two slits individually are independent events, classically we expect that if the two slits are opened together, the two probability distributions should add up giving the new probability distribution. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 49 / 70
  • 91. Double Slit Experiment: The Anomaly Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha What we observe in reality when both slits are opened together, is an interference pattern. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 50 / 70
  • 92. Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha There is no classical explanation to this observation. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
  • 93. Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha There is no classical explanation to this observation. However, using quantum mechanics, it can be explained. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
  • 94. Double Slit Experiment: Quantum Explanation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha 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 responsible for such aberrant observation. The wave nature of light is responsible for the interference pattern observed. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 51 / 70
  • 95. Double Slit Experiment: Complete Picture 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 Figure: Double Slit Experiment showing the anomaly - deviation of the observed result from the one predicted by Classical Physics. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 52 / 70
  • 96. Nobody understands Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum mechanics is a very counter-intuitive theory. The results of quantum mechanics is nothing like what we experience in everyday life. It is just that nature behaves very strangely at the level of elementary particles. And this strange way is described by quantum mechanics. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 53 / 70
  • 97. Nobody understands Quantum Mechanics Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Quantum mechanics is a very counter-intuitive theory. The results of quantum mechanics is nothing like what we experience in everyday life. It is just that nature behaves very strangely at the level of elementary particles. And this strange way is described by quantum mechanics. In fact, so strange is the theory that Richard Feynman once said Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 53 / 70
  • 98. 1st Postulate: State Space Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Postulate 1 Associated to any isolated system is a Hilbert Space called the state space. The system is completely defined by the state vector, which is a unit vector in the state space. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 54 / 70
  • 99. State Vector: Mathematical Realisation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Let us consider a quantum state |ψ = α |0 + β |1 Motivations for Quantum Computation Since a state is a unit vector, the norm of the vector must be unity, or in mathematical notation, Qubit ψ|ψ = 1 Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Hence, the condition that |ψ is a unit vector is equivalent to Reference |α|2 + |β|2 = 1 This condition is called the normalization condition of the state vector. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 55 / 70
  • 100. State: Why unit vector? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Let |ψ = α0 |0 + α1 |1 + . . . + αk−1 |k − 1 be a quantum state. As we shall see later, the superposition is not observable. When a state is observed, it collapses into one of the basis states. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 56 / 70
  • 101. State: Why unit vector? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation The square of the amplitude |αi |2 gives the probability that the system collapses to the state |i . Qubit Linear Algebra Uncertainty Principle Since the total probability is always 1, we must have: Postulates of Quantum Mechanics Next Presentation |αi |2 = 1 Reference This condition is satisfied only when the state vector is a unit vector. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 57 / 70
  • 102. 2nd Postulate: Evolution Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Postulate 2 Outline The evolution of a closed quantum system is described by a unitary transformation. That is, if |ψ1 is the state of the system at time t1 and |ψ2 at time t2 , then: |ψ2 = U(t1 , t2 ) |ψ1 where U(t1 , t2 ) is a unitary operator. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 58 / 70
  • 103. 2nd Postulate: Evolution Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Postulate 2 Outline The evolution of a closed quantum system is described by a unitary transformation. That is, if |ψ1 is the state of the system at time t1 and |ψ2 at time t2 , then: |ψ2 = U(t1 , t2 ) |ψ1 where U(t1 , t2 ) is a unitary operator. Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Figure: Quantum systems evolve by the rotation of the Hilbert Space Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 58 / 70
  • 104. Time Evolution: Schrodinger Equation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The time evolution of a closed quantum system is given by the Schrodinger Equation: Introduction Motivations for Quantum Computation Qubit i d|ψ dt = H |ψ Linear Algebra Uncertainty Principle where H is the hamiltonian and it is defined as the total energy (kinetic + potential) of the system. Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 59 / 70
  • 105. Time Evolution: Schrodinger Equation Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline The time evolution of a closed quantum system is given by the Schrodinger Equation: Introduction Motivations for Quantum Computation Qubit i d|ψ dt = H |ψ Linear Algebra Uncertainty Principle where H is the hamiltonian and it is defined as the total energy (kinetic + potential) of the system. Postulates of Quantum Mechanics Next Presentation Reference The connection between the hamitonian picture and the unitary 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 ) Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 59 / 70
  • 106. Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
  • 107. Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 60 / 70
  • 108. Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Postulates of Quantum Mechanics Next Presentation Reference After measurement, the state of the system collapses to either |0 or |1 with the said probability. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
