The Physics of Theoretical Computation

1. The Physics of Theoretical Computation Course being taught this semester at CMU Physics & Computer Science

2. The Goals • Replacing ad-hoc models of computation with: • Logic that reflects basic laws of physics • Reversibility, possible because both Physics & Computation are temporal processes amenable to reversibility • Conservation laws • The object is to develop models of computation fundamentally governed by physics so that mathematics could apply to varioius aspects

3. However… • Woke up at 7:30 this AM and decided to change the subject of my talk…

4. Avogadro Computation • Nature is computing the future at a good pace! • We would like, someday, to be able to force most atoms, in a lump of matter, into computing what we will then be interested in computing. • How might this be possible?

5. Heat! • Reversibility makes it possible to: • Imagine computers that dissipate very little! • Almost all computational steps can avoid all dissipation • After all, QM operates without dissipation! • The object is simply to make nature do our computation instead of doing its own computation • Non-reversible computation suffers from unavoidable Landauer dissipation: Log 2 k t

6. 2 Designs for Computing Matter 1. We make a nearly perfect, simple crystal • Someday, the crystal ought to weigh a reasonable fraction of a kilo! • Then we make that crystal compute • We make a purpose built crystal, where every atom in the crystal is exactly as specified. This is much more difficult to do but this crystal computes very much faster!

7. A Salt Like Architecture • As an example, we could have something similar to the Salt model • It would be a 3D RUCA that is also a Universal Constructor • It might use spin states for cell states • A 6 phase clock could be light pulses from the 6 faces of the cubic crystal

8. The Goals of the Salt Models • A family of Computational Models that: • are computationally universal • Are Universal Constructors • have things in motion • Subject to as many laws of physics as possible • Conservation of Spin (angular momentum) • Reversibility (Conservation of Information) • Goals of exact conservation of other quantities • Supports particles and waves???

9. Face Centered Cubic 3x3x3 Na+ Subarray

10. FCC offset 1 unit in x, y & z Cl- Subarray

11. The two put together… CUBIC • Known as • Table Salt • Na+ Cl-

12. Idealized Newtonian Billiard Balls x velocity = 1, y velocity = 1

13. A B A A B A B A B B The Billiard Ball Model The Billiard Ball Model was dreamt up to answer critics that claimed that there was no possible physical realization of conservative logic because it seemed (to the critics) that any implementation had to violate the 2nd law of thermodynamics!

14. AB The Feynman-Ressler Gate B A A A AB B AB AB A Two BBM Gates and 2 reflectors make one Feynman-Ressler Gate, also known as the Selector Gate. The schematic diagram for the Feynman-Ressler Gate is on the right.

15. SALT Molecular Computing This work was supported by a grant from the National Science Foundation

16. Gliders Interact When two gliders meet, depending on their phase and orientation, they may interact in such a way as to produce two new gliders on paths that lie on a plane orthogonal to the plane of the original paths.

17. Testing the Crystal • We introduce 1 test CPU. It builds 2 copies • This process continues giving exponential growth to the CPU build process • Next, Each CPU tests cells in its local area • Errors are reported back • When all crystal errors are known, a similar process fills the good parts of the crystal with a network of CPUs

18. Each CPU makes 2 copies

19. The Result? • Cheap matter that computes as an array of RUCA systems • Very much faster for some problems • Much slower than dedicated logic (10-8) • Real gates and wires as opposed to CA • However – Very cheap and easy to build and test

20. We Build a Planar Seed • We start with a conventional planar system • It is put into a nutrient bath • Local sites select atoms to add to the crystal • As it grows, new circuits are tested and if faulty those atoms are returned to solution and the local process starts over again • A full Avogadro crystal might take a number of yeare to complete

21. Growing a Designed CrystalEvery Atom is as Specified!

22. We know that Non-Dispative Logic is Possible! • CL does not violate basic laws of physics • Reversibility implicit in all well formed CL circuits → conservation of information • Conservation of bits (the signaling tokens) • The Billiard Ball Model (a version of CL) • Not ruled out are: • Conservation of energy • Conservation of momentum • Conservation of angular momentum

23. The Mystery of Reversibility • Aspects of reversibility are counter intuitive • 40 or so years ago, 99% of all computer scientists were certain that any reversible process had to be trivial. • Back then when Wolfram was asked “Why didn’t you demonstrate any reversible CA during your lecture?” he replied “Because they are all trivial!” • Today he can’t believe he said that.

24. Reversible Logic & Garbage • RL circuits can always get rid of Garbage • Cleanup can be local in time & space • Every function that conserves bits and information can be implemented with nothing extraneous needed or produced • Almost everyone misunderstands issues as to when a reversible computer must dissipate! • IO is a good example

25. Food and Waste • Reversible CA needs sources of constants and ways to get rid of garbage • 2 Methods: • Pipelines to the outside • Caches of • known data (constants) --“Food” • Garbage left behind -- “Waste”

26. 2nd Order Systems • Natural representation of Dynamic State • Systems of Reversible Difference Equations • Computational equivalent to Differential Equations • When programmed properly: • Exactly reversible despite roundoff & truncation error • Otherwise as accurate as ordinary difference equations • Reversible Cellular Automata • Similar capabilities as compared to arbitrary CA • Does not interfere with capabilities or universality

27. The Point • The simplest of particle interactions • Can be made to compute! • Ballistic computing could involve very little dissipation.

28. Physics as Computation • Particles or states remember by not decaying • Particles communicate by moving • Particles or states compute by interacting • The Billiard Ball Model is a good example Computation is really a fundamental process in our world. We normally think of it as implemented in some technologies (I.e. Silicon ICs, Magnetic Recording). But if we ask “What are the fundamental atomic actions within computation?”, they are more than coincidentally similar to the fundamental processes in physics.

29. Bits as Particles or States • Think of a bit as a particle or a state. • It shouldn’t decay on its own. • It needs to get from here to there. • In a wire, a bit is like a wave • It needs to interact with other bits • Interaction takes place in gates In QED, which is the model that explains most of the physics (other than gravity), the fundamental processes involve a particle (electron or photon) going from here to there, and interactions, where a photon is absorbed or emitted by an electron.

30. Particles as Bits • Think of a particle as a bit in a computer. • It shouldn’t decay. • It gets from here to there. <here|there> • In a wire, a bit is like a wave • It needs to compute (interact) with other bits • Interaction takes place in gates In a sense, what physics is doing is taking the present state of things and computing the future state. When nothing is happening, a particle is like a bit in memory or moving down a wire. In a QED type of interaction, we have a process that is similar to what happens in a gate.

31. How to do dissipationless IO • A data swap of 2 equal size blocks is always a reversible process. • INPUT: From a jump drive, swap input data from the jump drive with a equal size block of data from the computer. • OUTPUT: At the end of the computation, swap the results into the jump drive with the original data from the computer • Computer restored to original state

32. THE END