Quantum Computing Part 1: Quantum Mechanics Overview

Quantum ComputingPart 1: Quantum Mechanics Overview

Technological limits Gordon Moore’s law: • The observation made in 1965 by Gordon Moore, co-founder of Intel, that the number of transistors per square inch on integrated circuits had doubled every year since the integrated circuit was invented. • Moore predicted that this trend would continue for the foreseeable future. • In subsequent years, the pace slowed down a bit, but data density has doubled approximately every 18 months, and this is the current definition of Moore's Law, which Moore himself has blessed. • Most experts, including Moore himself, expect Moore's Law to hold for at least another two decades.

Technological limits • But all good things may come to an end… • We are limited in our ability to increase • the density and • the speed of a computing engine. • Reliability will also be affected • to increase the speed we need increasingly smaller circuits (light needs 1 ns to travel 30 cm in vacuum) • smaller circuits  systems consisting only of a few particles subject to Heissenberg uncertainty

Energy/operation • If there is a minimum amount of energy dissipated to perform an elementary operation, then to increase the speed, thus the number of operations performed each second, we require a liner increase of the amount of energy dissipated by the device. • The computer technology vintage year 2000 requires some 3 x 10-18 Joules per elementary operation. • Even if this limit is reduced say 100-fold we shall see a 10 (ten) times increase in the amount of power needed by devices operating at a speed 103 times larger than the sped of today's devices.

Power dissipation, circuit density, and speed In 1992 Ralph Merkle from Xerox PARC calculated that a 1 GHz computer operating at room temperature, with 1018 gates packed in a volume of about 1 cm3 would dissipate 3 MW of power. • A small city with 1,000 homes each using 3 KW would require the same amount of power; • A 500 MW nuclear reactor could only power some 166 such circuits.

Talking about the heat… The heat produced by a super dense computing engine is proportional with the number of elementary computing circuits, thus, with the volume of the engine. • The heat dissipated grows as the cube of the radius of the device. To prevent the destruction of the engine we have to remove the heat through a surface surrounding the device. Henceforth, • our ability to remove heat increases as the square of the radius • while the amount of heat increases with the cube of the size of the computing engine.

Quantum; Quantum mechanics • Quantum is a Latin word meaning some quantity. • In physics it is used with the same meaning as the word discrete in mathematics, i.e., some quantity or variable that can take only sharply defined values as opposed to a continuously varying quantity. • Quantum mechanics is a mathematical model of the physical world I'll bet any quantum mechanic in the service would give the rest of his life to fool around with this gadget. - Chief Engineer Quinn, Forbidden Planet, 1956

Heissenberg uncertainty principle • Heisenberg uncertainty principle says we cannot determine both the position and the momentum of a quantum particle with arbitrary precision. • In his Nobel prize lecture on December 11, 1954 Max Born says about this fundamental principle of Quantum Mechanics : ``... It shows that not only the determinism of classical physics must be abandoned, but also the naive concept of reality which looked upon atomic particles as if they were very small grains of sand. At every instant a grain of sand has a definite position and velocity. This is not the case with an electron. If the position is determined with increasing accuracy, the possibility of ascertaining its velocity becomes less and vice versa.''

Quantum Weirdness Quantum mechanics displays some weird features that cannot be duplicated by any classical system. Can Weirdness be useful? Quantum Computing Quantum Encryption Quantum teleportation Quantum Global Positioning

Electromagnetic Weirdness The theory was weird! Light was propagated as a wave in a vacuum - no underlying medium. In 1867 Maxwell proposed a theory unifying electricity, magnetism, and optics. Only when Marconi invented the radio transmitter and receiver did the weirdness become accepted as normal, or even obvious.

some remarkable quotes Quantum Weirdness The quantum is the greatest mystery we've got. Never in my life was I more up a tree than today. -John Wheeler If someone says that he can think about quantum physics without becoming dizzy, that shows only that he has not understood anything whatever about it. - Niels Bohr A picture may seem extraordinarily strange to you and after some time not only does it not seem strange but it is impossible to find what there was in it that was strange. - GERTRUDE STEINconcerning modern art. John Wheeler and friends Niels Bohr and big money. Gertrude Stein by Picasso

Classical versus Quantum Experiments • Classical Experiments • Experiment with bullets • Experiment with waves • Quantum Experiments • Two slits Experiment with electrons • Stern-Gerlach Experiment

Experiment with bullets detector Gun wall H1 H2 wall (a) (b) Figure 1: Experimentwith bullets

Experiment with bullets detector P1(x) Gun wall H1 H2 wall (a) H1 is open H2 is closed (b) Figure 1: Experimentwith bullets

Experiment with bullets detector Gun P2(x) wall H1 H2 wall (a) H1 is closed H2 is open (b) Figure 1: Experimentwith bullets

Experiment with bullets detector P1(x) Gun P2(x) wall (c) H1 H2 wall (a) H1 is open H2 is open (b) Figure 1: Experimentwith bullets

