Quantum Information Processing QIP 1 Basic Concepts Dan

Quantum Information Processing QIP 1 –Basic Concepts Dan C. Marinescu Computer Science Division, EECS Department University of Central Florida Email: dcm@cs. ucf. edu 10/15/2021 UTFSM May-June 2012 1

Contents n n n Laws of physics and the limits of solid-state technology The mathematical model A bit or a qubit of history The lectures are available at: http: //www. cs. ucf. edu/~dcm/Chile 2012/Chile. Index. html 10/15/2021 UTFSM May-June 2012 2

Moore’s Law 10/15/2021 Microprocessor Year 4004 1971 2, 250 8008 1972 2, 500 8080 1974 5, 000 8086 1978 29, 000 286 1982 120, 000 386 1985 275, 000 486 1989 1, 180, 000 Pentium 1993 3, 100, 000 Pentium II 1997 7, 500, 000 Pentium IV 2000 42, 000 Itanium 2002 220, 000 Itanium II 2003 410, 000 UTFSM May-June 2012 # transistors 3

Limits of solid-state technology n n n To increase the clock rate we have to pack transistors as densely as possible because the speed of light is finite. The power dissipation increases with the cube of the clock rate. When we double the speed of a device its power dissipation increases 8 (eight) fold. The computer technology vintage year 2000 requires some 3 x 10 -18 Joules/elementary operation. An exponential growth cannot be sustained indefinitely; sooner or later one will hit a wall. Revolutionary rather than evolutionary approach to information processing and to communication: ¨ Quantum computing and communication quantum information. ¨ DNA Computing biological information. 10/15/2021 UTFSM May-June 2012 4

Quantum information processing n n Quantum information Information encoded as the state of atomic or sub-atomic particles. A happy marriage between three of the greatest scientific achievements of the 20 th century: quantum mechanics ¨ stored program computers ¨ information theory ¨ n In 1985 Richard Feynman wrote: “. . it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms and quantum behavior holds sway. ” 10/15/2021 UTFSM May-June 2012 5

Information can be encoded as the photon polarization n n Light is a form of electromagnetic radiation. The electric and magnetic field oscillate in a plane perpendicular to the direction of propagation and ¨ are perpendicular to each other. ¨ n A photon is characterized by its vector momentum (the vector momentum determines the frequency) and ¨ polarization. ¨ n n n 10/15/2021 In the classical electromagnetic theory light is described as having an electric field which oscillates either vertically, the light is x-polarized, or horizontally, the light is y-polarized in a plane perpendicular to the direction of propagation, the z-axis. The two basis vectors are |h> and |v> Information encoding: 0 vertically polarized photon 1 horizontally polarized photon UTFSM May-June 2012 6

Information can be encoded as the spin of the electron n n Spin the intrinsic angular momentum; it takes discrete values (the spin quantum number s. ) Two classes of quantum particles: ¨ fermions - spin one-half particles (e. g. , electrons). n s=+1/2 and s=-1/2 ¨ bosons - spin one particles (e. g. , photons). n s=+1, s=0, and s=-1 The electron has spin s = ½; the spin projection can assume the values + ½ referred to as spin up, and - ½ referred to as spin down. The information can be encoded 0 spin up 1 spin down 10/15/2021 UTFSM May-June 2012 7

Quantum systems n Quantum concepts such as: Uncertainty, ¨ Superposition, ¨ Entanglement, ¨ No-cloning ¨ n do not have a correspondent in classical physics. Heisenberg’s Uncertainty Principle: the position and the momentum of a quantum particle cannot be determined with arbitrary precision. h=6. 6262 x 10 -34 Joule x second Planck’s constant 10/15/2021 UTFSM May-June 2012 8

Non-determinism is a basic tenet of quantum physics “Liebe Gott würfelt nicht” (Dear God does not play dice) - Albert Einstein 10/15/2021 UTFSM May-June 2012 9

The qubit n n n Qubit a unit of quantum information, the correspondent of a bit of classical information. Information is physical a qubit must be encoded as some property of a quantum particle. There are multiple physical embodiments of a qubit Photon – information encoded in the polarization ¨ Electron – information encoded in the spin ¨ Electron – information encoded in electric charge (quantum dots) ¨ Trapped ions in a cavity ¨ n n The physics of each embodiment is different: optics, condensed matter physics, electromagnetic field theory, etc. We need a mathematical model to describe quantum information regardless of the physical embodiment of a qubit. 10/15/2021 UTFSM May-June 2012 10

