Physics becomes the. computer. Computing Beyond Silicon Summer School. Norm Margolus

Transcription

1 Computing Beyond Silicon Summer School Physics becomes the computer Norm Margolus

2 Physics becomes the computer 0/ 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

3 Looking at nature as a computer

4 Looking at computation as physics

5 Looking at nature as a computer

6 Introduction As we zoom in on a digital image,

7 Introduction As we zoom in on a digital image, we begin to notice that there isn t an infinite amount of resolution:

8 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.

9 Introduction Something similar happens in nature. A box full of particles doesn t have an infinite number of possible configurations:

10 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.

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

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

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

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

15 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.

16 Introduction dq=tds Physics studies macro properties of finite information systems Basic quantities such as Entropy and Energy are informational: Entropy MAX =Info MAX KineticE MAX =Ops MAX

17 Introduction dq=tds Physics studies macro properties of finite information systems Basic quantities such as Entropy and Energy are informational: Entropy MAX =Info MAX KineticE MAX =Ops MAX (996, with Levitin)

18 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

19 What is Info? Info = p log p i i equally probable states, Ω Info = log = i = log Ω Ω Ω i number of bits system can hold, given its constraints system with 2 n 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

20 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

21 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)

22 Infinite Entropy? EM radiation in a cavity (periodic boundaries) 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!)

23 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

24 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

25 Statistical Dynamics A A B XOR A B U U U U XOR XOR XOR XOR 00 = 00 0 = 0 0 = = 0 a00 + b0 + c0 + d a00 + b0 + c + d0 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

26 Statistical Dynamics A A B XOR A B U U U U XOR XOR XOR XOR 00 = 00 0 = 0 0 = = 0 a 00 + b 0 + c 0 + d a 00 + b 0 + c + d0 Better to use square roots of probabilities (amplitudes) Evolution preserves vector length Lets us analyze system in other bases

27 Energy Basis U : Χ τ Χ L Χ Χ N E U E τ 0 0 = Χ + Χ + L + Χ N 0 0 N = Χ + Χ + L + Χ N = E 0 ( ) ( ) Suppose U τ represents one 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!

28 Energy Basis A NOT A U 0 =, U = 0 τ τ 0 + E =, E 2 0 = 0 Uτ E0 = E0, Uτ E = E 2 Example: suppose computer only has one bit, and U τ just flips it. Form new 2-state basis by adding and subtracting configs Magnitudes of amplitudes of energy states don t change with time

29 E n U E τ n E E j k Energy Basis U : Χ τ Χ L Χ Χ N 0 0 = = N N N m m mm, e e e 2 π in / N = e E = 2 π inm / N 2 π inm / N 2 π im ( k j )/ N = e = δ N m n Χ Χ 2 π ikm ( jm )/ N m, m + Χ Χ m m jk, In general: use complex amplitudes to form new orthogonal basis a is like a column vector of components a is like a row vector of complex conjugates

30 Energy Basis E U E τ Χ n m n = = = N N N m e 2 π inm / N e E e n τ n 2π = 2π N τ n m 2 π in / N = e E For a cycle: 2 π inm / N 2 π inm / N n Χ Χ m n,. m + U : Χ τ Χ L Χ Χ N 0 0 Energy basis is Fourier Transform of config basis E n cycles with a frequency of ν n =ν(n/n), where ν=/τ We will call hν n the Energy of the state E n, i.e. E n =hν n

31 Energy Basis E Χ e.g., Ψ = α E + β E, For Χ, energies are so n m Ψ 0 τ = = = α + β 2πij / N 2πik / N e E j e Ek 2 2 j k E = α E + β E m ν ν ν ν 0,, 2, 3, K,( N ) h N E = N N h N h ν 2 m n e 2 π inm / N 2 π inm / N e E j k h N Χ m n,. h N U : Χ τ Χ L Χ Χ N 0 0 Interpret coefficients in energy basis as probs Energy of any state is independent of time X n is composed of equally spaced energies, E n =nhν E=hν/2, or ν=2e/h

32 t t What is Energy? U : Χ τ Χ L Χ Χ N x x ν=2e/h, so energy is rate of change of configurations CA lattice can change one spot at a time for reversible rules Should count changes as bit changes (i.e., energy is extensive!)

33 t : t l + τ : What is Energy? M h ν l = = num particles particle energy ν = 2Mνl = 2E/ h Conservation Law: number of ones constant Constrains number of spots that can change in lattice update period τ l Focus on energy of the spots that can change If each particle is assigned an energy hν l max change is still 2E/h

34 What is Action? rest frame moving frame 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: 2E r t r /h = 2(Et-px)/h If x/t=c, then E=cp so that Et=px (comp stops)

35 What does QM add? A NOT A U U NOT NOT 0 = 0 2 = A NOT NOT A 2 2 U U 0 = NOT NOT U 0 NOT ( 2 2 )= U U = NOT NOT U 0 0 NOT ( )=+ 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-E min )/h

36 U U A B C θ θ 4 NOT XOR + are universal! 0 = cos θ 0 sin θ = sin θ 0 + cos θ A θ A θ = π/ 2 : U = U θ = π/ 4 : U = U θ θ NOT NOT -π/8 XOR -π/8 π/8 XOR XOR π/8 A B AC C

37 4 NOT XOR + are universal! U U θ θ 0 = cos θ 0 sin θ = sin θ 0 + cos θ A θ A θ = π/ 2 : U = U θ = π/ 4 : U = U θ θ NOT NOT A B C -π/8 XOR -π/8 π/8 XOR XOR π/8 A B AC C No probabilities Superposition configurations of different No probabilities

38 U U NOT XOR + are universal! θ θ 00 0 = cos θ 0 sin θ = sin θ 0 + cos θ A θ A c = cos( π / 8 ) s = sin( π / 8 ) -π/8 XOR -π/8 π/8 XOR XOR π/8 0 0 c 00 + s 0 c 00 + s c 00 + s 0 c 00 + s 0 00

39 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!

40 What does this mean? Classical: a 0 + b b 0 a NOT Quantum: a 0 + b NOT a + b a b

41 What does this mean? Classical: a 0 + b b 0 a NOT Quantum: a 0 + b NOT a + b a b New York Boston

42 Conclusions On a large scale, often can t tell if micro finitestate is QM or CM Entropy, Energy and Action all have comp meaning: others must Significant for comp and for physics for more information, see