Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP

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

2. Anyonic Walks: Motivation • 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) Quantum simulation

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

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

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

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

7. Approximating Jones Polynomials “trace” 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)]

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

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

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

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

12. CRW vs QW CRW QW P(x,T) Quantum spread ~T2, classical spread~T [Nayak, Vishwanath, quant-ph/0010117; Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)]

13. QW with more coins dim=2 dim=4 Variance =kT2 More (or larger) coins dilute the effect of interference (smaller k) New coin at each step destroys speedup (also decoherence) Variance =kT New coin every two steps? [Brun, Carteret, Ambainis, PRL (2003)]

14. If walk is time/position independent then it is either: classical (variance ~ kT) or quantum (variance ~ kT2) Decoherence, coin dimension, etc. give no richer structure... Is it possible to have time/position independent walk with variance ~ kTafor 1<a<2? Anyonic quantum walks are promising due to their non-local character. QW vs RW vs ...?

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. Ising anyons QW Hilbert space: 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. Trace & Kauffman’s brackets Trace (in pictures) TIME

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

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 ~T2 Variance ~T step, T The variance appears to be close to the classical RW.

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

23. Anyonic QW & SU(2)k probability P(x,T=10) index k position, x 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

24. Conclusions • Possible: quant simulations with FQHE, • p-wave sc, topological insulators...? • Asymptotics: Variance ~ kTa • 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? Thank you for your attention!