Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 IPQI-2010-Anu Venugopalan 1 Real computers are physical systems 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???? IPQI-2010-Anu Venugopalan 2 Moore’s law [www.intel.com] IPQI-2010-Anu Venugopalan 3 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] IPQI-2010-Anu Venugopalan 4 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 IPQI-2010-Anu Venugopalan 5 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) IPQI-2010-Anu Venugopalan 6 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 h p The wave corresponding to a quantum system is described by a wave function or state vector IPQI-2010-Anu Venugopalan 7 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 IPQI-2010-Anu Venugopalan 8 Quantum Mechanics _______________________________________________________ Quantum Mechanics Nature…….. – most successful working theory of 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 d i | H | dt Linear Deterministic Unitary evolution Linear superposition principle Some conceptual problems in QM: quantum measurement, entanglements, nonlocality ___________________________________ Quantum Measurement d ˆ : a , | | c i | i i | H | A i i dt Basic postulates of quantum measurement Measurement on | yields eigenvalue a i with probability | ci | 2 Measurement culminates in a collapse or reduction of | to one of the eigenstates, {| i } ‘non unitary’ process…. Some conceptual problems in QM: quantum measurement, entanglements, nonlocality _________________________________________ Macroscopic Superpositions i d | H | dt linear superposition principle Schrödinger's Cat | atom cat | | alive | | dead 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?? Conceptual problems of QM: quantum measurement, entanglements, nonlocality ___________________________________ Quantum entanglements – a uniquely quantum mechanical phenomenon associated with composite systems | AB c1 | 1 | 1 c2 | 2 | 2 | A | B | EPR A 1 2 {| A | B | A | B } B The Qubit ______________________________________ ‘Bit’ : fundamental concept of classical computation & info. - 0 or 1 ‘Qubit’ : fundamental concept of quantum computation & info 0 1 | |2 | |2 1 1 Normalization 0 - can be thought of mathematical objects having some specific properties Physical implementations - Photons, electron, spin, nuclear spin IPQI-2010-Anu Venugopalan 13 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 Cn The space of ‘n-tuples’ of complex numbers, (z1, z2, z3,………zn) Elements of vector spaces IPQI-2010-Anu Venugopalan vectors 14 Quantum mechanics & Linear Algebra ___________________________________ Vector : V column matrix The standard quantum mechanical representation for a vector in a vector space : : ‘Ket’ zz1 . 2 . zn Dirac notation The state of a closed quantum system is described by such a ‘state vector’ described on a ‘state space’ IPQI-2010-Anu Venugopalan 15 Quantum mechanics & Linear Algebra _____________________________________________ Associated to any quantum system is a complex vector space known as state space. The state of a closed quantum system is a unit vector in state space. A qubit, has a two-dimensional state space C2. 0 1 Most physical systems often have finite dimensional state spaces IPQI-2010-Anu Venugopalan 0 0 1 1 2 2 ... d 1 d 1 0 1 2 : d 1 ‘Qudit’ Cd 16 Linear Algebra & vector spaces ___________________________________ •Vector space V, closed under scalar multiplication & addition •Spanning set: A set of vectors in V : v1 , v3 , v3 ....... vn such that any vector v in the space V can be expressed as a linear combination: v ai vi i Example: For a Qubit: Vector Space IPQI-2010-Anu Venugopalan C2 17 Linear Algebra & vector spaces ___________________________________ Example: For a Qubit: Vector Space C2 0 1 v1 ; v2 1 0 v1 and v1 span the Vector space C2 a1 v ai vi a1 v1 a2 v2 a2 i IPQI-2010-Anu Venugopalan 18 Linear Algebra & vector spaces ___________________________________ A particular vector space could have many spanning sets. Example: For C2 1 1 w1 ; w2 1 1 w1 and w2 also span the Vector space C2 a1 a1 a2 a1 a2 v w1 w2 2 2 a2 IPQI-2010-Anu Venugopalan 19 Linear Algebra & vector spaces ___________________________________ v1 , v3 , v3 ....... vn are A set of non zero vectors, linearly dependent if there exists a set of complex numbers a1 , a2 ......an with ai 0 for at least one value of i such that a1 v1 a2 v2 ....................an vn 0 A set of nonzero vectors is linearly independent if they are not linearly dependent in the above sense IPQI-2010-Anu Venugopalan 20 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 ; IPQI-2010-Anu Venugopalan 0 and 1 are the