Topological Quantum Computation

1. Topological Quantum Computation PaulFendley, UVa BaisFest, UvA

2. Or how to compute by tying knots…

3. What is quantum computation? • Why is it interesting? • What’s that picture got to do with it?

4. All good things must come to an end

5. Once transistors get small enough, the effects of quantum mechanics become important. A quantum computer is an attempt at making quantum effects your friend. And perhaps destroy commerce on the web…

6. In an ordinary computer: • Information is stored in a series of bits: 01101000010111010001 • Each bit is in one of two states, labeled 0 or 1. • A program is specific way to flip the bits, depending on the values of the other bits.

7. A quantum computer has many more possibilities for storing information. This is a consequence of the fundamentally weird properties of quantum mechanics.

8. What kind of ``states’’ can you have? Classical Quantum Live cat: Cat no more: Live cat: Cat no more:

9. The fundamentally weird thing about quantum mechanics: If you manage to prepare such a state, and then do an experiment to see if the cat is alive or no more, you will sometimes measure the cat alive, and sometimes no more. Any quantum state can be added to another with arbitrary coefficient.

10. It gets weirder. Say we have two cats. Each can be in a different state:

11. Entanglement • You can prepare a state so that once you check on one cat, it determines whether the other one is alive or no more:

12. The last slide with cat torture Classical Quantum Information is stored in any combination of any sequence of qubits: Information is stored in a sequence of bits: 01101000010111010001

13. This fundamental weirdness means quantum computers could do things that classical computers probably can’t. I said ``could” because quantum computers don’t really exist yet. But first let’s see the money app….

14. Encryption for the web relies on the fact that it is very difficult to factor large numbers. 1807082088687404805951656164405905566278102516769401349170127021450056662540244048387341127590812303371781887966563182013214880557 = 39685999459597454290161126162883786067576449112810064832555157243 x 45534498646735972188403686897274408864356301263205069600999044599 A quantum computer could factor large numbers in polynomial time. So far: 15= 5 x 3

15. If you could do this, you could decrypt every internet communication. This possibility makes it much easier to get research in quantum computation funded. But more importantly (for me at least), it would be a way of simulating quantum systems.

16. With any silver lining… • Errors! • are bad! • are really really hard to avoid in a quantum computer. The parameter a can vary just slightly: it’s a feature and a bug!

17. To avoid errors ClassicalComputer: Quantum Computer: No-cloning theorems mean that you cannot make a copy of a quantum state without destroying it. Just be redundant 01101000010111010001 01101000010111010001 01101000010111010001

18. Now back to that picture…

19. We need to have redundancy in a quantum system.

20. We want to store information topologically. • Put more precisely, we want a quantum system where different states have different topological characteristics. • Small errors will not change this characteristic.

21. Example: a chain of atoms that prefer sharing one electron with one neighbor, two electrons with the other. They have two choices for doing this:

22. Two kinds of defects Two single bonds in a row: To change this into two double bonds in a row, just add an electron: You can’t get rid of the defect without changing many bonds. Moving an electron just moves the defect:

23. Fractional Charge Two defects have only one less electron. 11 electrons No defects: Two defects: 10 electrons Each defect has charge -½ !!

24. So how do we make a topological quantum computer? • We need to do much more than store information: we need to write programs. • To avoid errors, the program needs to manipulate the qubits topologically!

26. Spacetime braids • The path of a particle is called a worldline. time

27. Exchanging two particles gives a braid in 3d spacetime. From above: In spacetime:

28. How does this let us write quantum computer programs?

29. Topological defects can have strings attached. No defects: Two defects: A string of changed bonds connect the two defects.

30. A 2d example: spins/cats on a square grid that favor having an even number of arrows pointing into each point. To make two defects, we need to flip a string of spins.

31. A topological quantum computer needs defects with a very special property. There needs to be more than one type of defect, like we had in the chain. Moreover, braiding the defects must cause them to change type!

32. Because of the strings, the defects can change state when braided, even if they’re far apart! time This is essentially the same as flux metamorphosis.

33. Thinking of the defects as particles: From above: In spacetime:

34. We flip qubits by braiding defects! This is a purely topological operation – the result doesn’t depend on the details, just on which defects are braided in which order.

35. Defects with this behavior seem to exist only in two dimensions. In one dimension, they can’t go around each other without crossing. In three dimensions, either the paths can be deformed to give nothing, or they are confined. Such particles are called non-Abelian anyons, or non-Abelions. Should we call such defects morphons instead?

36. Although fractional charge has been observed, it is difficult to find real systems with morphons. One major complication is that in most models, the defects are not deconfined, i.e. there is a string tension. Only one experimental system - the fractional quantum Hall effect - (probably) exhibits non-Abelian braiding. A new strong possibility is in (the surface of) topological insulators. It may be possible in the future to engineer optical lattices to behave in this way.

37. Conclusions • Topological quantum computation requires physics, mathematics and computer science. • Hopefully soon it will require engineering too. • Is it just crazy enough? • Many mysteries need to be solved…

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39. Mariken van Nieumeghen is a 1974Dutchdrama film directed by Jos Stelling. It is based on the early Dutch book "Die waerachtige ende seer wonderlycke historie van Mariken van Nieumeghen die meer dan seven jaren met den duvel woonde ende verkeerde" (The true and wonderous story of Mariken van Nieumeghen who lived with the devil for more than seven years) , dating 1518.