Where Are We, Really?

Parallel Universes, Fact or Fiction

Lecture 3: Science’s Parallel Worlds – the Many-Worlds

Interpretation of Quantum Reality

Max Tegmark (1967 - )

Max Tegmark

- Professor of Physics at MIT

- Classified parallel universe theories

into 4 major categories or “levels”

Tegmark’s Parallel Universe Levels

Level Description Assumptions

1 Regions beyond our Infinite space, same laws of physics

cosmic horizon – subject of next lecture!

2 Multiple post-Big Bang Inflation, possibly different physical

“bubbles” constants or dimensions in different

“bubbles” – subject of next lecture!

3 The “many worlds” of Quantum physics, quantum

quantum physics computing; can coexist with Level 1

or Level 2 – subject of this lecture!

4 Other mathematical String theory and M-theory; whatever

structures is mathematically possible is

physically realizable – subject of

next lecture!

Tegmark’s Parallel Universe Levels

Level Description Assumptions

1 Regions beyond our Infinite space, same laws of physics

cosmic horizon – subject of next lecture!

2 Multiple post-Big Bang Inflation, possibly different physical

“bubbles” constants or dimensions in different

“bubbles” – subject of next lecture!

3 The “many worlds” of Quantum physics, quantum

quantum physics computing; can coexist with Level 1

or Level 2 – subject of this lecture!

4 Other mathematical String theory and M-theory; whatever

structures is mathematically possible is

physically realizable – subject of

next lecture!

Continuity vs. Quantization

Truly Continuous

Continuity vs. Quantization

1 2 3 4 5 6 7 8 9 …

Discrete or Quantized

Founders of Quantum Theory

Niels Bohr (1885-1962)Max Planck (1858-1947) Albert Einstein (1879-1955)

Planck: Blackbody Radiation

- Box made of perfectly radiation-absorbing material

- Box has a small hole

- Put box in furnace, measure energy radiating out

of hole

Anatomy of a Wave

- A = Wavelength of the wave

- B = Amplitude of the wave

- Frequency (n) of the wave = number of wavelengths / unit time

- Wavelength = 1 / Frequency; Frequency = 1 / Wavelength

Blackbody Radiation SpectrumRadiation intensity (E) measured at hole in box

= 1/frequency (1/n)

Radiation

escapes from

hole in box

Box absorbs

radiation

perfectly, heating

interior

=E (E = f(n) – continuous

function

of frequency)

Blackbody Radiation SpectrumRadiation intensity (E) measured at hole in box

= 1/frequency (1/n)

Radiation

escapes from

hole in box

Box absorbs

radiation

perfectly, heating

interior

=E (E = f(n) – continuous

function

of frequency)

Planck: E = f(hn) –

quantized function

Einstein: Photoelectric Effect

Potassium – 2.0 eV needed to eject electron

Bohr’s Atomic Model

- Electron “orbits” nucleus of atom

- When atom absorbs photon (E = hn), electron “jumps” to

orbit farther away from nucleus

- When atom emits photon, electron “jumps” back

Transition N=2 to N=3

Werner Heisenberg

(1901-1976)

Erwin Schrodinger

(1887-1961)

Founders of Quantum Mechanics

Max Born (1882-1970)

Consequences of Quantization

- Limits knowledge precision (Heisenberg’s Uncertainty Principle)

… Example: position and momentum

- Measurement unavoidably alters what is being measured

- Particles in multiple states at once … until we measure them

- Particles coordinate their properties instantaneously,

regardless of distance in space and time

- The universe at its most basic level is probabilistic instead

of deterministic

… Observed phenomena (e.g., interference) can only be

described by wave-based mathematics

I think it is safe to say that no one really understands

quantum mechanics. Do not keep saying to yourself,

if you can possibly avoid it, “How can it possibly be

like that?” No one knows how it can possibly be

like that.

-- Richard Feynman

Nobel Prize, 1965

Richard Feynman (1918-1988)

The more success the quantum theory has had,

the sillier it looks.

-- Albert Einstein

I do not like it, and I am sorry I ever had anything

to do with it.

