Physics becomes the computer Norm Margolus Physics becomes the - - PowerPoint PPT Presentation

Computing Beyond Silicon Summer School

Physics becomes the computer

Norm Margolus

Physics becomes the computer

Emulating Physics

» Finite-state, locality, invertibility, and conservation laws

Physical Worlds

» Incorporating comp-universality at small and large scales

Spatial Computers

» Architectures and algorithms for large-scale spatial computations

Nature as Computer

» Physical concepts enter CS and computer concepts enter Physics

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Looking at nature as a computer

Looking at computation as physics

Looking at nature as a computer

Introduction

As we zoom in on a digital image,

Introduction

As we zoom in on a digital image, we begin to notice that there isn’t an infinite amount of resolution:

Introduction

As we zoom in on a digital image, we begin to notice that there isn’t an infinite amount of resolution: We begin to see the pixels.

Introduction

Something similar happens in nature. A box full of particles doesn’t have an infinite number of possible configurations:

Introduction

Something similar happens in nature. A box full of particles doesn’t have an infinite number of different configurations: the number of distinct configurations is finite.

Introduction

Similarly, the rate at which a finite system can transition from one distinct state to another is also finite.

Introduction

Similarly, the rate at which a finite system can transition from one distinct state to another is also finite.

Introduction

Similarly, the rate at which a finite system can transition from one distinct state to another is also finite.

Introduction

Similarly, the rate at which a finite system can transition from one distinct state to another is also finite.

Introduction

Similarly, the rate at which a finite system can transition from one distinct state to another is also finite. Thus a finite physical system is much like a computer.

Introduction

• Physics studies macro

properties of finite information systems

• Basic quantities such

as Entropy and Energy are informational:

dQ=TdS

EntropyMAX=InfoMAX KineticEMAX=OpsMAX

Introduction

• Physics studies macro

properties of finite information systems

• Basic quantities such

as Entropy and Energy are informational:

dQ=TdS

EntropyMAX=InfoMAX KineticEMAX=OpsMAX (1996, with Levitin)

In this talk…

Review:

• info (Entropy) in physics

Discuss:

• statistical description of

computation (→ QM)

• energy and action in comp
• what does QM add?
• physics as computation

What is Info?

• number of bits system can

hold, given its constraints

• system with 2n possible

states can represent n bits

• focus on classical info:

» survives in macro limit » substitute micro dynamics when QM is invisible » ordinary macro quantities have classical info interp

Info = −∑ p p

i i i

log Info = − =

=

∑ 1

1

1 Ω

Ω Ω

Ω i

log log

‰ equally probable states,

What is Entropy?

• Formal parameter in

thermo (irreversibility)

• Boltzmann and Gibbs

understood as counting

• Mixing neat → mess
• Mixing mess → mess
• Entropy is log of #states

that fit with constraints

Classical Entropy

• For particles in a box,

can introduce some coarseness

• This allows relative

probabilities to be calculated

• (Also do the same

thing for momentum)

Infinite Entropy?

• Thermo of EM radiation in

cavity led to QM

• General state is a

superposition of waves with integer num peaks

• Any amplitude, can put

unit of energy into any wave (infinite info!)

• Planck proposed E = nhν

(finite info!)

EM radiation in a cavity (periodic boundaries)

Looking at nature as a computer

• With QM, every finite

system has finite state

• Dynamics of finite state

systems is familiar

• Develop QM from

computer viewpoint!

• Begin by discussing

computer logic in statistical situations

Looking at computation as physics

• With QM, every finite

system has finite state

• Dynamics of finite state

systems is familiar

• Develop QM from

computer viewpoint

• Begin by discussing

computer logic in statistical situations

Statistical Dynamics

• To give a complete

dynamics, we say what happens to each state in a fixed time

• Weighted sum of

states (superposition) describes an ensemble

• Probability of initial

state applies to corresponding final state

XOR

B A A A⊕B

U U U U

XOR XOR XOR XOR

00 00 01 01 10 11 11 10 = = = = a b c d a b c d 00 01 10 11 00 01 11 10 + + + → + + +

Statistical Dynamics

• Better to use square

roots of probabilities (amplitudes)

• Evolution preserves

vector length

• Lets us analyze

system in other bases

XOR

B A A A⊕B

U U U U

XOR XOR XOR XOR

00 00 01 01 10 11 11 10 = = = = a b c d a b c d 00 01 10 11 00 01 11 10 + + + → + + +

Energy Basis

U

N τ : Χ

Χ Χ Χ

1 1

→ → → →

−

L

E N U E N E

N 1 1 1 2

1 1 = + + +

( )

= + + +

( )

=

−

Χ Χ Χ Χ Χ Χ L L

τ

• Suppose Uτ represents
• ne clock period of a

reversible computer

• Add together all

configs in orbit

• This state has equal

prob for any config

• Time evolution leaves

this state unchanged!

Energy Basis

• Example: suppose

computer only has one bit, and Uτ just flips it.