The mathematical model of a qubit A mathematical model of a physical system should describe: n 1. 2. 3. 4. 5. n The state of the system The means to observe the system The dynamics of the system, the evolution from one state to another The measurement process How to describe a system consisting of two sub-systems The postulates of quantum mechanics establish a correspondence between the physical system and mathematical objects of the model The state postulate ¨ The dynamics postulate ¨ The measurement postulate ¨ n The model State of a quantum system vector in an n-dimensional Hilbert space 2. Observables the means to observe the system, linear operators 3. System dynamics Hermitian (self-adjoint) operators 4. System measurements measurement operators 1. 10/15/2021 UTFSM May-June 2012 11

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More about the measurement postulate n n The numerical outcome of a measurement of observable A of a quantum system in state is an eigenvalue of the operator A used to measure the observable A; immediately after the measurement the quantum state of the system is an eigenvector |ai> coresponding to the eigenvalue of the operator A. A measurement is a projection of the current state. In a projective measurement the measurement operators are selfadjoint and idempotent. The number of such operators is equal to the dimension of the Hilbert space. Since orthogonal measurement operators commute, they correspond to simultaneous observables. von Neumann gave an axiomatic definition of the measurement. A von Neumann measurement is a conditional expectation onto a maximal Abelian subalgebra of the algebra of all bounded operators acting on the Hilbert space. 10/15/2021 UTFSM May-June 2012 13

The state of a quantum system after a measurement n n n Consider a quantum system in state The state of the system after we apply the projector Pi the is The completeness of the set of projectors leads to a normalization condition and after the mesaurement the state becomes a normaized pure state: 10/15/2021 UTFSM May-June 2012 14

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n n 10/15/2021 UTFSM May-June 2012 Two stage von Neumann-type measurement. We measure the observable Q of the quantum system S using a macroscopic apparatus M. At time t the quantum system is in a superposition state and the apparatus is in state |m>. qi is the eigenvalue corresponding to the eigenvector ai of observable Q. At the end of the first phase the composite system S + M is in state shown at the left. Then, as a result of the measurement, the system collapses. 16

Quantum states n n n Quantum states are represented as vectors in an n-dimensional Hilbert space Hn a vector space over C the set of complex numbers. A canonical orthonormal basis in Hn can be expressed as the ket vectors |0>, |1>, |2>, ……|n-1> or as the bra vectors <0|, <1|, <2|, ……<n-1| The matrix representations of these vectors are: 10/15/2021 UTFSM May-June 2012 17

One qubit n n n Mathematical abstraction Vector in a two dimensional complex vector space (Hilbert space) Dirac’s notation ket column vector bra row vector bra dual vector (transpose and complex conjugate) 10/15/2021 UTFSM May-June 2012 18

Orthonormal bases 10/15/2021 UTFSM May-June 2012 19

One qubit are complex numbers 10/15/2021 UTFSM May-June 2012 20

The Boch sphere representation of one qubit n n A qubit in a superposition state is represented as a vector connecting the center of the Bloch sphere with a point on its periphery. The two probability amplitudes can be expressed using Euler angles. 10/15/2021 UTFSM May-June 2012 21

A bit versus a qubit n A bit Can be in two distinct states, 0 and 1 ¨ A measurement does not affect the state. ¨ n A qubit ¨ can be in state or in any other state that is a linear combination of the basis state ¨ when we measure the qubit we find it n in state with probability 10/15/2021 UTFSM May-June 2012 22

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Other possible states of a qubit 10/15/2021 UTFSM May-June 2012 24

A different basis for one qubit 10/15/2021 UTFSM May-June 2012 25

The measurement of a qubit 10/15/2021 UTFSM May-June 2012 26

A measurement is a projection of the surrent state 10/15/2021 UTFSM May-June 2012 27

Two qubits n Represented as a vector in a 2 -dimensional Hilbert space with four basis vectors with n When we measure a pair of qubits we decide that the system it is in one of four states 00, 01, 10, and 11 with probabilities 10/15/2021 UTFSM May-June 2012 28

A measurement of two qubits n n Before a measurement the state of the two qubits is uncertain. If we measure both qubits the state can be either 00, 01, 10, or 11. What if we observe only the first qubit, what conclusions can we draw? We expect that the system to be left in an uncertain sate, because we did not measure the second qubit that can still be in a continuum of states. The first qubit can be ¨ 0 with probability ¨ 1 with probability 10/15/2021 UTFSM May-June 2012 29

