1. Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP Ubergurgl, June 2010 Anyonic quantum walks: The Drunken Slalom

2. Random evolutions of topological structures arise in: Statistical physics (e.g. Potts model): Entropy of ensembles of extended object Plasma physics and superconductors: Vortex dynamics Polymer physics: Diffusion of polymer chains Molecular biology: DNA folding Cosmic strings Kinematic Golden Chain (ladder) Anyonic Walks: Motivation Quantum simulation

3. Anyons Bosons Fermions Anyons 3D 2D View anyon as vortex with flux and charge. Two dimensional systems Dynamically trivial (H=0). Only statistics.

4. Ising Anyon Properties Define particles: Define their fusion: Define their braiding: Fusion Hilbert space:

5. Ising Anyon Properties Assume we can: –Create identifiable anyons pair creation –Braid anyons Statistical evolution: braid representation B –Fuse anyons time

6. Approximating Jones Polynomials Knots (and links) are equivalent to braids with a “trace”. [Markov, Alexander theorems] “trace”

7. Approximating Jones Polynomials Is it possible to check if two knots are equivalent or not? The Jones polynomial is a topological invariant: if it differs, knots are not equivalent. Exponentially hard to evaluate classically –in general. Applications: DNA reconstruction, statistical physics… [Jones (1985)] “trace”

8. Approximating Jones Polynomials [Freedman, Kitaev, Wang (2002); Aharonov, Jones, Landau (2005); et al. Glaser (2009)] With QC polynomially easy to approximate: Simulate the knot with anyonic braiding Take “Trace” “trace”

9. Classical Random Walk on a line Recipe: 1)Start at the origin 2)Toss a fair coin: Heads or Tails 3)Move: Right for Heads or Left for Tails 4)Repeat steps (2,3) T times 5)Measure position of walker 6)Repeat steps (1-5) many times Probability distribution P(x,T): binomial Standard deviation: -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

10. QW on a line Recipe: 1)Start at the origin 2)Toss a quantum coin (qubit): 3)Move left and right: 4)Repeat steps (2,3) T times 5)Measure position of walker 6)Repeat steps (1-5) many times Probability distribution P(x,T):... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

11. QW on a line Recipe: 1)Start at the origin 2)Toss a quantum coin (qubit): 3)Move left and right: 4)Repeat steps (2,3) T times 5)Measure position of walker 6)Repeat steps (1-5) many times Probability distribution P(x,T):... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

12. CRW vs QW QW CRW Quantum spread ~T 2, classical spread~T [Nayak, Vishwanath, quant-ph/0010117; Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)] P(x,T)

13. QW with more coins Variance =kT 2 More (or larger) coins dilute the effect of interference (smaller k) New coin at each step destroys speedup (also decoherence) Variance =kT [Brun, Carteret, Ambainis, PRL (2003)] New coin every two steps? dim=2 dim=4

14. QW vs RW vs...? If walk is time/position independent then it is either: classical (variance ~ kT ) or quantum (variance ~ kT 2 ) Decoherence, coin dimension, etc. give no richer structure... Is it possible to have time/position independent walk with variance ~ kT a for 1<a<2? Anyonic quantum walks are promising due to their non-local character.

15. Ising anyons QW QW of an anyon with a coin by braiding it with other anyons of the same type fixed on a line. Evolve with quantum coin to braid with left or right anyon.

16. Ising anyons QW Evolve in time e.g. 5 steps What is the probability to find the walker at position x after T steps?

17. Hilbert space: Ising anyons QW P(x,T) involves tracing the coin and anyonic degrees of freedom:  add Kauffman’s bracket of each resulting link (Jones polynomial) P(x,T), is given in terms of such Kauffman’s brackets: exponentially hard to calculate! large number of paths.

18. TIME Trace (in pictures) Trace & Kauffman’s brackets

19. Ising anyons QW Evaluate Kauffman bracket. Repeat for each path of the walk. Walker probability distribution depends on the distribution of links (exponentially many). A link is proper if the linking between the walk and any other link is even. Non-proper links Kauffman(Ising)=0

20. Locality and Non-Locality Position distribution, P(x,T): z(L): sum of successive pairs of right steps τ(L): sum of Borromean rings Very local characteristic Very non-local characteristic

21. Ising QW Variance The variance appears to be close to the classical RW. step, T Variance ~T ~T 2

22. Ising QW Variance Assume z(L) and τ(L) are uncorrelated variables. local vs non-local step, T

23. Anyonic QW & SU(2) k The probability distribution P(x,T=10) for various k. k=2 (Ising anyons) appears classical k=∞ (fermions) it is quantum k seems to interpolate between these distributions position, x index k probability P(x,T=10)

24. Possible: quant simulations with FQHE, p-wave sc, topological insulators...? Asymptotics: Variance ~ kT a 1<a<2 Anyons: first possible example Spreading speed (Grover’s algorithm) is taken over by Evaluation of Kauffman’s brackets (BQP-complete problem) Simulation of decoherence? Conclusions Thank you for your attention!