-- Erwin Schrodinger

Level 3: The Many Worlds of

Quantum Physics

- Key concepts: superpositions, entanglement, decoherence

- Superpositions represent the results of events or actions in

the world with more than one possible outcome

- Entanglement occurs when the different elements involved

in a superposition evolve unobserved over time – without

interacting with anything else outside the superposition

- Decoherence causes a superposition to break down

due to interaction with the world outside the superposition

|X> = a vector for the quantum state of some thing (X)

y = the wavefunction representing how the quantum state

changes over time

Quantum Mechanics Notation

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

a2 = probability

that material

does not decay

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

a2 = probability

that material

does not decay

b2 = probability

that material

does decay

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

a2 = probability

that material

does not decay

b2 = probability

that material

does decay

Y = a(|Alive>Cat) + b(|Dead>Cat)

Superposition: Schrodinger’s Cat

The cat is in a superposition

of two states: alive and dead

|Alive>Cat

|Dead>Cat

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Superposition

- Hilbert space

…Vectors for different

outcome states are

orthogonal coordinates

|Alive>Cat

|Dead>Cat

a

b

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Superposition

- Hilbert space

…Vectors for different

outcome states are

orthogonal coordinates

... a and b are

amplitudes for

each outcome

|Alive>Cat

|Dead>Cat

a

b

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

Superposition

|Alive>Cat

|Dead>Cat1

a

b

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

Superposition

1

Unit circle represents all

points in Hilbert space

that satisfy a2 + b2 = 1

… Whatever the state

of the cat is, it has to

be on this circle

|Alive>Cat

|Dead>Cat

a

b

Amplitudes:

a = cos 45o = 1/ 2

b = sin 45o = 1/ 2

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

Superposition

a2 = ½

b2 = ½

|Alive>Cat

|Dead>Cat

1

a

b

45o

Amplitudes:

a = cos 45o = 1/ 2

b = sin 45o = 1/ 2

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

Superposition

a2 = ½

b2 = ½ This unit vector in Hilbert space

points to the state of reality

in the superposition (the state

of the cat)

|Alive>Cat

|Dead>Cat

1

a

b 1

a

b

45o 30o

Amplitudes:

a = cos 45o = 1/ 2

b = sin 45o = 1/ 2

Amplitudes:

a = cos 30o = 3 / 4 = 3 / 2

b = sin 30o = 1/ 4 = 1/2

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

|Dead>Cat

|Alive>Cat

Superposition

a2 = ½

b2 = ½ a2 = ¾

b2 = ¼

|Alive>Cat

|Dead>Cat

1

a

b 1

a

b

45o 30o

Amplitudes:

a = cos 45o = 1/ 2

b = sin 45o = 1/ 2

Amplitudes:

a = cos 30o = 3 / 4 = 3 / 2

b = sin 30o = 1/ 4 = 1/2

Y = a(|Alive>Cat ) + b(|Dead>Cat )

Unitary: a2 + b2 = 1

|Dead>Cat

|Alive>Cat

Superposition

a2 = ½

b2 = ½ a2 = ¾

b2 = ¼

What happens when an Observer

looks in the box?????

Copenhagen Interpretation of

Superposition

- Developed by Niels Bohr and Erwin Schrodinger in the 1920’s

- Suppose we do the Schrodinger’s Cat experiment with result:

… Y = a(|Alive>Cat ) + b(|Dead>Cat )

- The superposition “collapses” – the highest amplitude

outcome is actually observed on average

… A lower amplitude outcome might be observed

(example: quantum tunneling)

- What causes the superposition to “collapse”?

… It’s a mystery!

Electrons Tunneling through a

Resistance Barrier

Quantum Tunneling

Disk Drive Read Head Spin Valve

1 1 0 1 1 0 1DataMagnetic fields

in disk mediaMedia motion

Quantum Tunneling

Disk Drive Read Head Spin Valve

Magnetic fields

in head

High resistance

state

Low resistance

state

1 1 0 1 1 0 1Data

- High amplitude: head doesn’t conduct current at all

(high resistance state)

Magnetic fields

in disk mediaMedia motion

Insulator

Quantum Tunneling

Disk Drive Read Head Spin Valve

Direction of

tunneling

current in head

Magnetic fields

in head

High resistance

state

Low resistance

state

1 1 0 1 1 0 1Data

- High amplitude: head doesn’t conduct current “at all”

(high resistance state)

- Low amplitude: head conducts tiny current if magnetic field

directions in head are aligned (low resistance state –

tunneling current)