Experiment with Waves detector wall H1 is closed H2 is closed wave source H1 H2 wall (a) Figure 2: Experiments with waves

Experiment with Waves detector I1(x) wall H1 is open H2 is closed wave source H1 H2 wall (a) (b) Figure 2: Experiments with waves

Experiment with Waves detector wall H1 is closed H2 is open wave source H1 H2 I2(x) wall (a) (b) Figure 2: Experiments with waves

Experiment with Waves detector I1(x) I2(x) (b) (c) wall H1 is open H2 is open wave source H1 H2 This is a result of interference wall (a) Figure 2: Experiments with waves

Two Slit Experiment Results intuitively expected P1(x) H1 detector H2 P2(x) source of electrons wall (c) (a) (b) wall Are electrons particles or waves? Figure 3: Two slit experiment

Two Slit Experiment Results observed P1(x) H1 detector H2 P2(x) source of electrons wall (c) (a) (b) wall Figure 3: Two slit experiment

Two Slit Experiment With Observation light source Interference disappeared! P1(x) detector P2(x) source of electrons (c) (b) wall Now we add light source H1 H2 ⇨ “Decoherence” wall (a) Figure 4: Two slit experiment with observation

Stern-Gerlach Experiment S N 1922 - a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Stern-Gerlach experiment with spin-1/2 particles

Conclusions From the Experiments • Limitations of classical mechanics • Particles demonstrate wavelike behavior • Effect of observations cannot be ignored • Evolution and measurement must be distinguished Can we use these phenomena practically? Quantum computing and information

Quantum Weirdnesses Leonard Mandel 1. Complementarity: Every particle is a wave, and every wave is a particle. It just depends on how you measure it. classical particle classical wave quantum photon

Quantum Weirdnesses 3. Entanglement: The quantum state of a two particle system in which the states of the individual particles are inseparable even though the two particles are separated.

A revolutionary approach to computing and communication • We need to consider a revolutionary rather than an evolutionary approach to computing. • Quantum theory does not play only a supporting role by prescribing the limitations of physical systems used for computing and communication. • Quantum properties such as • uncertainty, • interference, and • entanglement form the foundation of a new brand of theory, the quantum information theory where computational and communication processes rest upon fundamental physics.

Milestones in quantum physics 1900 - Max Plank presents the black body radiation theory; the quantum theory is born. 1905 - Albert Einstein develops the 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.

Milestones in computing and information theory 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''. Church Thesis. 1946 - A report co-authored by John von Neumann outlines the von Neumann architecture. 1948 - Claude Shannon publishes ``A Mathematical Theory of Communication’’. 1961 - Rolf Landauer decrees that computation is physicaland studies heat generation. 1973 - Charles Bennet studies the logical reversibility of computations. 1981 - Richard Feynman suggests that physical systems including quantum systems can be simulated exactly with quantum computers. 1982 - Peter Beniof develops quantum mechanical models of Turing machines. 1984 - Charles Bennet and Gilles Brassard introduce quantum cryptography. 1985 - David Deutsch reinterprets the Church-Turing conjecture. 1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters discover quantum teleportation. 1994 - Peter Shor develops a clever algorithm for factoring large numbers.

Deterministic versus probabilistic photon behavior

The puzzling nature of light Each detector detects single photons. Why? What is a hidden information that controls this In an attempt to solve this puzzle we design this setup • If we start decreasing the intensity of the incident light we observe the granular nature of light. • Imagine that we send a single photon. • Then either detector D1 or detector D2 will record the arrival of a photon. • If we repeat the experiment involving a single photon over and over again we observe that each one of the two detectors records a number of events. • Could there be hidden information, which controls the behavior of a photon? • Does a photon carry a gene and one with a ``transmit'' gene continues and reaches detector D2 and another with a ``reflect'' gene ends up at D1?

The puzzling nature of light (cont’d) Why all these detectors detect light? • Consider now a cascade of beam splitters. • As before, we send a single photon and repeat the experiment many times and count the number of events registered by each detector. • According to our theory we expect the first beam splitter to decide the fate of an incoming photon; • the photon is either reflected by the first beam splitter or transmitted by all of them. • Thus, only the first and last detectors in the chainare expected to register an equal number of events. • Amazingly enough, the experiment shows that all the detectors have a chance to register an event.

