question - university of toronto · 2010-03-16 · base collector emitter • collector-emitter...

Question: How can an object be two things at the same time?

Daniel James

PHY189

4March2010

Storing Numbers (and letters)

• “off” and “on”: 0 and 1 • binary numbers: imagine counting if you only had two fingers....

• Letters: use code numbers (e.g. “E” →69 in the “ASCII” code)

binary digit or “bit”

Claude Shannon (1916 - 2001)

128 64 32 16 8 4 2 1 64+4+1 =69 69→0 1 0 0 0 1 0 1

DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

Sum (A+B, but 1+1 =0)

Carry (1 if A and B= 1, else 0)

A B

Boolean “AND”

Boolean “XOR”

George Boole (1815 - 1864)

Interactions can realize the Boolean operations needed to do arithmetic

Persuading machines to Add


base

collector

emitter

• collector-emitter current controlled by base-emitter voltage - amplifying electronic signals: radios, TVs, stereos etc.. - electronic Boolean logic ⇒ computers

Transistors



$$$

phys

ics/

eng

inee

ring

mat

h early theory (Turing, Shannon...)

WW-2 (Colossus, ENIAC, ...)

Glory days of device physics (transistors, integrated circuits...)

First computer revolution: large scale applications

Personal computers: Apples and PCs

invention of the internet

The 2nd computer revolution

1940

1950

1960

1970

1980

1990

2000

Growth of Computing

Year

Capability doubles every 18 months.

Gordon Moore (1929- ) co-founder of Intel Corp.

Mooreʼs Law


• processing speed and memory size improved by miniaturization • Can we make transistors smaller and smaller? • No, because of....

QUANTUM MECHANICS

And so on...?


quantum state engineering: the undiscovered country

Classical and Quantum Descriptions of Nature • Classical Dynamics: mass, position, velocity etc. • The configuration (“state”) of the system is the same as the observables

• Quantum physics:

“State” “Observables”

probabilities



• state of unobserved light bulb is indeterminate • quantum mechanics allows us to figure the odds whether it’s on or off.

But... • The mere possibility of observation is enough to collapse the indeterminate state (so light-bulbs are, in fact, either “on” or “off”, even if no-one is looking)*

*i.e. No, you can’t use quantum physics to dispute your electric bill with Toronto Hydro

But, again... • Microscopic objects (e.g. atoms, photons) can be insulated from the possibility of measurement so quantum behavior will prevail

Quantum Sytems: Examples


€

State of a qubit : ψqubit =a 0 + b 1

Example: a spin-1/2 particle in a magnetic field

spinʼs magnetic moment is either “up” or “down” with respect to the B-field

€

≡ 0( )

€

≡ 1( )

Atomic Qubits

€

4 2S1/2, mJ = −1/2 • “qubits” formed by two atomic levels:

€

3 2D5/2, mJ = −1/2


“When two systems, of which we know the states, ... enter into temporary physical interaction, ... they can no longer be described in the same way as before, viz. by endowing each of them with a [state] of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ -functions) have become entangled.”

-Erwin Schrödinger, 1935

Entanglement


Example: qubits

€

ψ = a 0 A + b 1 A( )qubit A

⊗ c 0 B + d 1 B( )

qubit B

(interaction region)

State of either system is well defined: “separable state”

The two systems are in an “entangled state”

€

≠ ′ a 0 A + ′ b 1 A( )⊗ ′ c 0 B + ′ d 1 B( )

€

′ ψ =α 00

+ β 01 + γ 10 + δ 11

joint state :

1 2 4 3 4

€

i .e . 0 ⊗ 0


• Two atoms, separated, in state:

Non-locality

€

ψ = 12 01 − 10( )

• Measure A: 50% probability: “1”, with 50% probability: “0” (just like tossing a coin!) • BUT: instantaneously B is projected into the opposite state. If information cannot travel faster than light, how does it “know” what state to be in?