Magnetic fields

in disk mediaMedia motion

Insulator

1101101

Entanglement

- When objects interact they share a single quantum

state until a measurement is made

… None of the individual objects can be fully described

without considering all the others

- Suppose the objects separate in space-time before the

measurement happens

… When one of them is measured, the properties of the

others instantaneously adjust to be consistent with it

- Entanglements only continue while the objects are isolated

from contact with anything else

… Any further interactions with other objects result in

decoherence

Entanglement

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

orA AB B

Entanglement

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

or

A B

A AB B

Entanglement

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

or

Entangled

A B

A B

A AB B

Entanglement

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

or

Entangled

A B

A B

A B

A AB B

Entanglement

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

or

Entangled

A B

A B

A B

A AB B

Entanglement and Decoherence

- Example: 2 electrons interact

… Each electron’s spin is either up ( ) or down ( )

… Spins must be opposite after the interaction

or

Entangled

Spin of A instantaneously set

regardless of distance between A & B

2 electrons are now decoherent

A B

A B

A B

A AB B

At any rate, I am convinced that [God] does not

play dice.

- Albert Einstein, in a

letter to Max Born

…[T]he laws of quantum mechanics itself cannot

be formulated … without recourse to

the concept of consciousness.

- Eugene Wigner

Nobel Prize, 1963

Hugh Everett III

Hugh Everett III

(1930-1982)

- Developed “relative state” formulation

of quantum mechanical superposition

in 1957 to address objections to

Copenhagen interpretation

- Left physics when ideas were not

accepted by Niels Bohr and other

mainstream physicists

- Everett’s theory began to gain support

starting in the early 1970’s

- Today Everett’s theory is the foundation

of quantum computing based on

later work by David Deutsch and others

Bryce Seligman deWitt

(1923-2004)

Bryce deWitt

- American theoretical physicist; worked

with John Wheeler at Institute of

Advanced Study at Princeton; Dirac

Prize, 1987

- Responsible for discovering Hugh

Everett’s thesis and popularizing it

as the many-worlds interpretation

of quantum physics in 1973

Everett’s “Many-Worlds” Interpretation

of Superposition

- Again, suppose we have the result of the Schrodinger’s Cat

experiment:

… The Observer’s state is entangled with the cat’s state

… Y = a(|”Alive”>Observer|Alive>Cat) + b(|”Dead”>Observer|Dead>Cat)

- Because of quantum decoherence the two parts of the

superposition separate into different universes

… Each universe has a version of the Observer, box and cat

… In one universe, the Observer sees a live cat

… In the other universe, the Observer sees a dead cat

-The superposition never “collapses”

… Universes can still interfere with each other if not too

different

Everett’s “Many-Worlds” Interpretation

of Superposition

Evidence for the Many-Worlds

Interpretation

- Quantum interference effects

… Multiple slit experiments

- Quantum computing

Quantum Interference Effects

- What is interference?

The Double-Slit Experiment

The Double-Slit Experiment- If a wave is aimed at two closely spaced slits an interference

pattern will result at a detector on the other side of the slits

Constructive

interference

Destructive

interference

The Double-Slit Experiment- If electrons are aimed one by one at two closely

spaced slits an interference pattern will

gradually build up on a detector on the other

side of the slits

- What is interfering with the electrons?

The Double-Slit Experiment

- Some observations

… If you put a detector near one or both slits, the interference

pattern disappears

… If you turn off the detector, the interference pattern comes

back

… If you set up a delayed means of determining what slit

the electron went through, but do nothing right now to

impact it, the interference pattern disappears

The Source of the Interference

in The Double-Slit Experiment

- Classical physics: Total probability conceptually depends on

adding individual probabilities that electron goes thru each

slit:

a2 + b2 = P(Right Slit) + P(Left Slit)

The Source of the Interference

in The Double-Slit Experiment

- Classical physics: Total probability conceptually depends on

adding individual probabilities that electron goes thru each

slit:

a2 + b2 = P(Right Slit) + P(Left Slit)

- Quantum physics: Total probability conceptually depends on

squaring the sum of the amplitudes for each electron!

(a + b)2 = a2 + b2 + 2ab =

P(Right Slit) + P(Left Slit) + interference

David Deutsch

David Deutsch (1953-)

- Affiliated with Oxford physics

department; Dirac Prize, 1998;

Fellow of the Royal Society,

2008

- Founder of quantum computing;

inventor of first quantum logic

gates, quantum algorithms,

quantum error correction

techniques and other key

aspects of quantum computation

- Major proponent of many-worlds interpretation of quantum

mechanics

- Although affiliated with Oxford, is not on faculty and has never

taught a course

David Deutsch’s Four-Slit Experiment

- Photons are shot one at a time at the apparatus

- Compare the interference pattern caused by 4 slits (a)

with the pattern caused by 2 slits (b)

x

- Notice what happens at point x

… “Something” is coming through 2 of the slits to interfere

with the photons coming through the other 2 slits and

prevent them from reaching point x

Let us take stock. We have found that when one

photon passes through this apparatus,

-- It passes through one of the slits, and then

something interferes with it, deflecting it in a way

that depends on what other slits are open;

-- The interfering entities have passed through

some of the other slits;

-- The interfering entities behave exactly like

photons …

… except that they cannot be seen.

