Transcript Jiannis Pachos

Anyonic quantum walks:
The Drunken Slalom
Gavin Brennen
Lauri Lehman
Zhenghan Wang
Valcav Zatloukal
JKP
Ubergurgl, June 2010
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
Anyons
•Two dimensional systems
•Dynamically trivial (H=0). Only statistics.
3D
2D
Bosons
  
Fermions
  ei 2 
  ei 2 
 U 
Anyons
View anyon as vortex with flux and charge.
Ising Anyon Properties
• Define particles:
• Define their fusion:
Fusion Hilbert space:
 ,   1 ,  ,  
• Define their braiding:
1,  , 
1   
   1
   
    1 
 
 
B
 
 
Ising Anyon Properties
• Assume we can:
1
– Create identifiable anyons
pair creation
 


time
– Braid anyons
Statistical evolution:
braid representation B
1
B
– Fuse anyons
    1 
B  ,   1   ,  
1

Approximating Jones Polynomials
“trace”
Knots (and links) are equivalent to braids with a “trace”.
[Markov, Alexander theorems]
Approximating Jones Polynomials
“trace”
Is it possible to check if two knots are equivalent or not?
The Jones polynomial is a topological invariant:
[Jones (1985)]
if it differs, knots are not equivalent.
Exponentially hard to evaluate classically –in general.
Applications: DNA reconstruction, statistical physics…
Approximating Jones Polynomials
4
t
1

 t
t
“trace”
1
4
t
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)]
Classical Random Walk on a line
-6
-5
-4
-3
-2
-1
0
1
2
3
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: x2 ~ T
4
5
6
QW on a line
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
1 1 1 


H
2 1  1 
Recipe:
1) Start at the origin
H 0  ( 0  1 )/ 2
2) Toss a quantum coin (qubit):
H 1  ( 0  1 )/ 2
3) Move left and right: S x,0  x  1,0 , S x,1  x  1,1
4) Repeat steps (2,3) T times
5) Measure position of walker
6) Repeat steps (1-5) many times
Probability distribution P(x,T):...
QW on a line
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
1 1 1 


H
2 1  1 
Recipe:
1) Start at the origin
H 0  ( 0  1 )/ 2
2) Toss a quantum coin (qubit):
H 1  ( 0  1 )/ 2
3) Move left and right: S x,0  x  1,0 , S x,1  x  1,1
4) Repeat steps (2,3) T times
5) Measure position of walker
6) Repeat steps (1-5) many times
Probability distribution P(x,T):...
CRW vs QW
P(x,T)
CRW
QW
Quantum spread
x2  x
2
~T2, classical spread~T
[Nayak, Vishwanath, quant-ph/0010117;
Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)]
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
[Brun, Carteret, Ambainis, PRL (2003)]
New coin every two steps?
QW vs RW vs ...?
• 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 ~ kTa for 1<a<2?
• Anyonic quantum walks are promising due to
their non-local character.
Ising anyons QW
bs 1
1
2
s 1
s
bs
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
Ising anyons QW
What is the probability to find the walker at position
x after T steps?
Ising anyons QW
Hilbert space: H(n)  Hqubit  Hanyons (n) Hposition (n)
2
~ 2
n
~n
P(x,T) involves tracing the coin and anyonic degrees of
freedom:


tr(B1 Ψ0 Ψ0 B2 )  (B2 B1 )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 & Kauffman’s brackets
0
B1
TIME
Trace
(in pictures)
B

2
0


tr(B1 Ψ0 Ψ0 B2 )  (B2 B1 )Markov
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
1
1




B


Locality and Non-Locality
Position distribution, P(x,T):
z(L)  τ(L)

( 1)
if L is proper
1
P(x, T)  T  
2 L  0 if L is non  proper
•z(L): sum of successive pairs of right steps
•τ(L): sum of Borromean rings
Very local
characteristic
Very non-local
characteristic
Ising QW Variance
Variance
~T2
~T
step, T
The variance appears to be close to the classical RW.
Ising QW Variance
local vs non-local
Assume z(L) and τ(L) are uncorrelated variables.
PAQW (x, T)  PRW (x, T)  δPQW (x, T)
step, T
Nproper
Ntotal
rτeven (x, T)  rτodd (x, T)
step, T
probability
P(x,T=10)
Anyonic QW & SU(2)k
index k
position, x
The probability distribution P(x,T=10) for various k.
a 1
k=2 (Ising anyons) appears classical
k=∞ (fermions) it is quantum
a 2
k seems to interpolate between these distributions
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!