QUANTUM COMPUTING What is it ? Jean V. Bellissard Georgia Institute of Technology & Institut Universitaire de France

Feynman ’ s proposal: Richard P. Feynman. Quantum mechanical computers. Optics News, 11(2):11-20, He suggested in 1982 that quantum computers might have fundamentally more powerful computational abilities than conventional ones. In 1994, Peter Shor showed that quantum mechanical computers can make factoring integers "easy", destroying the entire existing edifice of public-key cryptography, the current proposed basis for the electronic community of the future. Today, chips, based on trapped ions, are used by various groups to manipulate many more qubits than several years ago.

Flea: 1/8”= 3mm Cells: 1-20µm Proteins: 1-10nm atom:.05nm 1 : : : 1000

Challenge Can one use the tiniest objects (electrons, atoms, photons, ions, small molecules, Cooper pairs, anyons,…) provided by Nature to encode information to communicate it to store it to encrypt it to treat it to compute ?

Waves source interferences diffraction

Waves an electron in an atom or in a well (quantum dot)

Waves Schr ö dinger ’ s equation -  +V  =E 

Waves Probability of presence ∫||2∫||2 =1

Waves Inner product ∫  =

square well (quantum dots) harmonic well (ion traps) Coulomb well (atoms) Energies of the particle associated with the wave Waves

Information |1> |0> Elementary unit of quantum information: the “qubit ”

Information |1> |0> A laser beam can control or measure which qubit is actually activated

Information |1> |0> Orthogonality: if x,y  {0,1} =  xy

Information A laser beam can control or measure which qubit is actually activated Example : trapped ions (Cd + )

Information A laser beam can control or measure which qubit is actually activated Example : trapped ions (Cd + )

Information Each ion eigenstate is a vector |0> or |1> in a Hilbert space Any other vector is a linear combination of them  =  |0> +  |1> The family of ions is represented by tensor products  =|x 1 > |x 2 > … |x n > =|x 1 x 2 … x n >

Information Entanglement Bell’s state |00> + |11> __________ 2 1/2  00 =

Gates One qubit-gates are unitary operators U | x >U |x >

Gates Pauli gates 0 1 X = i Y = i Z = I = 0 1 X |0> = |1> X = |0> Z = (+1)|0> Z |1> = ( -1)|1> bit flipphase flip

Gates Hadamard gate 1 1 H = 2 -1/ H |x> = 2 -1/2 (|0> + (-1) x |1>) Fourier transform on one bit H | x > (|0> + (-1) x |1>) ————————— 2 1/2

Gates Commutation rules H X H = Z H Z H = X H XH = Z X Z = - Z X X2 X2 = 1 Z2 Z2 = 1 H2 H2 = 1

Gates 2-qubit gate CNOT | x > | y > | x > | x  y > CNOT 0 y x 1 1-y

Gates 3-qubit gate TOFFOLI 00,01,10 z x,y 11 1-z | x > | y > | x > | z  xy > Toffoli | z > | y >

Gates Theorem: Any unitary matrix in an N-qubit space can be approximated to any accuracy by a finite product of 1-qubit and CNOT gates

Circuits Measurement Device that measures a value of the bit x The part of the state corresponding to this line is lost.

Circuits Adding 2 numbers x 1 +2x 2 +2x 3 +… + y 1 +2y 2 +2y 3 +… = z 1 +2z 2 +2z 3 +… z 1 = x 1 +y 1 (mod 2) x 1 +y 1 =z 1 +2 c1c1 z 2 = x 2 +y 2 +c 1 (mod 2) x 2 +y 2 +c 1 =z 2 +2 c2c2 etc… carry

Circuits Adding 2 numbers |x 1 > |x 2 > |x 3 > |y 1 > |y 2 > |y 3 > |0> |z 1 > |z 2 > |z 3 > carry |c 1 > |c 2 > |c 3 >

Circuits Teleportation |  > |  00 > |  > H X Z

Circuits Teleportation |  > |  00 > |  > H X Z |x00>+|x11> √2

Circuits Teleportation |  > |  00 > |  > H X Z |xx0>+|x(1-x)1> √2

Circuits Teleportation |  > |  00 > |  > H X Z (|0x0>+(-) x |1x0>+|0 (1-x)1>+(-) x |1 (1-x)1>) 2

Circuits Teleportation |  > |  00 > |  > H X Z (|0xx>+(-) x |1xx>+|0 (1-x) x>+(-) x |1 (1-x)x>) 2

Circuits Teleportation |  > |  00 > |  > H X Z (|0x>+|1x>+|0 (1-x) >+|1 (1-x) >) |x> 2

Circuits Teleportation |  > |  00 > |  > H X Z (|00>+|11>+|01>+|10>) |x> 2

Circuits Teleportation The information sent by Alice is transmitted to Bob, provided they share one entangled qubit If Alice is an atom, molecule, … the state of it can be reproduced at Bob ’ s site Since the atoms are undiscernable, it is equivalent to transmit them at distance at the speed of light.

Fourier Transform The usual FFT requires a time O(N LnN) The Quantum FT is given by a circuit with n 2 /2 + 2n gates Since the N=2 n, the algorithm gives the result in a time (1 time unit/gate) O((LnN) 2 ) !!

Fourier Transform All encryption systems used today in computers and in communications (RSA) are using a Fourier transform to decompose integers into prime numbers. The speed of the QFT can jeopardize those systems

Funding for the research in Quantum Computing is provided by the Intelligence Advanced Research Projects Activity (IARPA); the National Security Agency (NSA), the Army Research Office (ARO), the NSF Physics at the Information Frontier (PIF) Program; the Defense Advanced Research Projects Agency (DARPA) Mems-Exchange and Optical Lattice Emulator Programs