Polarization experiment A all photons horizontally polarized photons randomly polarized 1/2 original intensity • Polarization of light: electric field vector is confined to a single direction. This is a quantum-mechanical property of photons. • Light source: Assume each photon has a random polarization • Filter A: Horizontally polarized

Polarization experiment A B all photons horizontally polarized photons randomly polarized no light • Polarization of light: electric field vector is confined to a single direction. This is a quantum-mechanical property of photons. • Light source: Assume each photon has a random polarization • Filter A: Horizontally polarized • Filter B: Vertically polarized

Polarization experiment A C B photons randomly polarized 1/8 original intensity • Polarization of light: electric field vector is confined to a single direction. This is a quantum-mechanical property of photons. • Light source: Assume each photon has a random polarization • Filter A: Horizontally polarized • Filter B: Vertically polarized • Filter C: polarized at 45 degrees

Polarization experiment explanation • Photon’s polarization state can be modeled by a unit vector pointing in the appropriate direction • Let the basis vectors be  and  • Any polarization can be expressed as where a and b are complex numbers such that

Measurement postulate of quantum mechanics • Any device measuring a quantum system has an associated orthonormal basis with respect to which the measurement takes place. • Measurement of a state transforms the state into one of these basis vectors. • The probability that the state is measured as basis vector u is the norm of the amplitude of the component of the original state in the direction of u.

Example: Let be the polarization state of a photon. Then this state will be measured as  with probability |a|2 and as with probability |b|2 . These are indeed probabilities, since . The measurement will also change the state to the result of the measurement. Its original value cannot be determined.

Back to the polarization experiment: • Original light source produces photons with random polarizations: • Thus 50% are measured as horizontal, and let through. (Why?) • Those photons are now in the horizontal state, None are let through by the vertical filter.

Now let the 45o filter be put behind the horizontal filter. The 45o filter measures photon polarization with respect to a new basis: • Photons in state  will be measured as with probability 1/2. So half will get through. • Likewise, filter B will let half of these through. • Total photons getting through all three filters: (1/2)3 = 1/8.

Quantum Mechanics Quantum mechanics vs. Classical Mechanics Formulated to explain the behavior of microscopic systems. Formulated to explain the behavior of macroscopic objects. Newton’s second law: Integrate twice → x(t). Two constants of integration →two additional pieces of information required to uniquely define the state of the system (e.g. xo and vo). The state of the system is defined by: FORCES, POSITIONS, VELOCITIES Knowledge of the initial state of the system can predict future states precisely.

Quantum Mechanics QM analogy to Newton’s 2nd law • Designed to describe the behavior observed for microscopic particles and systems. • Behavior: • discrete rather than continuous energy levels • photons, electrons and nuclei exhibit both wave and particle nature. • History: • 1925 Werner Heisenberg, Max Born and Pascual Jordon introduced matrix-based mathematical formalism to describe observed quantum mechanical phenomena • 1926 Erwin Schrödinger introduced a differential equation and its solution that equivalently describes equation observed quantum mechanical phenomena

The postulates of quantum mechanics (QM) Postulate I For any possible state of a system, there is a function y of the coordinates of the parts of the system and time that completely describes the system. Y Is called a wave function. For two particles system, The wave function square Y2 is proportional to probability. Since Y may be complex, we are interested in Y*Y, where Y* is the complex conjugate (i  -i) of Y. The quantity Y*Ydt is proportional to the probability of finding the particles of the system in the volume element, dt = dxdydz. that is the probability of finding the particle in the universe is 1  normalization condition.

The postulates of quantum mechanics (QM) Orthogonality of two wave functions Example: sinq and cosq are orthogonal functions. Fourier series expansion – sin(nq) and cos(nq) orthogonal functions

The Wavefunction QM analogy to Newton’s 2nd law Postulate I For any possible state of a system, there is a function y of the coordinates of the parts of the system and time that completely describes the system: • Y(t) = f + ig • f and g are real functions of coordinates and time • An abstract, complex quantity but related to physically measurable quantities • State is dependent on coordinates (spatial and spin) and time • The time-dependent Schrödinger equation: • A single integration with respect to time is required to obtain Y(t), so that only one constant of integration is required to predict future states of the system.

The Wavefunction dz dy dx How can we describe the state of a quantum mechanical system such as nuclear spins? Complex wavefunction: Y(t) = Y(t,t) ≡ description of all knowable information about the state of the system What if we want to know if the system is in a given state Y(t) at time t? The probability that the system is in the state given by Y(t) at time t is: P = Y*(t)Y(t) = │Y│2 For example, for a single particle at time t′, the wavefunction is Y(x,y,z,t′), and the probability that time t′, the particle is in a given volume of space (dxdydz) is given by: │Y(x,y,z,t′)│2 dxdydz Time dependence Spatial and spin coordinates (independent of time) probability density

Complex Conjugate Complex Conjugate Y*: Y = f + ig Y* = f – ig (replace i with –i) Y*Y = (f + ig) (f – ig) = f2 – ifg + ifg – (i2)g = f2 + g2 real, non-negative (as P should be!)

Normalization Condition Since the system must exist in some state at time, t, if we integrate over all coordinates of the system (t represents the generalized coordinates, which may include spatial coordinates and spin state), the probability density is 1. Normalization condition: ∫ Y*(t)Y(t)dt = 1 i.e. the probability of finding the particle somewhere in space is one. The Schrödinger equation describes the evolution in time of a given system:

Quantum Computing Part 1: Quantum Mechanics Overview