-- David Deutsch, The Fabric of Reality

Another Explanation?

- Interference effects don’t necessarily prove existence of

parallel universes

- Decoherence effects could explain why we don’t see

superpositions in the macrosopic world with no need for

parallel universes

Decoherence

- Superpositions persist only as long as they are completely

isolated

… The breakdown of a superposition due to interaction with

the world outside the superposition is called decoherence

- For macroscopic objects, this happens very quickly

- Perhaps decoherence is an alternative to many-worlds?

… Maybe superpositions “bleed” or “leak” into

surrounding environment rather than “collapsing”

Wojciech Zurek (1951-)

Wojciech Zurek

- Polish physicist at Los Alamos

National Laboratory and the Santa

Fe Institute

- Performed key research with Dieter

Zeh in late 1970’s and 1980’s that

elaborated how quantum decoherence

could explain why superpositions

are not seen in the macro world

Decoherence and Schrodinger’s Cat

- Suppose our original experiment with the cat:

… Y = a(|Alive>Cat ) + b(|Dead>Cat )

- Now suppose a molecule that can be in state X or Y interacts

with the cat

Decoherence and Schrodinger’s Cat

- Suppose our original experiment with the cat:

… Y = a(|Alive>Cat ) + b(|Dead>Cat )

- Now suppose a molecule that can be in state X or Y interacts

with the cat

… Y = a(|Alive>Cat |X>Molecule) + b(|Alive>Cat |Y>Molecule) +

c(|Dead>Cat |X>Molecule) + d(|Dead>Cat |Y>Molecule) +

e[a(|Alive>Cat ) + b(|Dead>Cat )](|X>Molecule) +

f[a(|Alive>Cat ) + b(|Dead>Cat )](|Y>Molecule) + …

Decoherence and Schrodinger’s Cat

- Suppose our original experiment with the cat:

… Y = a(|Alive>Cat ) + b(|Dead>Cat )

- Now suppose a molecule that can be in state X or Y interacts

with the cat

… Y = a(|Alive>Cat |X>Molecule) + b(|Alive>Cat |Y>Molecule) +

c(|Dead>Cat |X>Molecule) + d(|Dead>Cat |Y>Molecule) +

e[a(|Alive>Cat ) + b(|Dead>Cat )](|X>Molecule) +

f[a(|Alive>Cat ) + b(|Dead>Cat )](|Y>Molecule) + …

… a2 + b2 + c2 + d2 + e2 + f2 + … = 1

Decoherence and Schrodinger’s Cat

- Suppose our original experiment with the cat:

… Y = a(|Alive>Cat ) + b(|Dead>Cat )

- Now suppose a molecule that can be in state X or Y interacts

with the cat

… Y = a(|Alive>Cat |X>Molecule) + b(|Alive>Cat |Y>Molecule) +

c(|Dead>Cat |X>Molecule) + d(|Dead>Cat |Y>Molecule) +

e[a(|Alive>Cat ) + b(|Dead>Cat )](|X>Molecule) +

f[a(|Alive>Cat ) + b(|Dead>Cat )](|Y>Molecule) + …

… a2 + b2 + c2 + d2 + e2 + f2 + … = 1

- The wavefunction becomes much more complicated

- The Hilbert space has many more dimensions

- The overall probabilities are still unitary

- So the chance of the original superposition is greatly reduced

Quantum Computing

Quantum Computing

- Initially proposed by Richard Feynman in 1982

- Main component: qubit = a superposition of 0 and 1

… Y = a|0> + b|1> (a2 + b2 = 1)

- Preparing a qubit

… Normal mirror = white

… Semi-silvered mirror = grey (reflects 50%, transmits 50%)

Photon

in

Qubit

out

ab a3 a2 a1…R:

b qubits

Quantum Computing

- Several qubits together form a qubit register

- Qubit registers represent superpositions of numbers larger

than one qubit

... Example: 3-qubit register can represent any or all of the

following at once: 000, 001, 010, 011, 100, 101, 110, 111

(0) (1) (2) (3) (4) (5) (6) (7)

