Gavin Brennen
Lauri Lehman
Zhenghan Wang
Valcav Zatloukal
JKP
Ubergurgl, June 2010

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

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

• Define particles:
1 ,

,

• Define their fusion:
Fusion Hilbert space:

,
 
1 ,

,
  
1
 






 

1
 
   
1
 
 
• Define their braiding:
 

B
   

• Assume we can:
– Create identifiable anyons pair creation
  
– Braid anyons
Statistical evolution: braid representation B
B
– Fuse anyons
   
1
 
B

,
 
1
 
,
  


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

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. [Jones (1985)]
Exponentially hard to evaluate classically –in general.
Applications: DNA reconstruction, statistical physics…

1
4 t
4 t

1 t
Take “Trace”
 t
With QC polynomially easy to approximate:
Simulate the knot with anyonic braiding
[Freedman, Kitaev, Wang (2002); Aharonov, Jones, Landau (2005);
et al. Glaser (2009)]

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
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: x 2 ~ T

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Recipe:
1) Start at the origin
2) Toss a quantum coin (qubit):
H
H 1
0


(
(
0
0


1
1
H
)/
)/

2
2
1
2


1
1
S x,1 3) Move left and right: S x,0
4) Repeat steps (2,3) T times
5) Measure position of walker
 x
6) Repeat steps (1-5) many times
Probability distribution P(x,T):...
 1,0 ,  x  1,1

1
1



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Recipe:
1) Start at the origin
2) Toss a quantum coin (qubit):
H
H 1
0


(
(
0
0


1
1
H
)/
)/

2
2
1
2


1
1
S x,1 3) Move left and right: S x,0
4) Repeat steps (2,3) T times
5) Measure position of walker
 x
6) Repeat steps (1-5) many times
Probability distribution P(x,T):...
 1,0 ,  x  1,1

1
1



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

dim=2 dim=4
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?

• 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.

1 2 b s

1 s

1 s b s s

1 n

1 n
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.

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

Hilbert space: H(n)  H qubit
 H anyons
(n)  H position
(n)
2 ~ 2 n
~ n
P(x,T) involves tracing the coin and anyonic degrees of freedom: tr(B
1
Ψ
0
Ψ
0
B
2
 )  (B
2
 B
1
)
Markov
 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.

Trace
(in pictures) 
0
B
1
B
2


0 tr(B
1
Ψ
0
Ψ
0
B
2
 )  (B
2
 B
1
)
Markov

Repeat for each path of the walk.
Evaluate Kauffman bracket.
A link is proper if the linking between the walk and any other link is even.
Non-proper links
Kauffman(Ising)=0
1 1
Walker probability distribution depends on the distribution of links (exponentially many).

B
  
 

Position distribution, P(x,T):
P(x, T) 
1
2 T

L



(  1) z(L)  τ(L)
0 if L if is
L is non proper
 proper
•z(L): sum of successive pairs of right steps
•τ(L): sum of Borromean rings
Very local characteristic
Very non-local characteristic

~T 2
~T step, T
The variance appears to be close to the classical RW.

local vs non-local
Assume z(L) and τ(L) are uncorrelated variables.
P
AQW
(x, T)  P
RW
(x, T)  δP
QW
(x, T)
N proper
N total
 r
τ  even
(x, T)  r
τ  odd
(x, T)
 step, T step, T

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 a a


1
2 k seems to interpolate between these distributions

•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?
Thank you for your attention!