Quantum Mechanics for Quantum Information & Computation

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

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

IPQI-2010-Anu Venugopalan 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 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)

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 IPQI-2010-Anu Venugopalan

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

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

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….

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??

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

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

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 The space of ‘n-tuples’ of complex numbers, (z1, z2, z3,………zn) C n Elements of vector spaces vectors IPQI-2010-Anu Venugopalan

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’ IPQI-2010-Anu Venugopalan

Quantum mechanics & Linear Algebra_____________________________________________ Associated to any quantum system is a complex vector space known as state space. IPQI-2010-Anu Venugopalan The state of a closed quantum system is a unit vector in state space. A qubit, has a two-dimensional state space C2. Most physical systems often have finite dimensional state spaces ‘Qudit’ Cd

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 C2 IPQI-2010-Anu Venugopalan

Linear Algebra & vector spaces___________________________________ Example: For a Qubit: Vector Space C2 and span the Vector space C2 IPQI-2010-Anu Venugopalan

Linear Algebra & vector spaces___________________________________ A particular vector space could have many spanning sets. Example: For C2 and also span the Vector space C2 IPQI-2010-Anu Venugopalan

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 IPQI-2010-Anu Venugopalan

IPQI-2010-Anu Venugopalan 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 : C2 ;

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

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 IPQI-2010-Anu Venugopalan

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: IPQI-2010-Anu Venugopalan

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 IPQI-2010-Anu Venugopalan

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

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

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

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 aUnitary transformation A matrix/operator U is said to be Unitary if Unitary operators preserve normalization /inner products IPQI-2010-Anu Venugopalan

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

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

IPQI-2010-Anu Venugopalan Unitary operators & Matrices- examples___________________________________ Unitary operators acting on qubits The Quantum Hadamard Gate

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

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

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

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

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

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

Quantum Mechanics for Quantum Information & Computation