The measurement of the first qubit of the pair only n n Call the post-measurement state when we measure the first qubit and find it to be 0. Call the post-measurement state when we measure the first qubit and find it to be 1. 10/15/2021 UTFSM May-June 2012 30

The measurement of the second qubit of the pair only n n Call the post-measurement state when we measure the second qubit and find it to be 0. Call the post-measurement state when we measure the second qubit and find it to be 1. 10/15/2021 UTFSM May-June 2012 31

Bell states - a special state of a pair of qubits n If and When we measure the first qubit we get the post measurement state When we measure the second qubit we get the post measurement state 10/15/2021 UTFSM May-June 2012 32

This is an amazing result! n n n The two measurements are correlated, once we measure the first qubit we get exactly the same result as when we measure the second one. The two qubits need not be physically constrained to be at the same location and yet, because of the strong coupling between them, measurements performed on the second one allow us to determine the state of the first. An entangled pair is a single quantum system in a superposition of equally possible states. The entangled state contains no information about the individual particles, only that they are in opposite states. The important property of an entangled pair is that the measurement of one particle influences the state of the other particle. Einstein called that “Spooky action at a distance". Entanglement is an elegant, almost exact translation of the German term Verschrankung used by Schrodinger who was the first to recognize this quantum effect. 10/15/2021 UTFSM May-June 2012 33

Milestones in quantum physics n n n 1900 - Max Plank presents the black body radiation theory; the quantum theory is born. 1905 - Albert Einstein develops theory of the photoelectric effect. 1911 - Ernest Rutherford develops the planetary model of the atom. 1913 - Niels Bohr develops the quantum model of the hydrogen atom. 1923 - Louis de Broglie relates the momentum of a particle with the wavelength 1925 - Werner Heisenberg formulates the matrix quantum mechanics. 1926 - Erwin Schrodinger proposes the equation for the dynamics of the wave function. 1926 - Erwin Schrodinger and Paul Dirac show the equivalence of Heisenberg's matrix formulation and Dirac's algebraic one with Schrodinger's wave function. 1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pasqual Jordan obtain a complete formulation of quantum dynamics. 1926 - John von Newmann introduces Hilbert spaces to quantum mechanics. 1927 - Werner Heisenberg formulates the uncertainty principle. 10/15/2021 UTFSM May-June 2012 34

Milestones in computing and information theory n n n 1936 - Alan Turing dreams up the Universal Turing Machine, UTM. 1936 - Alonzo Church publishes a paper asserting that ``every function which can be regarded as computable can be computed by an universal computing machine''. 1945 - ENIAC, the world's first general purpose computer, the brainchild of J. Presper Eckert and John Macauly becomes operational. 1946 - A report co-authored by John von Neumann outlines the von Neumann architecture. 1948 - Claude Shannon publishes ``A Mathematical Theory of Communication’’. 1953 - The first commercial computer, UNIVAC I. 10/15/2021 UTFSM May-June 2012 35

Milestones in quantum computing n n n 1961 - Rolf Landauer decrees that computation is physical and studies heat generation. 1973 - Charles Bennett studies the logical reversibility of computations. 1982 - Richard Feynman suggests that physical systems including quantum systems can be simulated exactly with quantum computers. 1982 - Peter Benioff develops quantum mechanical models of Turing machines. 1984 - Charles Bennett and Gilles Brassard introduce quantum cryptography. 1985 - David Deutsch reinterprets the Church-Turing conjecture. 1993 - Bennett, Brassard, Crepeau, Josza, Peres, Wooters discover quantum teleportation. 1994 - Peter Shor develops a clever algorithm for factoring large numbers. 1996 – Lov Grover develops the quantum search algorithm. 1997 – Demonstration of quantum teleportation; De Martini at University of Rome; Zeillinger at Innsbruck 2001 – A group at IBM Research in San Jose build a quantum computer with seven qubits and factors the number 15 using Shor’s algorithm. 10/15/2021 UTFSM May-June 2012 36

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Next seminar – Tuesday June 4, 2012 n n n n n Classical gates Quantum gates Unitary transformations One-qubit gates Two-qubit gates, CNOT Three-qubit gates Fredkin and Toffoli gates Universal quantum gates Reversibility of quantum circuits Decoherere Di. Vincenzo’s criteria for realization of quantum circuits 10/15/2021 UTFSM May-June 2012 38