  • 109. Attempt at 3rd Postulate Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Unlike classical physics, measurement in quantum mechanics is not deterministic. Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle If we have a quantum state |ψ = α |0 + β |1 , then the probability of getting outcome |0 is |α|2 and that of |1 is |β|2 . Postulates of Quantum Mechanics Next Presentation Reference After measurement, the state of the system collapses to either |0 or |1 with the said probability. However, after measurement if the new state of the system is |0 (say), then further measurements in the same basis gives outcome |0 with probability 1. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 60 / 70
  • 110. 3rd Postulate: Measurement Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Postulate 3 Motivations for Quantum Computation Quantum measurements are described by a collection {Mm } of measurement operators. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ before experiment, then the probability that result m occurs is given by, Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference p(m) = ψ| Mm † .Mm |ψ And the state of the system after measurement is √ Ritajit Majumdar, Arunabha Saha (CU) Mm |ψ ψ|Mm †.Mm |ψ Introduction to Quantum Computation September 9, 2013 61 / 70
  • 111. So what is the Big Deal? Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha The postulate states that a quantum system stays in a superposition when it is not observed. When a measurement is done, it immediately collapses to one of its eigenstates. Hence we can never observe what the original superposition of the system was. We merely observe the state after it collapses. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra This inherent ambiguity provides an excellent security in Quantum Cryptography. Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 62 / 70
  • 112. Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
  • 113. Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
  • 114. Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Without opening the box, an observer cannot know whether the cat is alive or dead. So the cat may be said to be in a superposition of the two states. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference September 9, 2013 63 / 70
  • 115. Introduction to Quantum Computation Schrodinger’s Cat Ritajit Majumdar, Arunabha Saha This is a thought experiment proposed by Erwin Schrodinger. Place a cat in a steel chamber with a device containing a vial of hydrocyanic acid and a radioactive substance. If even a single atom of the substance decays, it will trip a hammer and break the vial which in turn kills the cat. Without opening the box, an observer cannot know whether the cat is alive or dead. So the cat may be said to be in a superposition of the two states. Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference However, when the box is opened, we observe deterministically that the cat is either dead or alive. We can, by no means, observe the superposition. Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 63 / 70
  • 116. 4th Postulate: Composite System Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction What we have seen so far was a single qubit system. What happens when there are multiple qubits? This is given by the last postulate: Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Postulate 4 Next Presentation The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have n systems, and the system number i is prepared in state |ψi , then the joint state of the total system is Reference |ψ1 ⊗ |ψ2 ⊗ ... ⊗ |ψn Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 64 / 70
  • 117. Introduction to Quantum Computation Two Qubit System Ritajit Majumdar, Arunabha Saha Outline Introduction Let us consider a two qubit system. Classically, with two bits, we can have 4 states - 00, 01, 10, 11. A quantum system is a linear superposition of all these four states. Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle So, a general two qubit quantum state can be represented as |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 10 Postulates of Quantum Mechanics Next Presentation Reference 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
  • 118. Measurement in Two Qubit System Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Measurement is similar to single qubit system. When we measure the two qubit system we get outcome j with probability |αj |2 and the new state will be |j . Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics That is for the system Next Presentation Reference |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 , we get outcome |00 with probability |α00 |2 and the new state of the system will be |00 . Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 66 / 70
  • 119. Introduction to Quantum Computation Partial Measurement Ritajit Majumdar, Arunabha Saha Outline So what if we want to measure only the first qubit? Or maybe only the second one? Introduction Motivations for Quantum Computation Qubit We take the same two qubit system, |ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 , Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation If only the first qubit is measured then we get the outcome |0 for the first qubit with probability |α00 |2 + |α01 |2 Reference 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
  • 120. Coming up in next talk... Introduction to Quantum Computation Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Einstein-Polosky-Rosen (EPR) Paradox Qubit Bell State Linear Algebra Uncertainty Principle Quantum Entanglement Density Matrix Notation of Quantum Mechanics Quantum Gates Postulates of Quantum Mechanics Next Presentation Reference and many more... Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 68 / 70
  • 121. Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Michael A. Nielsen, Isaac Chuang Quantum Computation and Quantum Information Cambridge University Press David J. Griffiths Introduction to Quantum Mechanics Prentice Hall, 2nd Edition Umesh Vazirani, University of California Berkeley Quantum Mechanics and Quantum Computation https://class.coursera.org/qcomp-2012-001/ Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle Postulates of Quantum Mechanics Next Presentation Reference Michael A. Nielsen, University of Queensland Quantum Computing for the determined http://michaelnielsen.org/blog/ quantum-computing-for-the-determined/ James Branson, University of California San Diego Quantum Physics (UCSD Physics 130) http://quantummechanics.ucsd.edu/ph130a/130_ notes/130_notes.html Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation September 9, 2013 69 / 70
  • 122. Introduction to Quantum Computation Reference Ritajit Majumdar, Arunabha Saha Outline Introduction Motivations for Quantum Computation Qubit Linear Algebra Uncertainty Principle N. David Mermin, Cornell University Lecture Notes on Quantum Computation Postulates of Quantum Mechanics Next Presentation http://www.wikipedia.org Ritajit Majumdar, Arunabha Saha (CU) Introduction to Quantum Computation Reference September 9, 2013 70 / 70