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 0/1 Spatial Computers » Architectures and algorithms for large-scale spatial computations Nature as Computer » Physical concepts enter CS and computer concepts enter Physics
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 dQ=TdS properties of finite information systems • Basic quantities such as Entropy and Energy are informational: Entropy MAX = Info MAX KineticE MAX = Ops MAX
Introduction • Physics studies macro dQ=TdS properties of finite information systems • Basic quantities such as Entropy and Energy are informational: Entropy MAX = Info MAX KineticE MAX = Ops MAX (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 Info = − ∑ p hold, given its constraints log p i i • system with 2 n possible i states can represent n bits ‰ equally probable states, • focus on classical info: Ω ∑ 1 1 Info = − » survives in macro limit log 1 Ω Ω = » substitute micro dynamics i = Ω log when QM is invisible » ordinary macro quantities have classical info interp
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 = nh ν EM radiation in a cavity (finite info!) (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 A • To give a complete A A ⊕ B dynamics, we say XOR B what happens to each state in a fixed time = U 00 00 XOR • Weighted sum of = U 01 01 XOR states (superposition) = U 10 11 XOR describes an ensemble = U 11 10 XOR • Probability of initial + + + state applies to a 00 b 01 c 10 d 11 → corresponding final + + + a 00 b 01 c 11 d 10 state
Statistical Dynamics A A • Better to use square A ⊕ B XOR B roots of probabilities (amplitudes) = U 00 00 XOR • Evolution preserves = U 01 01 XOR vector length = U 10 11 XOR = U 11 10 • Lets us analyze XOR system in other bases + + + a 00 b 01 c 10 d 11 → + + + a 00 b 01 c 11 d 10
Energy Basis • Suppose U τ represents τ : Χ → Χ → → Χ → Χ L U − 0 1 N 1 0 one clock period of a reversible computer ( ) 1 = Χ + Χ + + Χ L E • Add together all − 0 0 1 N 1 N configs in orbit ( ) 1 = Χ + Χ + + Χ • This state has equal L U E τ 0 1 2 0 N prob for any config = • Time evolution leaves E 0 this state unchanged!
Energy Basis • Example: suppose A A NOT computer only has one bit, and U τ just flips it. = = U 0 1 , U 1 0 τ τ • Form new 2-state basis by adding and subtracting + − 0 1 0 1 = = configs E , E 0 1 2 2 • Magnitudes of amplitudes of energy states don’t = = − U E E , U E E τ τ change with time 0 0 1 1
Energy Basis τ : Χ → Χ → → Χ → Χ L U − 0 1 N 1 0 ∑ 1 π = Χ 2 inm N / E e , n m • In general: use complex N m ∑ 1 = π Χ amplitudes to form new 2 inm N / U E e τ + n m 1 N m orthogonal basis − π = 2 in N / e E n • a 〉 is like a column ∑ 1 π − ′ vector of components = Χ Χ 2 i km ( jm )/ N E E e ′ j k m m N ′ m m , • 〈 a is like a row vector ∑ 1 = π − = δ 2 im k ( j )/ N e j k , N of complex conjugates m
Energy Basis τ : Χ → Χ → → Χ → Χ L U − 0 1 N 1 0 ∑ 1 = π Χ 2 inm N / E e , n m N m ∑ 1 Χ = − π • Energy basis is Fourier 2 inm N / e E . m n N n Transform of config basis ∑ • E n 〉 cycles with a frequency 1 = π Χ 2 inm N / U E e τ + n m 1 N of ν n = ν (n/N) , where ν = 1 / τ m − π = 2 in N / e E n • We will call h ν n the Energy For a cycle: of the state E n 〉 , i.e. E n =h ν n τ n π = π × n 2 2 τ N
Energy Basis ∑ 1 τ : Χ → Χ → → Χ → Χ = π Χ L 2 inm N / U E e , − 0 1 N 1 0 n m N m ∑ 1 Χ = − π 2 inm N / e E . m n • Interpret coefficients in N n energy basis as probs Ψ = α + β e.g. , E E , 0 j k • Energy of any state is Ψ = α − π + β − π 2 ij N / 2 ik N / e E e E τ j k independent of time = α + β 2 2 E E E j k • X n 〉 is composed of Χ equally spaced energies, For , energies are m E n =nh ν 1 ν ν ν ν h h h h − K 0 , , 2 , 3 , , ( N 1 ) N N N N • E=h ν /2 , or ν = 2 E/h ν h = so E 2
What is Energy? τ : Χ → Χ → → Χ → Χ L U − 0 1 N 1 0 t • ν = 2 E/h, so energy is rate of change of 1 0 0 1 0 configurations 0 1 1 0 1 x • CA lattice can change one spot at a time for t reversible rules 0 1 0 0 • Should count changes 1 1 0 as bit changes (i.e., 1 1 0 x energy is extensive!)
What is Energy? • Conservation Law: number of ones constant t : • Constrains number of spots that can change in lattice update period τ l + τ : t l • Focus on energy of the spots that can change = M num particles • If each particle is assigned an energy h ν l ν = h particle energy l ν = ν = 2 M 2 E h / max change is still 2E/h ∆ l
What is Action? • ν = 2 E/h, so Ω (t)=2Et/h • Action is amount of evolution (total ops for ideal computation) rest frame • Number of comp events in rest frame is rel scalar • Comp energy must transform like rel energy: 2E r t r /h = 2(Et-px)/h • If x/t=c , then E=cp so moving frame that Et=px (comp stops)
What does QM add? • Stat Comp is special case: A A NOT QM allows some new kinds of operations = − 1 1 U 0 0 1 NOT 2 2 = + • Any invertible evolution 1 1 U 1 0 1 NOT 2 2 which preserves vector length is okay A A NOT NOT • Probabilities can come = U U 0 and go! NOT NOT ( ) = − − 1 1 U 0 1 1 • Only need to add extra NOT 2 2 = single-bit operations U U 1 NOT NOT ( ) = + • ν ∆ =2(E-E min )/h + 1 1 U 0 1 0 NOT 2 2
4 NOT XOR + are universal! θ = π = = θ − θ / : 2 U U U 0 cos 0 sin 1 θ θ NOT θ A A θ = π = = θ + θ / 4 : U U U 1 sin 0 cos 1 θ θ NOT A A B ⊕ AC B π/8 XOR XOR π/8 - π/8 - π/8 XOR C C
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