computational basis states 21 Linear operators & Matrices ________________________________ 1 0 Computational Basis for a Qubit 0 ; 1 0 1 A linear operator between vector spaces V and W is defines as any function  Â: V W, which is linear in its inputs Aˆ ai vi ai Aˆ vi i i 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 IPQI-2010-Anu Venugopalan 22 Linear operators & Matrices __________________________________ Linear operators and Matrix representations are equivalent Examples: Four extremely useful matrices that operate on elements in C 2 1 0 0 I 0 1 IPQI-2010-Anu Venugopalan 0 1 x 1 0 0 i y i 0 1 0 z 0 1 The Pauli Matrices 23 Linear operators and matrices - some properties ____________________________________ v , w vw Inner product - A vector space equipped with an inner product is called an inner product space- e.g. “Hilbert Space” v , w v , w * Norm: v IPQI-2010-Anu Venugopalan v , v 0 v v 1 for a unit vecto r 24 Linear operators and matrices - some properties ____________________________________ Norm: v v v 1 for a unit vecto r Normalized form for any non-zero vector: v v A set of vectors with index i is orthonormal if each vector is a unit vector and distinct vectors are orthogonal i j ij IPQI-2010-Anu Venugopalan The Gram-Schmidt orthonormalization procedure 25 Linear operators and matrices - some properties ____________________________________ Outer Product w v : w v v : vector in inner product space V w : vector in inner product space W A linear operator from V to W w v v' w v v' v v' w IPQI-2010-Anu Venugopalan completeness relation i i I i 26 Linear operators and matrices - some properties ____________________________________ Eigenvalues and eigenvectors Diagonal Representation Aˆ v v v ˆ A i i i i c( ) det Aˆ Iˆ 0 i example diagonal representation for z IPQI-2010-Anu Venugopalan An orthonormal set of eigenvectors for  with corresponding eigenvalues i 1 0 0 0 1 1 z 0 1 27 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: C2. IPQI-2010-Anu Venugopalan 28 The Postulates of Quantum Mechanics ____________________________________ Evolution - How does the state, change with time? , of a quantum system Postulate 2: The evolution of a closed quantum system is described by a Unitary transformation ' U A matrix/operator U is said to be Unitary if U U I Unitary operators preserve normalization /inner products IPQI-2010-Anu Venugopalan 29 The Postulates of Quantum Mechanics - Unitary operators/Matrices _________________________________________ a b A c d Hermitian conjugation; taking the adjoint A† A* T a * c * * * b d A is said to be unitary if AA† A†A I We usually write unitary matrices as U. Example: 0 1 0 1 1 0 XX I 1 0 1 0 0 1 † IPQI-2010-Anu Venugopalan 30 Linear operators & Matrices – operations on a Qubit (examples) ___________________________________ The Pauli Matrices- Unitary operators on qubits - Gates 0 1 ˆ ˆ X ; X 0 1 ; Xˆ 1 0 x 1 0 NOT Gate 0 i ˆ ˆ Y ; Y 0 i 1 ; Yˆ 1 i 0 y i 0 1 0 ˆ Phase flip Z ; Ẑ 0 0 ; Ẑ 1 1 z Gate 0 1 IPQI-2010-Anu Venugopalan 31 Unitary operators & Matrices- examples ___________________________________ Unitary operators acting on qubits The Quantum Hadamard Gate 1 1 1 ˆ H 2 1 1 1 1 ˆ ˆ 0 1 ; H 1 0 1 H 0 2 2 IPQI-2010-Anu Venugopalan 32 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 IPQI-2010-Anu Venugopalan 33 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 IPQI-2010-Anu Venugopalan 34 Measuring a qubit _____________________________________ 0 1 Quantum mechanics DOES NOT allow us to determine and . We can, however, read out limited information about and . If we measure in the computational basis, i.e., 2 P (0) ; P (1) 0 and 1 2 Measurement unavoidably disturbs the system, leaving it in a state 0 or 1 determined by the outcome. IPQI-2010-Anu Venugopalan 35 More general measurements ____________________________________ Observable A (to be measured) corresponds to operator   has a set of eigenvectors with corresponding eigenvalues Aˆ : ai , | i To measure  on the system whose state vector is one expresses | in terms of the eigenvectors | ci | i | More general measurements ____________________________________ | ci | i 1.The measurement on state the eigenvalues, ai | yields only one of with probability | ci | 2 2.The measurement culminates with the state collapsing to one of the eigenstates, The process is non unitary | i Quantum Classical transition in a quantum measurement The collapse of the wavefunction following measurement Several interpretations of quantum mechanics seek to explain this transition and a resolution to this apparent nonunitary collapse in a quantum measurement. The quantum measurement paradox/foundations of quantum mechanics