A B

• Perhaps outcomes of measurements is pre-ordained by some ʻhiddenʼ variable? Bellʼs theorem provides a test for this hypothesis: the answer is NO (sort of!)


Whatʼs this got to do with computing?

• Moore’s law: conventional electronics cannot function with objects less than about 0.00001mm (and we’re pretty much there already...) • Can we turn a ‘bug’ into a ‘feature’: can quantum mechanics be exploited to enhance (or even revolutionize) computation?

Richard Feynman (1918-88)


Suppose we have quantum light bulbs (that can be “on” and “off” at the same time: QU-BITS)

state of one bulb: a + b

state of two bulbs: a + b

+c + d

three bulbs: 8 possibilities, four bulbs: 16 possibilities, and so on.....

Quantum Memories are BIG!


Quantum memories can do things classical computers cannot

- store all possible solutions of a problem at once, then down-select to the desired answer.



Killer Ap: factoring 15= = 5 x 3 77= = 11 x 7

133= = 19 x 7

2934331= = 3221 x 911 1633= = 71 x 23

27997833911221327870829467638722601621070446786955428537560009929326128400107609345671052955360856061822351910951365788637105954482006576775098580557613579098734950144178863178946295187237869221823983

3532461934402770121272604978198464368671197400197625023649303468776121253679423200058547956528088349

7925869954478333033347085841480059687737975857364219960734330341455767872818152135381409304740185467

= x

Factoring is DIFFICULT

RSA200

Classical ~ exp{AL}

# of instructions

# of bits, L, factored

0 200 400 600 800 10001101001000

1 1041 1051 1061 1071 1081 1091 10101 10111 10121 10131 10141 10151 10161 10171 10181 10191 10201 10211 10221 10231 1024 ~ 1020 instructions:

16 months (2003-05)

Quantum~ L3

~ 1012 operations: Hours ?


Why is Factoring ʻInterestingʼ? • Maybe you are a math whizz.... • Or, maybe you like to break codes.

RSA cryptosystem ʻpublic keyʼ encryption system used widely in internet communications relies on the difficulty of factoring for its security.


made from four blades (a), two tips (b) and a supporting structure (c). (from F. Schmidt-Kaler et al., Appl. Phys. B 77, 789 (2003)).

Trapped Ions ( the best Quantum Computer So Far)


Alice relays results to Bob

Bobʼs qubit is projected into a

pure state

To Bob

Transformation Bobʼs qubit stored

in memory

selected on the basis of Aliceʼs

message

Bobʼs qubit ends up identical to

Charlieʼs

Aliceʼs classical message

Charlieʼs unknown qubit

entangled-qubit source

Aliceʼs “Teleporter” (Bell state analyzer)

To Bob


Charlieʼs unknown qubit

entangled-qubit source

Aliceʼs “Teleporter” (Bell state analyzer)

To Bob

Teleportation Circuit



Alice relays results to Bob

Bobʼs qubit is projected into a

pure state

To Bob



Transformation Bobʼs qubit stored

in memory

selected on the basis of Aliceʼs

message

Bobʼs qubit ends up identical to

Charlieʼs

Aliceʼs classical message


• 8 qubits; maybe 12 or 15 possible soon • 20+ logic operations

• operating speeds: 100 kHz (too slow) • Factoring of the number 15 (!)

• need 1500 qubits operating at 100 MHz to do something really useful... • Solid state or not?

Whence, Where and Whither?



$$$

phys

ics/

eng

inee

ring

mat

h early theory (Turing, Shannon...)

WW-2 (Colossus, ENIAC, ...)

Glory days of device physics (transistors, integrated circuits...)

First computer revolution: large scale applications

Personal computers: Apples and PCs

invention of the internet

The 2nd computer revolution

1940

1950

1960

1970

1980

1990

2000

Growth of Computing 1980

1990

2000

Growth of Quantum Computing

2010

question - university of toronto · 2010-03-16 · base collector emitter • collector-emitter...

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