Why should anyone care about computing with anyons? Jiannis K. Pachos

Transcription

1 Why should anyone care about computing with anyons? Jiannis K. Pachos Singapore, January 2016

2 Computers ntikythera mechanism Robotron Z 9001 nalogue computer Digital computer: 0 & 1

3 Computational complexity Problems that can be solved in: -Polynomial time (quickly) -Exponential time (slowly)...with the input size. Classical computers: P: polynomially easy to solve NP: polynomially easy to verify solution Quantum computers: BQP: polynomially easy to solve with QC

4 Starship is threatened! Quantum Games test: penny in a box head up. Q puts hand in box, then P, then Q. If head up Q wins If head down P wins

5 Quantum lgorithms Prime number factorisation (public key encryption) quantum hackers exponentially better than classical hackers! Searching database: where is? ΙΠÃ ⅙ ⅖ű Quantum simulation: Efficient simulation of quantum systems Errors during QC are too catastrophic

6 Topological quantum computation Topology promises to solve the problem of errors that inhibit the experimental realisation of quantum computers and it is lots of fun :-)

7 Long introduction dvanced material How to get it: Download from web Lecture notes mazon nswer one of 3 questions

8 Geometry Topology Geometry Local properties of object Topology Global properties of object geom. topo.

9 Topological quantum effects haronov-bohm effect Magnetic flux and charge The phase is a function of winding number Topological effect: is the integer number of rotations

10 Particle statistics Exchange two identical particles: Statistical symmetry: Physics stays the same, but could change!

11 Particle statistics 3D Bosons Fermions Ψ Ψ Ψ i2π e Ψ 2D nyons nyons: vortices with flux & charge (fractional). haronov-bohm effect Berry Phase.

12 nyons and physical systems nyonic properties can be found in 2-dimensional topological physical systems: Fractional quantum Hall effect Topological insulators Cold-atom systems [FQHE, Lattice models: QD, LW, KHoney, TI & TSC ]

13 nyonic evolutions Create anyons from the vacuum (in pairs) Braid anyons (exchange them) Fuse anyons (bring them together/measure)

14 nyons and knots Initiate: Pair creation of anyons time Braiding Measure: do they fuse to the vacuum?

15 nyons and knots

16 nyons and knots ssume I can generate anyons in the laboratory. The state of anyons is efficiently described by their world lines. Create, braid, fuse. The final quantum state of anyons is invariant under continuous deformations of strands.

17 The Reidemeister moves Theorem: Two knots can be deformed continuously one into the other iff one knot can be transformed into the other by local moves:

18 Skein relations = d

19 Skein and Reidemeister d Reidemeister move (II) is satisfied. Similarly (III).

20 Kauffman bracket The Skein relations give rise to the Kauffman bracket: Skein( )= 1 1

21 Jones polynomials The Skein relations give rise to the Kauffman bracket: Skein( )= To satisfy move (I) one needs to define Jones polynomial: w (L) is the writhe of link. For an oriented link it is the sum of the signs for all crossings + 1 1

22 Jones polynomials The Skein relations give rise to the Kauffman bracket: Skein( )= To satisfy move (I) one needs to define Jones polynomial: 1 1 w = 1

23 Jones polynomials If two links have different Jones polynomials then they are inequivalent => use it to distinguish links Jones polynomials keep: only topological information, no geometrical

24 Topology of knots and links re two knots equivalent? topo. lgorithmically extremely time consuming ( 60s) Common problem (speech recognition, ) Jones polynomials recognise if two knots are inequivalent. Solve: Λύνω : disentangle: lexander and Phrygia

25 Jones polynomial from anyons Braiding evolutions of anyonic states: Simulate the knot with braiding anyons Translate it to circuit model: <=> find trace of matrices

26 Jones polynomial from QC Evaluating Jones polynomials is a #P-hard problem. With quantum computers it is polynomially easy to approximate with additive error. Belongs to BQP class. [Freedman, Kitaev, Larsen, Wang (2002); haronov, Jones, Landau (2005); et al. Glaser (2009); Kuperberg (2009)] Do we need Jones poly to describe anyons or are Kauffman brackets enough?

27 Conclusions Jones poly are used for QI: encrypt quantum info quantum money Fundamental properties of anyons provides QC. nyonic systems are currently engineered...