- Calculation operations on a qubit register happen

in parallel to all numbers in the superposition

A Photonic Quantum Computer

An Electronic Quantum Computer

Applications of Quantum Computing

- Factorizing large numbers

… Classical computing: Divide by each possible factor

starting with 2 up until square root – becomes intractable

… Quantum computing: Try all factors in single parallel step

- Public key cryptography: RSA algorithm

… Invented by Ronald Rivest, Adi Shamir and Leonard

Adelman in 1978 (based on earlier work by Whitfield

Diffie and Martin Hellman in 1971)

… Encrypts secure information on Internet (https://)

… Uses large (512-bit or longer) numbers as keys

… Can break encryption by finding keys’ factors

Peter Shor

Peter Shor (1959-)

- Bell Labs, 1987-2003; Professor of

Mathematics, MIT, 2003-; MacArthur

Fellow, 1999

- Developed Shor’s algorithm for

using quantum computation to

find factors of very large numbers

- Algorithm works by creating a

superposition of all the possible

factors, trying them all in parallel,

and then using quantum interference

to get all the results except the desired factors to cancel

themselves out of the calculation

Breaking the Code – Shor’s Algorithm

- Suppose we factor a 512 bit number using Shor’s algorithm

…~10500 interference terms in superposition during key step

… Where will those calculations be done in reality?

… Entire observable universe has fewer than 10115 particles!

- Shouldn’t we conclude that the interference terms really get

determined in parallel universes?

What’s Wrong with This Picture?

- Decoherence limits (at present) most quantum computing

… Biggest number successfully factored by Shor’s algorithm:

56 (6 bits)

- Interference effects do not necessarily prove existence of

parallel universes

- Decoherence explains the main phenomenon that many-worlds

explains – no macroscopic superpositions

What’s Wrong with This Picture?

- Many-worlds theory is not falsifiable?

- Occam’s Razor

- Does the many-worlds theory describe reality, or only

knowledge of it?

- What is the meaning of probability and uncertainty in

many-worlds theory?

- Theory is too weird

Superpositions

- Suppose a scientist, John (J), is attempting to measure the

spin of an electron (E), which can either be up or down

- If everything were completely deterministic and John’s apparatus

were 100% accurate there would be two possibilities --

… |Ready>J|Spin-up>E |”Spin-up”>J|Spin-up>E

… |Ready>J|Spin-down>E |”Spin-down”>J|Spin-down>E

- But quantum mechanics says the electron is in a superposition:

… a(|Spin-up>E)+ b(|Spin-down>E)

… a2 and b2 are the probabilities of the respective states

- So when John is ready to start the experiment the initial state is:

… |Ready>J [a(|Spin-up>E)+ b(|Spin-down>E)]

- After the experiment the result is also a superposition:

… a(|”Spin-up”>J|Spin-up>E ) + b(|”Spin-down”>J|Spin-down>E)

Copenhagen Interpretation of

Superposition at Outcome of Experiment

- This interpretation was developed by Niels Bohr and Erwin

Schrodinger in the 1920’s

- Suppose we have the result of John’s experiment:

… a(|”Spin-up”>J|Spin-up>E ) + b(|”Spin-down”>J|Spin-down>E)

- The superposition “collapses” and only the highest probability

outcome is actually observed in the (single) universe

… If a2 > b2 then John would observe “Spin-up”

… If a2 < b2 then John would observe “Spin-down”

… If a2 = b2 then John may observe either “Spin-up” or

“Spin-down” but not both (one is randomly chosen)

- What causes the superposition to “collapse”?

… It’s a mystery!

The Source of the Interference

in The Double-Slit Experiment- State vectors: |Slit 1> + |Slit 2>

- Amplitude of electron thru Slit 1 = a; thru Slit 2 = b

- Detector is oriented in the y-axis direction

-To get the wavefunction, multiply by the complex conjugate:

|Y> = (eiay + eiby)(e-iay + e-iby)

= eiaye-iay + eiaye-iby + eibye-iay + eibye-iby

= 2 + eiaye-iby + eibye-iay

= 2 + ei(a-b)y + ei(b-a)y

= 2 + 2 cos (a-b) y

= 2[1 + cos (a-b) y]

Amplitude

y

2

Zeroes of 2[1 + cos (a-b) y] represent

dark places on interference pattern

(where no electrons hit screen)

If one hole is shut,

no interference

The Double-Slit Experiment and the

Many-Worlds Interpretation

- a2 = P(Right Slit)

- b2 = P(Left Slit)

- Outcome of experiment is a superposition:

… Y = [a(|”Right Slit”>Observer|Right Slit>Electron ) +

b(|”Left Slit”>Observer|Left Slit>Electron)]2

- The two parts of the outcome superposition both exist

simultaneously and interfere with each other

… Where do they exist? – In parallel universes?