• Form new 2-state basis by

adding and subtracting configs

• Magnitudes of amplitudes
• f energy states don’t

change with time

E E

1

1 2 1 2 = + = − , U U

τ τ

1 1 = = ,

NOT

A A

U E E U E E

τ τ 1 1

= = − ,

Energy Basis

• In general: use complex

amplitudes to form new

• rthogonal basis
• a〉 is like a column

vector of components

• 〈a is like a row vector
• f complex conjugates

E N e U E N e e E E E N e N e

n inm N m m n inm N m m in N n j k i km jm N m m m m im k j N m j k

= = = = = =

∑ ∑ ∑ ∑

+ − − ′ ′ ′ −

1 1 1 1

2 2 1 2 2 2 π τ π π π π

δ

/ / / ( )/ , ( )/ ,

, Χ Χ Χ Χ U

N τ : Χ

Χ Χ Χ

1 1

→ → → →

−

L

Energy Basis

• Energy basis is Fourier

Transform of config basis

• En〉 cycles with a frequency
• f νn=ν(n/N), where ν=1/τ
• We will call hνn the Energy
• f the state En〉, i.e. En=hνn

E N e N e E U E N e e E n N

n inm N m m m inm N n n n inm N m m in N n n

= = = = = ×

∑ ∑ ∑

− + −

1 1 1 2 2

2 2 2 1 2 π π τ π π

π π τ τ

/ / / /

, . Χ Χ Χ

For a cycle:

U

N τ : Χ

Χ Χ Χ

1 1

→ → → →

−

L

Energy Basis

• Interpret coefficients in

energy basis as probs

• Energy of any state is

independent of time

• Xn〉 is composed of

equally spaced energies, En=nhν1

• E=hν/2, or ν=2E/h

U

N τ : Χ

Χ Χ Χ

1 1

→ → → →

−

L

e.g. For energies are so , , , , , , , , ( )

/ /

Ψ Ψ Χ

2 2 2 2

2 3 1 2 = + = + = + − =

− −

α β α β α β ν ν ν ν ν

τ π π

E E e E e E E E E h N h N h N N h N E h

j k ij N j ik N k j k m

K E N e N e E

n inm N m m m inm N n n

= =

∑ ∑

−

1 1

2 2 π π / /

, . Χ Χ

What is Energy?

• ν=2E/h, so energy is

rate of change of configurations

• CA lattice can change
• ne spot at a time for

reversible rules

• Should count changes

as bit changes (i.e., energy is extensive!)

1 1 1

t x

1 1 1 1 1

t x

1 1

U

N τ : Χ

Χ Χ Χ

1 1

→ → → →

−

L

What is Energy?

• Conservation Law:

number of ones constant

• Constrains number of

spots that can change in lattice update period τl

• Focus on energy of the

spots that can change

• If each particle is

assigned an energy hνl max change is still 2E/h

M h M E h

l l

= = = = num particles ν ν ν particle energy /

∆

2 2 t : t

l

+ τ :

What is Action?

• ν=2E/h, so Ω(t)=2Et/h
• Action is amount of

evolution (total ops for ideal computation)

• Number of comp events

in rest frame is rel scalar

• Comp energy must

transform like rel energy: 2Ertr/h = 2(Et-px)/h

• If x/t=c, then E=cp so

that Et=px (comp stops)

rest frame moving frame

What does QM add?

• Stat Comp is special case:

QM allows some new kinds of operations

• Any invertible evolution

which preserves vector length is okay

• Probabilities can come

and go!

• Only need to add extra

single-bit operations

• ν∆=2(E-Emin)/h

A A

U U

NOT NOT

1 1 1

1 2 1 2 1 2 1 2

= − = +

NOT NOT

U U U

NOT NOT NOT

1 1

1 2 1 2

= −

( ) = − A A

NOT

U U U

NOT NOT NOT

1 1

1 2 1 2

= +

( ) = +

θ

XOR + are universal!

XOR XOR XOR

• π/8
• π/8

π/8 π/8

A A A B C A B⊕AC C

U U

θ θ

θ θ θ θ 1 1 1 = − = + cos sin sin cos

4 NOT

θ π θ π

θ θ

= = = = / : / :

NOT NOT

2 4 U U U U

θ

XOR + are universal!

XOR XOR XOR

• π/8
• π/8

π/8 π/8

A A A B C A B⊕AC C

U U

θ θ

θ θ θ θ 1 1 1 = − = + cos sin sin cos

4 NOT

θ π θ π

θ θ

= = = = / : / :

NOT NOT

2 4 U U U U

No prob- abilities No prob- abilities

1 2 444 3 444

Superposition of different configurations

θ

XOR + are universal!

XOR XOR XOR

• π/8
• π/8

π/8 π/8

A A 1 1

U U

θ θ

θ θ θ θ 1 1 1 = − = + cos sin sin cos

4 NOT

c s = = cos( / ) sin( / ) π π 8 8 001 c s 001 011 + c s 001 011 +

1 2 1 2

001 011 +

1 2 1 2

001 011 + c s 001 011 + c s 001 011 + 001

What does QM add?

• No new kinds of

computations; at most reduces effort required

• Distinction is basis

dependent

• Fundamental Q: If

speedup is exponential, then distinction is real!

What does this mean?

a b b a 1 1 + → −

NOT

a b a b a b 1 2 2 1 + → + + −

NOT

Classical: Quantum:

What does this mean?

Classical: Quantum:

Boston New York

a b b a 1 1 + → −

NOT

a b a b a b 1 2 2 1 + → + + −

NOT

Conclusions

• On a large scale, often

can’t tell if micro finite- state is QM or CM

• Entropy, Energy and

Action all have comp meaning: others must

• Significant for comp

and for physics

for more information, see http://www.ai.mit.edu/people/nhm/looking-at-nature.pdf