PHYSICAL COMPUTATION FROM MECHANICAL COMPUTATION TO QUANTUM COMPUTATION Daniel F. Bento PT-Comunicações - SAPO Faculdade de Ciências Universidade de Lisboa 

 Pen and Paper  Chalk and Slate  Tables It was mostly associeted with numbers By 2000 BC, computing started to evolve with abstract concepts Number Mathematical object Basic Arithmetic (on numbers) Unary operations (sign) Binary operations (+, -, *, /) DANIEL BENTO - 2014 COMPUTING? 

ABACUS 2700 – 2300 BC – Sumerian Abacus Adopted in many places, exemple: • Babylonia • Egypt • Persia • Greek • Roman • Chinese • Indian • Japanese • Korean • Native American • Russian DANIEL BENTO - 2014 

Logarithm Properties: log(ab) = log(a) + log(b) log(a/b) = log(a) – log(b) log(a^b) = blog(a) log(a^(1/b)) = (1/b) log(a) - Very useful during science evolution. Because… it simplifies a lot of calculations; (sum/subtractions insted of multiplications /divisions) Exemple: We want: 1670*12, so 10^log(1670*12) log(1670*12) = log(1670) + log(12) ~ 3.2227 + 1.079 = 4.3017 10^4.3017 ~ 20030.87 ~ 1670*12 = 20040 DANIEL BENTO - 2014 TABLES AND LOG TABLES Napier (1614) – Logarithm Method Euler (1730) - Natural Logarithm Kepler used it a lot for Astronomy 

• Wilhelm Schickard • Blaise Pascal • Leibniz • Charles Xavier Thomas DANIEL BENTO - 2014 Computational Machines Schickard’s calculating machine Thomas Arithmometer – First digital mechanical calculator 

DANIEL BENTO - 2014 THEY WERE NOT PROGRAMMABLE 

DANIEL BENTO - 2014 SCIENCE OF COMPUTATION 

AND: A & B OR: A + B NOT: ~A NAND: A|B NOR: A - B XOR: A ○ B XOR: A ● B LOGIC GATES NOT and NAND are Universal Gates DANIEL BENTO - 2014 

DANIEL BENTO - 2014 TURING MACHINE AND VON NEUNMANN ARCHITECTURE Turing Machine Implementation Von Neumann Architecture Von Neunmann Alen Turing 

Electronic Numerical Integrator And Computer  Turing-Complete  Digital  Programmable Numbers: - 17.500 Vacum tubes - 7.200 Crystal diodes - 1.500 Relays - 70.000 Resistors - 10.000 Capacitors Arithmetic counting pulses 10 by 10 digits take 2800 microseconds DANIEL BENTO - 2014 ENIAC – FIRST COMPUTER 

The correlator in the ALMA Array Operations Site Technical Building Radio Astronomy - 134 million processors - 17 quadrillion operations per second - Solid State Disks - Twice the normal airflow needed - Seismic activity in account ALL THIS TO PROCESS DATA FROM THE 66 ANTENNAS DANIEL BENTO - 2014 SCIENTIFIC COMPUTATION - ALMA 

- 25 Petabytes of data proccessed per year - 40 world facilities DANIEL BENTO - 2014 SCIENTIFIC COMPUTATION - CERN Particle Physics 

QUANTUM MECHANICS DANIEL BENTO - 2014 

WAVE EQUATION – 1 DIMENSION 1 = sin() 2 = cos() DANIEL BENTO - 2014 

= (() + ) WAVE EQUATION – SUM PARTIAL I PARTIAL II PARTIAL III PARTIAL IV SUM DANIEL BENTO - 2014 

BY: DAISHIN UEYAMA WAVE EQUATION – 2 DIMENSIONAL INTERFERENCE DANIEL BENTO - 2014 

why we need this theory? important unsolved problems DANIEL BENTO - 2014 

BLACK BODY RADIATION DANIEL BENTO - 2014 

WAVE PARTICLE DUALITY DANIEL BENTO - 2014 

solution!? Plank quantum theory Max Planck (1858-1947) DANIEL BENTO - 2014 

Edwin Schrödinger NO FORMULAS HERE! WAVE FUNCTION DANIEL BENTO - 2014 

the famous Schrödinger’s cat DANIEL BENTO - 2014 

MENTAL EXPERIMENT DANIEL BENTO - 2014 

DANIEL BENTO - 2014 

Introducted by Yurin Manin (1980) and Feynman (1982) • Two-state quantum system – qubits • Operate by unitary evolution (time-dependente Schrödinger equation) • Operate in qubits as bits in classical mechanics • It needs a large number of qubits for entanglement • It is extremely sensitive • Most of the problems are solved in the same time as classical computers, but: • Classical Prime Factorization ~ exp(2L^1/3 ln(L^2/3)) • Quantum Prime Factorization ~ 300L^3 where L = ln(N), N number of digits classical method is faster for L = 10, quantum method is faster for L > 200 (10^9 years to about 8 hours) QUANTUM COMPUTATION DANIEL BENTO - 2014 

• In classical mechanics the initial and final states are linked • In quantum mechanics the final state is random at one time at one direction • Atoms can bound infinite states UNITARY EVOLUTION DANIEL BENTO - 2014 

BLOCH SPHERE DANIEL BENTO - 2014 Geometric representation of two-level quantum mechanical system qubit - Hilbert space - North and South poles are chosen for the |0> and |1> (electron spin up and spin down) - It can be extended to n-level quantum space 

QUANTUM LOGIC GATES Fredkin Gate: CSWAP Gate 3-bit gate Universal for Classical Computation Toffoli Gate: CCNOT Gate 3-bit gate Universal for Classical Computation CNOT Gate 2-bit gate NOT Gate SWAP Gate Hadamard Gate Pauli-X Gate – Rotation on Bloch Sphere around X-Axis by π radians = Classic NOT GATE Pauli-Y Gate – Rotation on Bloch Sphere around Y-Axis by π radians Pauli-Z Gate – Rotation on Bloch Sphere around Z-Axis by π radians DANIEL BENTO - 2014 

DANIEL BENTO - 2014 QUANTUM CRYPTOGRAPHY 

DANIEL BENTO - 2014 D-WAVE QUANTUM COMPUTER 128 qubit using quantum annealing 

DANIEL BENTO - 2014 NEW FORMS OF COMPUTING • Metamaterials • DNA • Spintronics 

THANKS FOR LISTENING! Daniel Bento [email protected] www.overdestiny.com twitter.com/danielfbento DANIEL BENTO - 2014