Modular Arithmetic

- Sometimes called “clock arithmetic”

- A system where numbers “wrap around” after they reach a

certain value (the modulus)

… Example: Add 4 hours to 9:00 (on a 12 hour clock) = 1:00,

so 9 + 4 = 13 = 1 mod 12

… But adding 16 hours to 9:00 also = 1:00, so 9 + 16 = 25

= 1 mod 12 too! So modular arithmetic is periodic

- Closely related to finding the remainder in division

n = pq

(p, q prime)

e = f(p,q)

d = f(e,p,q)

Public key:

(n,e)Private key:

(n,d)

Large

random

number

Generate

keys

Alice

Alice

WebMerchant.comm

“$500 credit

approved”

Alice’s

public key

Encrypt

C = me mod n

C

DFCD3454

BBEA788…

m

“$500 credit

approved”

Internet

(https://)

Decrypt

Alice’s

private key

m = Cd mod n

Public-Key Cryptography - RSA

d is the modular inverse of e

Public-Key Cryptography - RSA

C = md mod nm

C

mm = Ce mod n(n, e) (n, d)

n = pq

(p, q prime)

calculate

e, d

e = f(p,q)

d = f(e,p,q)

Public-Key Cryptography - RSA

- Key generation program:

… Choose 2 prime numbers (p, q)

… n = pq; f(n) = (p – 1)(q – 1)

… Choose e such that 1 < e < f(n) and greatest common

denominator of e and f(n) is 1

… Solve for d = e-1 mod f(n); i.e., find d given (d * e) mod f(n) = 1

… Public key = (n, e); private key = (n, d)

C = md mod nm

C

mm = Ce mod n

(n, e) (n, d)

n = pq

(p, q prime)

calculate

e, d

Breaking the Code – Shor’s Algorithm

- Suppose n is the number we want to factor

- xy mod n is periodic when the greatest common divisor of x

and n is 1, so the remainder of dividing xy / n is 1 whenever

y cycles completely through its period

- Suppose the period of xy mod n is r

… x0 = 1 mod n (since x0 = 1); xr = 1 mod n; x2r = 1 mod n; etc.

- Then:

... (xr/2)2 = xr = 1 mod n

... (xr/2)2 - 1 = 0 mod n, so [(xr/2)2 – 1] / n has no remainder

… If r is even, (xr/2 – 1)(xr/2 + 1) is an integer multiple of n

… As long as |xr/2| is not 1, one of these must have a common

factor with n

- Shor’s algorithm uses quantum computing to find r

Breaking the Code – Shor’s Algorithm

|y>

- Make y’s from the integers 1 thru g -1 such that n2 <= g <= 2n2

- Prepare a superposition |y> of all the y’s in R1

|yb> |y3> |y2> |y1>…R1: R2:

b qubits b qubits

…

Prepare |y>

Breaking the Code – Shor’s Algorithm

|y>

|xy mod n> = |k>

- Calculate xy mod n on |y> and place the resulting superposition

|k> in R2

|yb> |y3> |y2> |y1>…

|k>

|kb> |k3> |k2> |k1>…R1: R2:

b qubits b qubits

Breaking the Code – Shor’s Algorithm

|y>

|xy mod n> = |k>

- Calculate xy mod n on |y> and place the resulting superposition

|k> in R2

- The qubits in R1 (|y>) and R2 (|k>) are now entangled

|yb> |y3> |y2> |y1>…

|k>

|kb> |k3> |k2> |k1>…R1: R2:

b qubits b qubits

Breaking the Code – Shor’s Algorithm

- Measure R2

… Sets value of k

… Also sets R1 to superposition of states consistent

with k: c, c + g/r, c + 2g/r, etc. where c is the lowest integer

such that xc mod n = k

|c + jg/r>

Measure R2

|c+

jg/rb>

|c+

jg/r3>

|c+

jg/r2>

|c+

jg/r1>…

k

kb k3 k2 k1…R1: R2:

J=0, 1, 2, …

Breaking the Code – Shor’s Algorithm

- Performing a quantum discrete Fourier transform (QDFT) on R1

and measuring R1 then yields g/r

g/r

QDFT(R1); Measure R1

g/rb g/r3 g/r2 g/r1…

k

kb k3 k2 k1…R1: R2: