1. IPQI-2010-Anu Venugopalan 1 Quantum Mechanics for Quantum Information & Computation Anu Venugopalan Guru Gobind Singh Indraprastha Univeristy Delhi _______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-2010) Institute of Physics (IOP), Bhubaneswar January 2010

2. IPQI-2010-Anu Venugopalan 2 Computer technology in the last fifty years- dramatic miniaturization Faster and smaller – - the memory capacity of a chip approximately doubles every 18 months – clock speeds and transistor density are rising exponentially...what is their ultimate fate???? Real computers are physical systems

3. IPQI-2010-Anu Venugopalan 3 Moore’s law [www.intel.com]

4. IPQI-2010-Anu Venugopalan 4 The future of computer technology If Moore’s law is extrapolated, by the year 2020 the basic memory component of the chip would be of the size of an atom – what will be space, time and energy considerations at these scales (heat dissipation…)? At such scales, the laws of quantum physics would come into play - the laws of quantum physics are very different from the laws of classical physics - everything would change! [“There’s plenty of room at the bottom” Richard P. Feynman (1969) Feynman explored the idea of data bits the size of a single atom, and discussed the possibility of building devices an atom or a molecule at a time (bottom-up approach) - nanotechnology]

5. IPQI-2010-Anu Venugopalan 5 Quantum Mechanics _______________________________ At the turn of the last century, there were several experimental observations which could not be explained by the established laws of classical physics and called for a radically different way of thinking This led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature

6. IPQI-2010-Anu Venugopalan 6 Some key events/observations that led to the development of quantum mechanics… ___________________________________ Black body radiation spectrum (Planck, 1901) Photoelectric effect (Einstein, 1905) Model of the atom (Rutherford, 1911) Quantum Theory of Spectra (Bohr, 1913) Scattering of photons off electrons (Compton, 1922) Exclusion Principle (Pauli, 1922) Matter Waves (de Broglie 1925) Experimental test of matter waves (Davisson and Germer, 1927)

7. IPQI-2010-Anu Venugopalan 7 Quantum Mechanics ___________________________________  Matter and radiation have a dual nature – of both wave and particle  The matter wave associated with a particle has a de Broglie wavelength given by  The wave corresponding to a quantum system is described by a wave function or state vector

8. IPQI-2010-Anu Venugopalan 8 Quantum Mechanics ___________________________________ Quantum Mechanics is the most accurate and complete description of the physical world – It also forms a basis for the understanding of quantum information

9. Quantum Mechanics _______________________________________________________ Quantum Mechanics – most successful working theory of Nature…….. The price to be paid for this powerful tool is that some of the predictions that Quantum Mechanics makes are highly counterintuitive and compel us to reshape our classical (‘common sense’) notions......... Schrödinger Equation Linear superposition principle  Linear  Deterministic  Unitary evolution

10. Some conceptual problems in QM: quantum measurement, entanglements, nonlocality ___________________________________ Quantum Measurement Basic postulates of quantum measurement Measurement on yields eigenvalue with probability Measurement culminates in a collapse or reduction of to one of the eigenstates, ‘non unitary’ process….

11. Some conceptual problems in QM: quantum measurement, entanglements, nonlocality _________________________________________ Macroscopic Superpositions linear superposition principle Schrödinger's Cat Such states are almost never seen for classical (‘macro’) objects in our familiar physical world….but the ‘macro’ is finally made up of the ‘micro’…so, where is the boundary??

12. Conceptual problems of QM: quantum measurement, entanglements, nonlocality ___________________________________ Quantum entanglements – a uniquely quantum mechanical phenomenon associated with composite systems A B

13. IPQI-2010-Anu Venugopalan 13 The Qubit ______________________________________ Normalization Physical implementations - Photons, electron, spin, nuclear spin ‘Bit’ : fundamental concept of classical computation & info. - 0 or 1 ‘Qubit’ : fundamental concept of quantum computation & info - can be thought of mathematical objects having some specific properties

14. IPQI-2010-Anu Venugopalan 14 Quantum Mechanics & Linear Algebra ___________________________________ Linear Algebra: The study of vector spaces and of linear operations on those vector spaces. Basic objects of Linear algebra Vector spaces C n The space of ‘n-tuples’ of complex numbers, (z1, z2, z3,………zn) Elements of vector spacesvectors

15. IPQI-2010-Anu Venugopalan 15 Quantum mechanics & Linear Algebra ___________________________________ Vector :column matrix The standard quantum mechanical representation for a vector in a vector space : : ‘Ket’Dirac notation The state of a closed quantum system is described by such a ‘state vector’ described on a ‘state space’

16. IPQI-2010-Anu Venugopalan 16 Quantum mechanics & Linear Algebra _____________________________________________ Associated to any quantum system is a complex vector space known as state space. A qubit, has a two-dimensional state space C 2. The state of a closed quantum system is a unit vector in state space. Most physical systems often have finite dimensional state spaces ‘Qudit’ C d

17. IPQI-2010-Anu Venugopalan 17 Linear Algebra & vector spaces ___________________________________ Vector space V, closed under scalar multiplication & addition Spanning set: A set of vectors in V : such that any vector in the space V can be expressed as a linear combination: Example: For a Qubit: Vector Space C 2

