Gavin BrennenLauri Lehman

Zhenghan Wang Valcav Zatloukal

JKP

Ubergurgl, June 2010

Anyonic quantum walks:Anyonic quantum walks:The Drunken SlalomThe Drunken Slalom

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

Anyons

2ie

Bosons

Fermions

U

ei 2

Anyons

3D

2D

View anyon as vortex with flux and charge.

•Two dimensional systems •Dynamically trivial (H=0). Only statistics.

Ising Anyon Properties• Define particles:

• Define their fusion:

• Define their braiding:

,,1

1

1

1

B

,,1,Fusion Hilbert space:

Ising Anyon Properties• Assume we can:

– Create identifiable anyons pair creation

– Braid anyons Statistical evolution: braid representation B

– Fuse anyons

time

,1,B

1

1

1

B

1

1

Approximating Jones Polynomials

Knots (and links) are equivalent to braids with a “trace”.

[Markov, Alexander theorems]

“trace

”

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

”

Approximating Jones Polynomials

4

1

t4 t

[Freedman, Kitaev, Wang (2002); Aharonov, Jones, Landau (2005);et al. Glaser (2009)]

With QC polynomially easy to approximate: Simulate the knot with anyonic braiding

tt

1

Take “Trace”

“trace

”

Classical Random Walk on a line

Recipe:1)Start at the origin2)Toss a fair coin: Heads or Tails3)Move: Right for Heads or Left for Tails4)Repeat steps (2,3) T times5)Measure position of walker6)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

T~x2

QW on a line

Recipe:1)Start at the origin2)Toss a quantum coin (qubit):

3)Move left and right:4)Repeat steps (2,3) T times5)Measure position of walker6)Repeat steps (1-5) many times

Probability distribution P(x,T):...

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

2)/10(1H

2)/10(0H

1,1xx,1S,1,0xx,0S

11

11

2

1H

QW on a line

Recipe:1)Start at the origin2)Toss a quantum coin (qubit):

3)Move left and right:4)Repeat steps (2,3) T times5)Measure position of walker6)Repeat steps (1-5) many times

Probability distribution P(x,T):...

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

1,1xx,1S,1,0xx,0S

2)/10(1H

2)/10(0H

11

11

2

1H

CRW vs QW

QW

CRW

Quantum spread ~T2, classical spread~T [Nayak, Vishwanath, quant-ph/0010117;

Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)]

22 xx

P(x

,T)

QW with more coins

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?

dim=2

dim=4

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

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.

sb1s1 2 s 1s 1n n

1sb

Ising anyons QWEvolve in time e.g. 5 steps

What is the probability to find the walker at position x after T steps?

Hilbert space:

n~2~2

(n)H(n)HHH(n)

n

positionanyonsqubit

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.

Markov122001 )B(B)BΨΨtr(B

TIM

E

Trace (in pictures)

Markov122001 )B(B)BΨΨtr(B

0

1B

2B

0

Trace & Kauffman’s brackets

Ising anyons QWEvaluate 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

B

1

1

Locality and Non-LocalityPosition distribution, P(x,T):

L

τ(L)z(L)

T propernonisLif0

properisLif1)(

21

T)P(x,

•z(L): sum of successive pairs of right steps•τ(L): sum of Borromean rings

Very localcharacteristic

Very non-localcharacteristic

Ising QW Variance

The variance appears to be close to the classical RW.

step, T

Vari

an

ce

~T

~T2

Ising QW Variance

Assume z(L) and τ(L) are uncorrelated variables.

local vs non-local

T)(x,rT)(x,rN

NT)(x,δPT)(x,PT)(x,P oddτevenτ

total

properQWRWAQW

step, T step, T

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 quantumk seems to interpolate between these distributions

position, x

index k

pro

bab

ility

P(x

,T=

10

)

2a

1a

•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?

Conclusions

Thank you for your attention!