quantum computing what is it ? jean v. bellissard georgia institute of technology & institut...

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, 1985.

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

1:100

1:1000

ChallengeChallenge

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

?

WavesWaves

source

interferences

diffraction

WavesWaves

an electron in an atomor

in a well (quantum dot)

WavesWaves

Schrödinger’s equation

-+V=E

WavesWaves

Probability of presence

∫||2=1

WavesWaves

Inner product

∫=<|>

square well (quantum dots)

harmonic well (ion traps)

Coulomb well (atoms)

Energies of the particle associated with the wave

WavesWaves

InformatioInformationn

|1>

|0>

Elementary unit of quantum information: the “qubit ”


|1>

|0>

A laser beam can control or measure which qubit is actually activated


|1>

|0>

Orthogonality: if x,y{0,1}

<x|y> = xy


A laser beam can control or measure which qubit is actually activated

Example : trapped ions (Cd+)


Each ion eigenstate is a vector

|0> or |1>

in a Hilbert space

Any other vector is a linearcombination of them

=|0> + |1>

The family of ions is represented by tensor products

=|x1> |x2> … |xn>

=|x1 x2 … xn>

QuickTime™ and aGIF decompressorare needed to see this picture.


Entanglement

Bell’s state

QuickTime™ and aGIF decompressorare needed to see this picture.|00> + |11>

__________21/200 =

GatesGatesOne qubit-gates

are

unitary operators

U| x > U |x >

GatesGatesPauli gates

0 1X = 1 0

0 -iY = i 0

1 0Z = 0 -1

1 0I = 0 1

X |0> = |1>X |1> = |0>

Z |0> = (+1)|0>Z |1> = ( -1)|1>

bit flip phase flip

GatesGatesHadamard gate

1 1H = 2-1/2

1 -1

H |x> = 2-1/2 (|0> + (-1)x |1>)

Fourier transform on one bit

H| x >(|0> + (-1)x |1>)

—————————21/2

GatesGatesCommutation rules

H X H = Z H Z H = X

H X H = Z

X Z = - Z X

X2 = 1 Z2 = 1 H2 = 1

GatesGates2-qubit gate

CNOT

| x >

| y >

| x >

| xy >

CNOT

0

y

x

1

1-y

GatesGates3-qubit gate

TOFFOLI

00,01,10

z

x,y

11

1-z

| x >

| y >

| x >

| zxy >

Toffoli

| z >

| y >

GatesGates

Theorem:

Any unitary matrix in an N-qubit space can be approximated to any accuracy by a finite productof 1-qubit and CNOT gates

CircuitsCircuitsMeasurement

• Device that measures a value of the bit x

• The part of the state corresponding to this line is lost.

CircuitsCircuitsAdding 2 numbers

x1+2x2+2x3+…+

y1+2y2+2y3+…= z1+2z2+2z3+…

z1= x1+y1 (mod 2)

x1+y1=z1 +2 c1

z2= x2+y2 +c1(mod 2)

x2+y2+c1=z2 +2 c2

etc…

carry

CircuitsCircuitsAdding 2 numbers

|x1>

|x2>

|x3>

|y1>

|y2>

|y3>

|0>

|0>

|0>

|z1>

|z2>

|z3>

carry

|c1>

|c2>

|c3>

CircuitsCircuitsTeleportation

| >

| 00>| >

H

X Z


| >

| 00>| >

H

X Z

|x00>+|

x11>

√2


| >

| 00>| >

H

X Z

|xx0>+|x(1-

x)1>

√2


| >

| 00>| >

H

X Z

(|0x0>+(-) x|1x0>+|0 (1-x)1>+(-) x|1 (1-

x)1>)

2


| >

| 00>| >

H

X Z

(|0xx>+(-) x|1xx>+|0 (1-x) x>+(-) x|1 (1-

x)x>)

2


| >

| 00>| >

H

X Z

(|0x>+|1x>+|0 (1-x) >+|1 (1-x) >)

|x>

2


| >

| 00>| >

H

X Z

(|00>+|11>+|01>+|10>) |x>

2


• 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 TransformFourier Transform

• The usual FFT requires a timeO(N LnN)

• The Quantum FT is given by a circuit with n2/2 + 2n gates

• Since the N=2n, the algorithm gives the result in a time (1 time unit/gate)

O((LnN)2) !!

Fourier TransformFourier Transform

All encryption systems used today incomputers and in communications (RSA)

are using a Fourier transform to decompose integers into prime numbers.

The speed of the QFT can jeopardizethose 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

quantum computing what is it ? jean v. bellissard georgia institute of technology & institut...

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