Cosmological Computation Computers in a weird universe Patrick Rall Ph70 May 10, 2016

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Some Weird Stuff Time Travel Chapter 20 Expanding Universe Chapter 21 Anthropic Principle Chapter 18

Back on Earth: Some types of problems Easy problems Arithmetic Grocery shopping Taxes Halting problem Hard problems Cracking Crypto Messy room Buying things Very hard problems Traveling salesman

Computing with Time Travel Forward in time: easy We are doing it right now! Travel close to the speed of light Hang out near a black hole Problem: turn 2 n into n? Need exponentially large speed Need exponentially amount of fuel

Computing with Time Travel Backward in time: harder Closed Timelike Curves (CTCs) Solution to Einstein’s Eqn’s, but: Don’t appear to occur naturally Nobody knows how to make them Even if made, how to keep open? Can these exist? What happens to computers if they do?

Computing with CTCs? Start Compute... Get result Send result back Start T = 1 hr T = 0 hr T = 1 hr T = 0 hr T = 1 hr T = 0 hr T = 1 hr T = 0 hr T = 1 hr T = 0 hr Get result Done!...

Grandfather paradox...

Grandfather paradox resolved? 50%

Time travel algorithm Find a document in a messy room Pick up a document If it is not the one you want, go to next document, repeat If it is the one you want, stop Not found Found! Not found... Stationary state of chain: found! Solve search problems instantly!

CTC Debate If CTCs work this way (David Deutsch 1991) Showed: CTCs can solve arbitrary search problems Cryptography Mathematical proofs Time is reusable, just like space Quantum computation = Classical computation Other proposals (2009): Bennett et al: CTCs can’t depend on outside information Lloyd et al: Postselect on time travel outcomes

Anthropic Principle

God’s coin toss (Nick Bostrom) P(heads) = ½ P(red | heads) = 1 P(red | tails) = ½ P(heads | red) = P(red | heads) P(heads) P(red) 1 ½ 1 ½ + ½ ½ = ⅓

Doomsday argument (Brandon Carter) Doom SoonDoom Late Pop = 80 billion Pop = 80 quadrillion Pop = 0

Computing with postselection Draw a lottery ticket If you lose, kill yourself Guaranteed to be alive, if you win! Therefore guaranteed to win if you are alive! Apply to probabilistic computing Can solve search problems! Can it do more? Quantum postselection likely more powerful than classical

Summary so far Given outlandish theory: Can it be true? What are the consequences for computation? Do those seem likely? Time travel computing Solves arbitrary search problems (under Deutsch’s theory) Classical computation = Quantum computation Anthropic computing (or computing with postselection) Solves arbitrary search problems Classical and quantum computation could still be different Search problems seem hard in this universe Still outlandish: An expanding universe?

Some basic cosmology Cosmological principle: The universe is homogeneous at large scales Density Affects curvature of the universe Flat, positive or negative curvature? Two parts Mass density: M Vacuum energy: Λ Cosmological constant Λ

Universe timeline Time Size Inflation: Exponential growth Slow, matter dominated growth Matter and Λ contribute equally Λ dominated growth: Exponential again You are here

Bits that wander off... Events P and Q Start of computation P End of computation Q Finite number of bits can be used in computation Lassoing bits around Fundamental limit One space dimension: 1/√Λ Two space dimensions: 1/Λ? Time P Q Can use bits here! P cannot use Q cannot use Nobody can use

Conclusions Computation is a fundamental part of our universe What can computer science theory tell us? Unstructured search problems seem hard Automate mathematical proofs Break cryptography Therefore: CTCs, Anthropic computing seem unlikely? Fundamental limits to computing: 1/√Λ? Relations between quantum and classical computation? Interested? Consider “Quantum Computation since Democritus”!

Thank you for your attention!