Quantum Computing

BCS Belgium Branch

Quantum Computing

• Basic Quantum Mechanics

• Quantum Algorithms

• Quantum Computer Hardware

Basic Quantum Mechanics

• Wave particle duality• Coherence• Interference• Young’s slits• Entanglement

• Observer matters

• Classical physics• Exact knowledge• Deterministic

Beam Splitter

• Split light beam with semi silvered mirror

A

B

Interference

• Split and recombine light beams

A

B

Interference

• Split and recombine light beams

A

B

Young’s slits

Diffraction patterns

Wave Interference

+ +

= =

EPR Paradox

• Quantum Teleportation made real

Teleportation diagrams Courtesy of IBM, Copyright, IBM Corp, 1995

Quantum Interference

• Superpositions of quantum states

• Wavefunctions are complex

• Modulus has a physical interpretation

Classical Bits are 0 or 1

• Classical CPUs use binary representation

• Only 0 or 1 is defined

• N-bit register contains one number from 2N

1

0

Qubits are the key

• Quantum CPU works on Qubits

• Represent 0 and 1• Or any mixture

• N-bit register may contain any subset of numbers from 2N

|1>

|0>

-|1>

-|0>

Hadamard Transform

• Controlled mixing• Given a system with

eigenstates |0> |1>• Forms

|0> + |1>

|0> - |1>

• Self inverse

1 11 -1

1 1 1 1 1 -1 1 -11 1 -1 -11 -1 -1 1

Quantum registers

• |0> + |1> is 0 and 1• Entangle 3 qubits• And you get

• |000> + |001> + |010> +|011> + |100> + |101> + |110> + |111>

• L operations• 2L different numbers

|0>+|1> |0>+|1>|0>+|1>

|0>+|1>|0>+|1>|0>+|1>

Deutsch ’85 Problem

• Given f(x) , x = {0,1}• Compute once• Decide if f(0)=f(1)• Impossible for classical CPU

H Uf H|0>

|0> - |1> |0> - |1>

answer

Conditional Test

• Classical

• Quantum

if (x)

if (qb)

False True

|1>

|0>

-|0>

-|1>

Simon’s Algorithm ‘93

• Given a periodic function of period r

• f(x)=f(x+r)

• Find period r in polynomial time

• Single step finds all possible values of r

• Bad news r, 2r, 3r,… Nr all solutions too!

• Good news GCD is easy on classical CPU

Factoring Composites

• Factoring is slow for conventional CPUs

• Simple example – factorise

• 35 = ? x ?

• 221 = ? ? x ? ?

• 29083 = ? ? ? x ? ? ?

• Multiplying is much easier

• 123 x 456 = ? ? ? ? ?

Factoring Composites 2

• Factoring is slow for conventional CPUs

• Simple example – factorise

• 35 = 5 x 7

• 221 = 13 x 17

• 29083 = 127 x 229

• Multiplying is much easier

• 123 x 456 = 56088

Periodicity Factorisation

• a < N chosen at random

• Prob( GCD(a, N) = 1 ) > 1 / log N

• GCD (a, N) = 1

• f(x) = a x mod N

• Find period r using quantum machine

• Factors are GCD ( a r/2 mod N + 1, N )

• Fast periodicity determination => Factors

Shamir’s Twinkle

• Hardware accelerator for classical CPUs

• Optoelectronic device ~1000x faster

• 512bit RSA keys vulnerable

• Past dedicated hardware triumphs include

• WWII Colossus just beats a Pentium

• Turing’s Bombe still 60x better

Grover’s Algorithm ‘96

• Find a match in N unsorted records

• Classical brute force time ~ N/2

• Quantum algorithm time ~ N1/2

Alice|0> := -|0>

Bob|k> := -|k>H

H

Searching

• Grovers algorithm using Q comparisons

• N1/2 = 1 / sin ( / 2(2Q+1))

• N ~ 4 (2Q+1)2 / 2

Q N

1 4

2 10

3 20

4 33

Is Life a Quantum Computer?

• Q=1, N=4 DNA uses a 4 base code• UCAG

• Q=3, N=20 Life uses ~20 amino acids

• Genetic code has supersymmetry• Weird coincidence or deep link to QM ??• Decoherence time seems too short

DNA Code of Life

DNA Computing

• Adleman - Travelling salesman problem

• Fast combinatorial solutions

• Hard to set up

• Answer fast

• Interpretation slow

DNA Turing Machine

• Universal computing machine

• Molecular computing

• DNA is program tape

• Enzymes are hardware

• Nature 22 Nov 2001

Quantum Computer Hardware

• Toy versions exist, algorithms work

• 2 qubit Chloroform CHCl3

• 3 qubit Trichlorethylene CHCl=CCl2

• Other possibilities– Josephson Junctions– Ion traps, BECs

Chloroform 2 Qubits

• Nuclear spin resonance

• 1H – hydrogen

• 13C – carbon

• Complex chemical analysis

• Simple molecules

• Program by RF pulses

Programming

• Hardware dependent

• RF pulses

• Tuned lasers

• Read back output

• NMR spectrum

NMR Animation, Courtesy of IBM, Copyright, IBM Corp, 1995

The Race for More Qubits

Date Group Compound Qubits

1998 IBM Chloroform 2

1998 IBM/MIT Trichloroethylene 3

Mar 2000 LANL Crotonic acid 7

Aug 2000 MIT Fluorine 5

References

There is a lot of information available on the web from the various research groups that have specialised in quantum algorithm development and design of hardware for quantum computers.

  I created this talk from a number of sources, but the ones below and their links contain additional material at a range of different levels from the basics up to and including the latest work.

Weblinks

• General reviews at many levels

http://www.qubit.org/Intros_Tuts.html

http://www.qubit.org/people/wim/quantum.html

http://www.howstuffworks.com/quantum-computer2.htm 

• Scientific American review article http://www.sciam.com/1998/0698issue/0698gershenfeld.html

http://www.sciam.com/explorations/091696explorations.html

  

• Physics and computer science http://theory.caltech.edu/people/preskill/ph229/#reference

  

• University level algorithms http://www.hpl.hp.co.uk/brims/websems/quantum/ekert/sem.html

http://www.dagstuhl.de/DATA/Reports/98191/report_html.html

 

Weblinks

• Practical Hardware Developments

 

Quantum Experimental Kit http://stardec.hpcc.neu.edu/~bba/RES/QCOMP/QCOMP.html

 

NMR Quantum Computershttp://www.almaden.ibm.com/st/projects/quantum/nmr/

 

LANL 7 Qubit Machinehttp://www.lanl.gov/worldview/news/releases/archive/00-041.shtml 

Quantum Teleportationhttp://www.research.ibm.com/quantuminfo/teleportation/

Weblinks •  Other Related Articles

• EU Quantum Projectshttp://www.cwi.nl/projects/QAIP/

 • Shamir’s Twinkle

http://www.simovits.com/archive/twinkle.pdf • DNA Computing

http://www.csd.uwo.ca/~morey/dnatalk/kevin/dna/dnaerror.html • DNA Turing Machine

http://www.nature.com/nature/links/011122/011122-2.html • Speculative

http://www.consciousness.arizona.edu/hameroff/Davies-Physics-and-life.htm