18. IPQI-2010-Anu Venugopalan 18 Linear Algebra & vector spaces ___________________________________ Example: For a Qubit: Vector Space C 2 and span the Vector space C 2

19. IPQI-2010-Anu Venugopalan 19 Linear Algebra & vector spaces ___________________________________ A particular vector space could have many spanning sets. Example: For C 2 and also span the Vector space C 2

20. IPQI-2010-Anu Venugopalan 20 Linear Algebra & vector spaces ___________________________________ A set of non zero vectors, are linearly dependent if there exists a set of complex numbers for at least one value of i such that A set of nonzero vectors is linearly independent if they are not linearly dependent in the above sense

21. IPQI-2010-Anu Venugopalan 21 Linear Algebra & vector spaces ___________________________________ Any two sets of linearly independent vectors that span a vector space V have the same number of elements A linearly independent spanning set is called a basis set The number of elements in the basis set is equal to the dimension of the vector space V For a qubit, V : C 2 ;

22. IPQI-2010-Anu Venugopalan 22 Linear operators & Matrices ________________________________ Computational Basis for a Qubit A linear operator between vector spaces V and W is defines as any function   : V W, which is linear in its inputs Î: Identity operator Ô: Zero Operator Once the action of a linear operator  on a basis is specified, the action of  is completely determined on all inputs

23. IPQI-2010-Anu Venugopalan 23 Linear operators & Matrices __________________________________ Linear operators and Matrix representations are equivalent Examples: Four extremely useful matrices that operate on elements in C 2 The Pauli Matrices

24. IPQI-2010-Anu Venugopalan 24 Linear operators and matrices - some properties ____________________________________ Inner product - A vector space equipped with an inner product is called an inner product space- e.g. “Hilbert Space” Norm:

25. IPQI-2010-Anu Venugopalan 25 Linear operators and matrices - some properties ____________________________________ Norm: Normalized form for any non-zero vector: A set of vectors with index i is orthonormal if each vector is a unit vector and distinct vectors are orthogonal The Gram-Schmidt orthonormalization procedure

26. IPQI-2010-Anu Venugopalan 26 Linear operators and matrices - some properties ____________________________________ Outer Product vector in inner product space V vector in inner product space W A linear operator from V to W completeness relation

27. IPQI-2010-Anu Venugopalan 27 Linear operators and matrices - some properties ____________________________________ Eigenvalues and eigenvectors Diagonal Representation An orthonormal set of eigenvectors for  with corresponding eigenvalues i example diagonal representation for  z

28. IPQI-2010-Anu Venugopalan 28 The Postulates of Quantum Mechanics ____________________________________ Quantum mechanics is a mathematical framework for the development of physical theories. The postulates of quantum mechanics connect the physical world to the mathematical formalism Postulate 1: Associated with any isolated physical system is a complex vector space with inner product, known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space A qubit, has a two-dimensional state space: C 2.

29. IPQI-2010-Anu Venugopalan 29 The Postulates of Quantum Mechanics ____________________________________ Evolution - How does the state,, of a quantum system change with time? Postulate 2: The evolution of a closed quantum system is described by a Unitary transformation A matrix/operator U is said to be Unitary if Unitary operators preserve normalization /inner products

30. IPQI-2010-Anu Venugopalan 30 Hermitian conjugation; taking the adjoint A is said to be unitary if We usually write unitary matrices as U. The Postulates of Quantum Mechanics - Unitary operators/Matrices _________________________________________

31. IPQI-2010-Anu Venugopalan 31 Linear operators & Matrices – operations on a Qubit (examples) ___________________________________ The Pauli Matrices- Unitary operators on qubits - Gates NOT Gate Phase flip Gate

32. IPQI-2010-Anu Venugopalan 32 Unitary operators & Matrices- examples ___________________________________ Unitary operators acting on qubits The Quantum Hadamard Gate

33. IPQI-2010-Anu Venugopalan 33 The Postulates of Quantum Mechanics ____________________________________ Quantum Measurement The outcome of the measurement cannot be determined with certainty but only probabilistically Soon after the measurement, the state of the system changes (collapses) to an eigenstate of the operator corresponding to measured observable

34. IPQI-2010-Anu Venugopalan 34 The Postulates of Quantum Mechanics ____________________________________ Quantum Measurement Postulate 3:. Unlike classical systems, when we measure a quantum system, our action ends up disturbing the system and changing its state. The act of quantum measurements are described by a collection of measurement operators which act on the state space of the system being measure

35. IPQI-2010-Anu Venugopalan 35 Measuring a qubit _____________________________________ If we measure in the computational basis, i.e., and

36. More general measurements ____________________________________ Observable A (to be measured) corresponds to operator has a set of eigenvectors with corresponding eigenvalues To measure on the system whose state vector is one expresses in terms of the eigenvectors

37. More general measurements ____________________________________ 1.The measurement on state yields only one of the eigenvalues, with probability 2.The measurement culminates with the state collapsing to one of the eigenstates, The process is non unitary

38. Quantum Classical transition in a quantum measurement Several interpretations of quantum mechanics seek to explain this transition and a resolution to this apparent nonunitary collapse in a quantum measurement. The collapse of the wavefunction following measurement The quantum measurement paradox/foundations of quantum mechanics