Limits and Horizon of Computing Post silicon computing

Limits Theoretical limit: Some unsolvable problems Halting problem

Limits Theoretical limit: Some unsolvable problems Halting problem Practical Limits: Too slow to be worth it

Example Know there is a binary key of n digits that decrypts data… try every possible key 00000000 00000001 00000010 00000011 00000100 …

Example Know there is a binary key of n digits that decrypts data… try every possible key O(2n) n Possible keys 1 2 4 3 8 16 … 10 1024 20 ~1,000,000 30 ~1,000,000,000 40 ~1,000,000,000,000

Example Traveling Salesman Problem:

Example O(n!) Traveling Salesman Problem: Towns Routes 2 3 6 4 24 5 120 … 10 3,628,800 20 2,432,902,008,176,640,000

Classes Exponential and factorial growth:

Impossible For Any Significant Size Classes Exponential and factorial growth: Impossible For Any Significant Size Doable

Classes Polynomial: Work is O(nm) for some constant m O(1), O(logn), O(n), O(n*logn), O(n2), O(n3) Non-polynomial: More time than polynomial O(2n), O(n!)

P vs NP https://www.youtube.com/watch?v=YX40hbAHx3s

Other Hard Problems Factoring Integers – why RSA works! Many optimization problems

But Moore's Law! Moore's Law "solves" polynomial problems 18 months, 2x as fast 3 years, 4x as fast 6 years, 16x as fast

But Moore's Law! Moore's Law "solves" polynomial problems 18 months, 2x as fast 3 years, 4x as fast 6 years, 16x as fast O(n) : do 16x more work O(n2) : do 4x more work

But Moore's Law! More's law not much help with non- polynomial problems 2n doubles each time n increases by 1

But Moore's Law! More's law not much help with non- polynomial problems 2n doubles each time n increases by 1 18 months do +1 units of work 3 years do +2 units of work 6 years do +4 units of work

Silicon Reaching limits of ability to work with silicon…

Tiny tiny tiny Transistors are small http://htwins.net/scale2/ Modern chip: 14 nanometer scale Transistor ~30 atoms across 30 atoms!!!

New Materials Trick 1: New materials

Molecular Computation Trick 2: Molecular computation DNA Storage: 700 terabytes in one gram

Longer Term? Moore's law is going to break…

Longer Term? Moore's law is going to break… Even it can't help us with some problems…

Longer Term? Need something completely different

Quantum Mechanics Trick 3: Quantum Mechanics Rules that govern sub atomic physics Particles can pass through solid objects Particles can be entangled and read each other's "minds" across the universe Everything is random until it is observed… then it changes to match observation

Video Quantum Computers: What they are: https://www.youtube.com/watch?v=CMdHDHEuOUE How they work: https://www.youtube.com/watch?v=g_IaVepNDT4

Optimization Problem Each switch is on or off – make the highest total:

Classical Approach 6 switches, each on or off 26 or 32 possible states… try them one by one

Scaling

Quantum Approach Switches can be both on and off… Test all possible solution at once! Observing forces qubits to one state… …set up so desired answer is most likely state

100 A problem space is represented by 100 bits 2100 possible answers Conventional computer: 1,000,000,000 solutions checked per second 40 trillion years to solve

100 A problem space is represented by 100 bits 2100 possible answers Quantum computer with 100 bits Try all states at once Seconds (+ lots of setup time) Answer is only probably correct – need multiple runs to confirm…

Reality Solving 3 x 5 = 15 the hard way https://www.youtube.com/watch?v=Yl3o236gdp8#t=250